GRANDES DÉVIATIONS ET CONCENTRATION CONVEXE EN TEMPS CONTINU ET DISCRET

THÈSE DE DOCTORAT de l UNIVERSITÉ DE LA ROCHELLE Discipline : Mathématiques Spécialité : Probabilités Présentée par Yutao MA pour obtenir le grade de DOCTEUR de l UNIVERSITÉ DE LA ROCHELLE. Titre de la thèse : GRANDES DÉVIATIONS ET CONCENTRATION CONVEXE EN TEMPS CONTINU ET DISCRET soutenue publiquement le 25 Janvier 27 devant le jury composé de : M. Patrick CATTIAUX (Université de Paris X) - Président M. Quan-sheng LIU (Université de Bretagne-Sud) -Examinateur M. Nicolas PRIVAULT (Université de Poitiers) - Directeur de thèse M. Alain ROUAULT (Université de Versailles) - Rapporteur M. Feng-yu WANG (Université normale de Beijing, Chine) - Rapporteur M. Li-ming WU (Université de Clermont-Ferrand II) - Directeur de thèse

Résumé Cette thèse consiste en trois parties: principes de grandes déviations, inégalités de concentration convexe et inégalités fonctionnelles. Dans la première partie nous obtenons un principe de grandes déviations par rapport à la topologie τ pour les suites échangeables, et un principe de déviations modérées pour les fonctionnelles additives lipschitziennes des processus de Markov. Dans la deuxième partie nous généralisons la formule d Itô aux martingales progressives/rétrogrades. Par conséquent, nous obtenons des inégalités de concentration convexe pour des intégrales dirigées par des mesures aléatoires de Poisson et des mouvements browniens, des martingales normales, des processus symétriques stables ainsi que dans le modèle du gaz continu. Dans la troisième partie nous obtenons une inégalité FKG sur l espace de Wiener. Nous obtenons aussi une inégalité de trou spectral et une inégalité de concentration convexe pour les processus de naissance et de mort. Mots-clés: principe de grandes déviations, inégalité de concentration convexe, formule d Itô aux martingales progressives/rétrogrades, formule de Clark-Ocone, modèle du gaz continu, inégalité FKG, trou spectral, processus de naissance et de mort. Abstract This thesis consists of three parts, on the large deviation principle, on convex concentration inequalities and on functional inequalities. In the first part we obtain a large deviation principle for exchangeable sequence with respect to τ-topology and a moderate deviation principe for additive Lipschitzian functionals of Markov process. In the second part we generalize the Ito s formula to forward/backward martingales, and we obtain some convex concentration inequalities for the integrals with respect to Poisson random measure, Brownian motion, normal martingales, symmetric stable process, and also for the continuous gas model. In the third part we get a FKG inequality on classical Wiener space. We also prove spectral gap and convex concentration inequalities for birth-death process. Keywords: Large deviation principle, convex concentration inequality, Itô s formula for forward/backward martingale, Clark-Ocone formula, continuous gas, FKG inequality, spectral gap, birth-death process.

4

Remerciements Je tiens tout d abord à remercier Liming Wu pour l encadrement de mes études de Master en chine et de cette thèse de doctorat. Durant ces six années, il a été porteur d idées et de soutiens permanents sur les mathématiques et il m a aussi donné de nombreux de conseils et aides bénéfiques pour la vie courante. Je suis également redevable à Nicolas Privault pour sa direction de thèse. Durant ces trois années, j ai largement bénéficié de son encouragement dans mes choix, ainsi que de son aide pendant mes séjours en france. Nos discussions informelles m ont permis une plus grande réflexion sur le monde de la recherche et des publications. Je remercie très chaleureusement Alain Rouault et Feng-yu Wang les rapporteurs pour leurs suggestions constructives, ainsi que Patrick Cattiaux, Quan-sheng Liu qui me font l honneur de composer le jury. Mes remerciements s adressent aussi à l Ambassade de France en Chine pour m avoir accordé cette bourse en ces trois périodes. Merci à Magali Moreau, la responsable du service international à Poitiers, m a aidée à gérer toutes mes démarches administratives. Aldéric Joulin, Anthony Reveillac, Delphine David, Thomas Forget et Thomas Batard, m ont guidée attentivement dans mes loisirs et dans l apprentissage de la langue française qui est si particulière. Je leur témoigne toute mon amitié. Je n oublie pas non plus tous les membres du laboratoire de Mathématiques et Applications et de l Ecole Doctorale de l Université de La Rochelle, auxquels je témoigne ma profonde sympathie. Mille mercis à mes copines Liangzhen Lei, Xiaoqun Zhang et Jing Deng.

À mes parents et mon mari

Table des matières Introduction et résultats principaux 11.1 Chapitre 1: PGD pour des suites échangeables................ 13.2 Chapitre 2: PDM pour des fonctionnelles lipschitziennes.......... 16.3 Chapitre 3: Inégalités de concentration convexe............... 21.4 Chapitre 4: Domination convexe pour des intégrales stables......... 34.5 Chapitre 5: Modèle du gaz continu...................... 35.6 Chapitre 6: Inégalité FKG sur l espace de Wiener.............. 39.7 Chapitre 7: Processus de naissance et de mort................ 43 I Principes de grandes déviations 51 1 LDP for exchangeable sequences 53 1.1 Introduction and Main result......................... 53 1.2 Preliminaries.................................. 56 1.2.1 Several elementary lemmas....................... 56 1.2.2 General lower bound of large deviation................ 62 1.3 Proof of the main result............................ 64 2 MDP for Lipschitzian functionals 67 2.1 Introduction................................... 67 2.2 Motivation: to beat the irreducibility assumption.............. 68 2.3 Main result................................... 69 2.4 Proof of the main results............................ 7 2.4.1 Some preparations........................... 7 2.4.2 Proof of the results........................... 75 2.5 Applications: Two typical models....................... 79 2.5.1 Linear States Space Model(LSSM).................. 79 2.5.2 Scalar Nonlinear state Space Model(SNSSM)............ 8 II Inégalités de concentration convexe 83 3 Convex concentration inequalities 85 7

8 TABLE DES MATIÈRES 3.1 Introduction................................... 85 3.2 Notation..................................... 88 3.3 Convex concentration inequalities for martingales.............. 89 3.4 Application to point processes......................... 96 3.5 Application to Poisson random measures................... 99 3.6 Clark formula.................................. 12 3.7 Normal martingales............................... 15 3.8 Appendix.................................... 18 4 A convex domination principle 113 4.1 Introduction................................... 113 4.2 Main result................................... 114 4.3 Proof of Theorem 4.2.1............................. 115 4.3.1 Forward-backward stochastic calculus................. 115 4.3.2 Integrability of convex functions.................... 116 4.3.3 Proof of Theorem 4.2.1......................... 117 5 The model of continuous gas 121 5.1 Introduction................................... 121 5.2 Main Results.................................. 122 5.3 Proof of the theorems.............................. 124 5.4 Stochastic domination............................. 127 III Inégalités fonctionnelles 131 6 FKG inequality on the Wiener space 133 6.1 Introduction................................... 133 6.2 Analysis on the Wiener space......................... 134 6.3 FKG inequality on the Wiener space..................... 138 6.4 The discrete case................................ 143 7 Spectral gap of birth-death process 145 7.1 Introduction................................... 145 7.2 Representation of ( L) 1 Lip(ρ)....................... 147 7.3 Application to spectral gap.......................... 151 7.4 convex concentration inequality........................ 153 7.4.1 Two classic examples.......................... 157 Bibliographie 161

1 TABLE DES MATIÈRES

Chapitre Introduction et résultats principaux Cette thèse de Doctorat d Université a été élaborée sous la direction des professeurs Nicolas Privault et Liming Wu au sein du laboratoire de Mathématiques et Applications de l Université de La Rochelle. Elle a bénéficié d une bourse de l Ambassade de France en Chine sur une durée de trois ans. Elle est consacrée à l étude des principes de grandes déviations, de la concentration convexe, et des inégalités fonctionnelles. Nous présentons ici les éléments qui ont motivé cette étude ainsi qu un résumé des principaux résultats obtenus. Introduction La théorie du principe de grandes déviations (PGD en abrégé) trouve son origine dans les travaux de Donsker et Varadhan [31], I-IV(1975). Ainsi, depuis une vingtaine d années l obtention de principes de grandes déviations et le développement d outils théoriques ont bénéficié d un engouement soutenu jusqu à devenir une branche des probabilités, et ce grâce à la multitude de leurs applications possibles, que ce soit dans le domaine des mathématiques ou de la physique. Les inégalités fonctionnelles comprennent beaucoup de types d inégalités, par exemple les inégalité de Poincaré, inégalité de Sobolev logarithmiques, inégalité FKG etc.. Elles permettent des estimations effectives de queues de distributions. Ainsi elles apparaissent fréquemment en statistiques et en estimation a priori. Les inégalités de concentration convexe ont été premièrement introduites par Hoeffding dans [42] (1963). Puis elles ont été appliquées pour obtenir des estimations de queues de distribution et des inégalités fonctionnelles. Prenons deux exemples. Shao [84] (2) a démontré comment obtenir des inégalités classiques par les inégalités de concentration convexe, par exemple, l inégalité maximale de Rosenthal et l inégalité de Kolmogorov. Par ailleur, Bentkus [7] (24) a obtenu des estimations de queues de distributions pour 11

12 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX martingales discrètes bornées. Dans la suite de cette partie introductive, nous présentons les résultats obtenus au cours de la préparation de cette thèse. Ils sont généralement présentés sous forme de projets d articles soumis à des revues internationales. Que le lecteur veuille bien excuser alors l utilisation de la langue anglaise. Ce chapitre comprend en outre un résumé en français de chacun de ces articles, en explicitant leur contenu et esquissant les démonstrations. Cette thèse comprend trois parties principales: principes de grandes déviations, inégalités de concentration convexe et inégalités fonctionnelles. La première partie traite du PGD dans les Chapitres 1 et 2. Dans un premier temps, au cours d un travail élaboré en collaboration avec Qiongxia Song et Liming Wu, nous avons établi un PGD pour les suites échangeables. Nous donnons des conditions suffisantes et nécessaires pour le PGD par rapport à la topologie τ, qui est telle que pour toute fonction f mesurable bornée et toute mesure µ, µ µ(f) est continue. Le deuxième chapitre est consacré à l étude du principe de déviations modérées pour les processus de Markov, traité en collaboration avec Qiongxia Song. Notre méthode repose sur celle de Wu dans [92]. Nous utilisons le théorème de perturbation analytique d opérateurs de Kato pour vérifier la condition C 2 régularité, qui donne le principe de déviations modérées pour les fonctionnelles additives lipschitziennes. Note que, dans notre travail, le processus de Markov n est pas nécessairement irréductible et nous remplaçons cette condition par la contractibilité du semigroupe. La deuxième partie, quant à elle, est une étude des inégalités de concentration convexe dans les Chapitres 3, 4, 5. Le troisième chapitre, élaboré en collaboration avec Thierry Klein et Nicolas Privault, concerne des inégalités de concentration convexe. Tout d abord nous généralisons la formule d Itô aux martingales progressives/rétrogrades. Considérant la somme d une martingale progressive et d une martingale rétrograde, nous obtenons des inégalités de concentration convexe pour cette somme. Par conséquent, nous obtenons des inégalités de concentration convexe pour les intégrales dirigées par un mouvement brownien et une mesure aléatoire de Poisson non nécessairement indépendants ainsi que pour les intégrales par rapport à des martingales normales. Puis nous appliquons la formule de Clark-Ocone pour obtenir des inégalités de concentration convexe pour les variables aléatoires de carré intégrable sur certains espaces probabilisés. Puis avec Aldéric Joulin, dans le Chapitre 4, nous considérons des processus α stables symétriques. Nous poursuivons les idées du Chapitre 3 pour obtenir des inégalités de concentration convexe pour les intégrales stochastiques browniennes et stables corrélées. C est le cas pour les sauts non-bornés et de variance infinie.

.1. CHAPITRE 1: PGD POUR DES SUITES ÉCHANGEABLES 13 Le cinquième chapitre est consacré au modèle du gaz continu. En appliquant de nouveau des idées du Chapitre 3, nous obtenons les inégalités de concentration convexe en espérance, pour la covariance et de type brownien pour la mesure de Gibbs du gaz continu. De plus, nous prouvons que la mesure de Gibbs est stochastiquement dominée par la mesure correspondante de Poisson. L objectif de la troisème partie est d étudier des inégalités fonctionnelles dans les Chapitres 6, 7. Dans le Chapitre 6, avec Nicolas Privault, nous prouvons une inégalité FKG sur l espace de Wiener classique. A l aide de la formule de Clark-Ocone et du calcul de Malliavin, nous donnons une nouvelle approche permettant d obtenir l inégalité FKG sur l espace de Wiener classique muni d une certaine relation ordre. Le dernier chapitre, en collaboration avec Wei Liu, est relative aux inégalités de trou spectral et de concentration convexe des processus de naissance et de mort. Nous considérons des processus de naissance et de mort de générateur L et de mesure invariante µ. Etant donnée une fonction ρ strictement croissante, définissons une norme lipschitzienne Lip(ρ) par rapport à ρ. D abord, nous obtenons une représentation de ( L) 1 Lip(ρ). Puis comme application, nous avons le trou spectral de L dans L 2 (µ). De plus, nous obtenons une inégalité de concentration convexe pour des martingales de saut pur et ensuite pour des fonctionnelles des processus de naissance et de mort..1 Principe de grandes déviations pour des suites échangeables par rapport à la topologie τ: conditions suffisantes et nécessaires Dans cette partie nous présentons les résultats que nous avons obtenus, en collaboration avec Qiongxia Song et Liming Wu, concernant un principe de grandes déviations pour des suites échangeables par rapport à la topologie τ. Ce travail est publié dans la revue Statistics and Probability Letters, 77(3), 239-246, 27. Soient (X k ) k une suite échangeable de variables aléatoires à valeurs dans un espace polonais E, c est-à-dire que (X, X 1,, X n ) a la même loi que (X σ(), X σ(1),, X σ(n) ) quelle que soit la permutation {σ (1),, σ (n) } de {1,, n}, n 1.

14 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX Considérons les mesures empiriques (ou d occupation) L n := 1 n 1 δ Xk ( où δ désigne la mesure de Dirac en ), n 1, n k= qui sont des éléments aléatoires de M 1 (E) l espace des mesures de probabilité sur E muni de la σ-algèbre B(M 1 (E)) = σ(ν ν(f) f bb), où bb est l espace des fonctionnelles B-mesurables bornées. Les mesures empiriques du processus sont R n := 1 n 1 δ (Xk,X n k+1, ), n 1, k= et appartient à M 1 (E N ) l espace des mesures de probabilité sur E N muni de la σ-algèbre B(M 1 (E N )) = σ(q Q(F ) F bf N, N ), où F N = σ(x k ; k N). Par le théorème de de Finetti, la loi P de (X k ) k est un mélange des mesures produit homogènes, i.e., il existe une mesure de probabilité m sur M 1 (E) telle que P = ρ N dm(ρ). (.1.1) M 1 (E) Ainsi le PGD pour les suites échangeables se réduit certainement au cas indépendant et identiquement distribué comme le lecteur peut l imaginer. Toutefois cette réduction présente des difficultés, comme démontré dans [24], [26], [85], [91], [97]. Les suites échangeables sont appliquées aux statistiques de Bayes et aux statistiques de bootstrap, voir les travaux récents de Chen ([14]), Trashorras ([86]) et Chaganty ([13]). Le lecteur peut se référer à Aldous [1] (1985) pour d autres applications. Maintenant nous présentons les résultats qui ont motivé ce travail. De Acosta [26] (1995) a prouvé une bonne borne supérieure du PGD pour 1 n n k=1 X k avec une fonction convexe de taux qui, en général, n est pas exacte. Quand E est fini, le troisième auteur Wu [91] (1991) a obtenu le premier PGD complet pour R n et a trouvé que sa fonction de taux n était pas convexe (donc on ne pouvait pas utiliser le théorème de Ellis-Gartëner). Puis, ce sujet a été étudié avec succès par Dinwoodie et Zabell [32] (1992) et Daras [24] (1997) et Trashorras [85] (22). Par exemple, Daras [25] (22) a obtenu le PGD de R n par rapport à la topologie de la convergence faible quand E est compact. La motivation directe de ce travail est l article récent de Wu [98] (24), où les PGDs pour L n sur M 1 (E) et pour R n sur M 1 (E N ) par rapport à la topologie de la convergence faible ont été bien caractérisés par la compacité du support de m sous la topologie de la convergence faible. Dans cet article, nous allons caractériser le PGD correspondant par rapport à la topologie τ. La topologie τ sur M 1 (E) est la topologie la plus faible sur M 1 (E) telle que quelle que soit la fonction f bb, ν ν(f) est continue, où ν(f) := fdν. La limite projective de E

.1. CHAPITRE 1: PGD POUR DES SUITES ÉCHANGEABLES 15 la topologie τ sur M 1 (E N ), noté par τ p, est définie comme la topologie la plus faible telle que pour toute F bf N, Q Q(F ) est continue sur M 1 (E N ). Le résultat principal de cette partie est le théorème suivant et exprimé dans le langage de [3]: Théorème.1.1. Soit S τ le support topologique de la mesure m apparaissant dans (.1.1) par rapport à la topologie τ. Les propriétés suivantes sont équivalentes: (a) Quand n tend vers l infini, la loi de L n satisfait au PGD sur M 1 (E) par rapport à la topologie τ, avec certaine fonction de taux I : M 1 (E) [, + ]. (Convention: le PGD implique nécessairement que la fonction de taux est bonne ou τ-inf-compacte, i.e., [I L] est τ-compact, L R +.) (b) La loi de R n satisfait au PGD sur M 1 (E N ) par rapport à la topologie τ p avec la fonction de taux donnée par J(Q) := inf ρ S τ h sp (Q ρ N ), Q M 1 (E N ), (.1.2) où h sp (Q ρ N ) est l entropie relative spécifique de Q par rapport à ρ N, i.e., 1 lim n h sp (Q ρ N ) := h(q n [,n 1] ρ n ) si Q M1(E s N ) + sinon, (.1.3) où M s 1(E N ) est l espace des Q M 1 (E N ) qui sont stationnaires, Q [,n 1] est la loi des n premières coordonnés (X k ) k n 1 de E N sous Q, h(q [,n 1] ρ n ) est l entropie relative de Q [,n 1] par rapport à ρ n (voir (.1.5) pour la définition). (c) Le support S τ = supp τ (m) est τ-compact dans M 1 (E). Dans ce cas, la fonction de taux dans le PGD de partie (a) est donnée par I(ν) = inf ρ Sτ h(ν ρ), ν M 1 (E). (.1.4)

16 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX où h(ν ρ) est l entropie relative de Kullback de ν par rapport à ρ, i.e., dν dν log E dρ h(ν ρ) := dρ, si ν << ρ dρ + sinon (.1.5) Remarque.1.2. Puisque S τ est compact par rapport à la topologie de la convergence faible quand il est τ-compact, nos PGDs dans parties (a) et (b) du Théorème.1.1 sont plus forts que ceux de [93]. Cette amélioration est utile car en pratique les observations sont discrètes Comparé avec [93], la nouvelle difficulté principale réside dans le fait que la topologie τ sur M 1 (E) n est pas métrisable et donc beaucoup d ingrédients techniques dans [93], n étant pas du tout valides, doivent être refaits ou adaptés. Heureusement, par le théoréme dans [81], (S τ, τ) est métrisable quand S τ est τ compact et en fait par (.1.1) les informations sur S τ sont suffisantes. Donc par des arguments classiques, nous avons établi un principe de grandes déviations. D autre part, le Lemme 1.2.3, qui permet de caractériser la compacité relative de S τ dans (M 1 (E), τ), joue un role clé pour la nécessité..2 Principe de Déviations modérées pour des fonctionnelles additives lipschitziennes des processus de Markov Ce travail, en collaboration avec Qiongxia Song, concernant un principe de déviations modérées pour des fonctionnelles lipschitziennes des processus de Markov, est publié dans la revue Acta Math. Sin. (série chinoise), 27 5(1): 33-42. Les estimations de déviations modérées, comme les estimations de grandes déviations, trouvent leur origine en Statistiques. Elles offrent des estimations plus précises que le théorème de la limite centrale. Soit (X k ) k un processus de Markov sur (Ω, F, P). Posons L n := 1 n n k=1 δ X k, où δ x désigne la mesure de Dirac en x, et considérons la limite de P ν (L n (f) m(f) A n ), (.2.1) où m est la mesure invariante du processus de Markov (X n ) n, ν est la distribution initiale de X, f est une fonctionnelle lipschitzienne, A n est un ensemble borelien de R.

.2. CHAPITRE 2: PDM POUR DES FONCTIONNELLES LIPSCHITZIENNES 17 Quand A n = A pour tout n, (.2.1) est l estimation de grandes déviations, quand quelque soit n, A n = 1 n A, (.2.1) est le théorème de la limite centrale. Si A n = λ(n) n A, où < λ(n), λ(n), λ(n) n, (.2.1) devient l estimation de déviations modérées. Dans ce cas, (.2.1) peut être écrit comme ( ) n P ν λ(n) (L n(f) m(f)) A. Ainsi, elle est une estimation entre le théorème de la limite centrale et l estimation de grandes déviations. Dans les années récentes, il y a eu beaucoup de résultats du principe de déviations modérées. Pour le cas indépendant et identiquement distribué, il existe des résultats optimaux (voir [22], (1991) et [55], (1992)). Pour une étude systématique du principe de déviations modérées pour les martingales, voir, cf. Puhaliski, 1993 et Djellout, 21. Récemment, plus en plus de travaux sont réalisés pour le cas dépendant. Dans Gao [37] (1993), le principe de déviations modérées pour les processus Doeblin récurrent de Markov a été bien discuté. Ensuite en 1996 dans [38], il a établi un principe de déviations modérées pour les martingales et les processus aléatoires mélangeants. Wu, dans son article [91] (1995), a donné un principe de déviations modérées pour les fonctionnelles additives lipschitziennes du processus de Markov avec un trou spectral. Il a considéré des fonctionnelles dans les espaces bb, C b (E) et L 2 (E). Il a introduit la C 2 -régularité et prouvé que cette condition a été suffisante et nécessaire pour le principe de déviations modérées par le théorème de perturbation analytique d opérateurs de Kato. De Acosta, dans son travail [27] (1997), a obtenu la borne inférieure du principe de déviations modérées pour les mesures empiriques de la chaîne de Markov. Wu, dans son article [92] (21) a obtenu le principe complet de déviations modérées sous l hypothèse de récurrence exponentielle et puis Djellout et Guillin (22) ont relaxé cette condition. Tous les résultats précédents sont sous l hypothèse fondamentale que le processus de Markov est irréductible (à part pour quelques résultats de [92]). Mais, comme on voit, beaucoup de processus de Markov ne sont pas irréductibles. Nous allons présenter deux exemples typiques comme application. En fait, le but le plus direct de ce travail est de remplacer la condition d irréductibilité par la contractibilité du semigroupe. Soit (E, d) un espace polonais connexe par arc, i.e., il satisfait: d(x, y) = inf γ lim sup δ τ: τ δ i= n d(γ(t i ), γ(t i+1 )), où γ est une courbe continue connectant x et y (i.e., γ() = x, γ(1) = y) et τ est une partition, τ = { = t < t 1 < < t n = 1}, τ := max 1 i n γ(t i ) γ(t i 1 ). E est la

18 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX σ-algèbre de Borel de (E, d). Nous désignons par M 1 (E) l espace des mesures de probabilité sur (E, E) et par be l espace des fonctionnelles mesurables bornées sur E. Pour une fonctionnelle mesurable f et une mesure µ, nous écrivons µ(f) = f, µ = E fdµ. Une fonction f : E R est dite une fonctionnelle lipschitzienne si f(x) f(y) f Lip := lim sup d(x,y)> d(x, y) Nous définissons un opérateur gradient sur E comme: f(x) := lim sup y x < +. f(x) f(y). d(x, y) Soit (Ω, F, (F k ) k, (X k ), (P x ) x E ) une chaîne de Markov de probabilité de transition P et P ν = E P xν(dx) pour une mesure initiale ν M 1 (E). Nous travaillons sous l hypothèse essentielle suivante: (H 1 ) P Lip < + ; N > et r [, 1) tels que P N f Lip r f Lip, quelle que soit la fonctionnelle lipschitzienne f. Il est facile de prouver que sous (H 1 ), P possède une mesure de probabilité unique invariante m telle que L := m(d(x, x )) <. Considérons (C Lip, ), l espace des fonctionnelles lipschitziennes sur (E, d) où la norme est définie comme = Lip + m( ). Nous étudions la déviation modérée de L n (f) sachant le processsus de Markov (X n ) n, où L n (f) a la représentation suivante: L n (f) = 1 n 1 f(x k ). (.2.2) n k= Le résultat principal de cette partie est le suivant: Théorème.2.1. Supposons que le noyau de transition P satisfait (H 1 ). Soit (λ(n)) n une suite positive satisfaisant: < λ(n), λ(n), λ(n) n. Si f : E R vérifie la condition: (C) f est lipschitzienne, f be et f(x)(1 + d(x, x)) c pour certain c >,

.2. CHAPITRE 2: PDM POUR DES FONCTIONNELLES LIPSCHITZIENNES 19 alors (i) P ν ( n[l n (f) m(f)] ) converge faiblement vers N(, σ 2 (f)) uniformément en ν B L quelque soit L L quand n tend vers l infini, où N(, σ 2 (f)) est la loi normale, B L := {ν M 1 (E)/ν(d(x, )) L} et σ 2 (f) est donnée par (ii) σ 2 (f) = m(f m(f)) 2 + 2 m(p k f(f m(f))). (.2.3) k=1 ( P ν ( ) n [L λ(n) n(f) m(f)] ) satisfait au PGD uniformément en ν B L pour tout L L, de vitesse λ 2 (n) et de fonction de taux I f (r) = tout ensemble borelien A R, nous avons ( ) n lim sup λ 2 (n) log sup P ν ( n ν B L λ(n) [L n(f) m(f)] A) r2. Autrement dit, pour 2σ 2 (f) inf I f (z); z A et lim inf n ( λ 2 (n) log sup P ν ( ν B L ) n λ(n) [L n(f) m(f)] A) inf z A o I f(z). Nous suivons la méthode de Wu ([92]). D abord nous rappelons la définition de C 2 régularité, qui est suffisante et nécessaire d avoir le principe de déviations modérées et le théorème de la limite centrale. Posons Λ t ɛ = ɛ log exp(tx/ɛ)dµ i ɛ. R Définition.2.2. Nous disons que (µ i ɛ, i A) ɛ sont C 2 -régulières à droite (resp. à gauche) uniformément en A, si [Λ i ɛ] (t) Λ (t) uniformément pour i A et t [, δ] (resp. t [ δ, ]), où Λ () est interprétée comme la dérivée seconde à droite Λ +() := lim t +(Λ (t) Λ +())/t [resp. Λ ()]. Si elles sont C 2 régulières uniformément en A à droite et à gauche en même temps, nous disons qu elles sont C 2 -régulières uniformément en A. Puis définissons l opérateur de perturbation de P par P f (x, dy) = e f P (x, dy).

2 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX Nous avons alors, par la formule de Feynman-Kac, (P f ) n g(x) = E x g(x n ) exp { n 1 } f(x k ). Donc soit µ ν ɛ la loi de L n (f) sous P ν avec ɛ = 1 n, la fonction Λν ɛ définie au-dessus est équivalente à 1 n log < ν, (P f ) n 1(x) >. Par conséquent, la C 2 régularité devient une propriété de l opérateur de perturbation P f. Nous vérifions que z P zf g est holomorphe bornée pour toute fonction lipschitzienne g. Puis appliquant le théorème de perturbation analytique d opérateurs de Kato à P f, nous finissons la preuve. La section suivante concerne deux exemples classiques comme application. k= Modèle d espace des états linéaires (MEEL) Le modèle (voir [63], Chap.1) est donné comme suivant: X = (X k ) k est un processus stochastique sur R d qui satisfait X k+1 = F X k + GW k+1, k. X est arbitraire, F est une matrice d d, et G est une matrice d p. (W k ) k sont variables aléatoires indépendantes et identiquement distribuées sur R p et indépendantes de X. Corollaire.2.3. Soit r sp (F ) := max{ λ /λ C est une valeur propre de F }. Supposons que r sp (F ) < 1, alors nous obtenuons le principe de déviations modérées pour toute fonctionnelle f satisfaisant la condition (C). Remarque.2.4. La mesure invariante m est la loi de X. Modèle d espace des états scalaires nonlinéaires (MEESN) Étudions le modèle non linéaire suivant dans R d (d 1): X (x) = x R d, X n+1 (x) = F (X n (x), W n+1 ), n, où (W n ) n est une suite de variables aléatoires à valeurs dans R k (k 1) indépendantes et identiquement distribuées, avec F : R d R k R d une fonction.

.3. CHAPITRE 3: INÉGALITÉS DE CONCENTRATION CONVEXE 21 Corollaire.2.5. Si F (x, w) F (y, w) r x y, r < 1, désigne une norme arbitraire dans R d. Alors le principe de déviations modérées est satisfait pour toute fonctionnelle f satisfaisant la condition (C). Remarque.2.6. La théorie classique demande toujours que la loi de (W n ) n soit intégrable et absolument continue par rapport à la mesure de Lebesgue. En fait, la continuité absolue est nécessaire pour l irréductibilité. Mais ici, nous posons seulement certaines hypothèses sur F dans les modèles MEEL et MEESN. Dès maintenant, nous nous tournons à la deuxième partie concernant les inégalités de concentration convexe..3 Inégalités de concentration convexe et calcul stochastique pour les martingales progressives/rétrogrades Ce travail est élaboré avec Thierry Klein et Nicolas Privault, et publié dans Electronic Journal of Probab., Vol 11, 26. En 1963, Hoeffding [42] a premièrement introduit l inégalité de concentration convexe pour les variables aléatoires de Bernoulli, i.e., pour toute fonction convexe φ, on a E[φ(S n )] E[φ(S n)], où S n = n k=1 X k est la somme de variables aléatoires (X k ) 1 k n indépendantes de Bernoulli de paramètre (p k ) 1 i n et Sn est une variable aléatoire binomiale B(n, p) où p est le moyen arithmétique de (p k ) 1 k n. Cette type d inégalité est très utile pour obtenir des estimations de queues de distributions comme démontré dans [42] (1963), et Bretagnolle [9] (1981) a donné une version fonctionnelle de ce résultat. Pinelis dans [73] (1994) et [74] (1998) a étudié un cas plus général où φ a été dans une famille plus grande. Shao [84] (2) a traité les variables négativement associées, et a montré comment obtenir quelques inégalités classiques, par exemple l inégalité maximale de Rosenthal et l inínégalité de Kolmogorov, par l inégalité de concentration convexe. Bentkus [7] (24) a utilisé les inégalités de concentration convexe pour obtenir des estimations de queues de distributions pour les martingales discrètes bornées. Klein dans sa thèse [53] (23) s est concentré sur les inégalités de concentration convexe. Il a prouvé des inégalités de concentration convexe pour les processus ponctuels à sauts positifs par le calcul stochastique. Dans cette section, nous généralisons les résultats de [53] à un cadre

22 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX plus général. Tout d abord, nous généralisons la formule d Itô aux martingales progressives/rétrogrades, qui sera utilisée très souvent dans cette thèse. Puis appliquant cette formule, nous obtenons des inégalités de concentration convexe pour les intégrales stochastiques dirigées par une mesure aléatoire de Poisson et par un mouvement brownien non nécessairement indépendants, et aussi pour les intégrales par rapport à des martingales normales. A l aide de la formule de Clark-Ocone, nous obtenons des inégalités de concentration convexe pour des fonctionnelles générales. Deux variables aléatoires F et G satisfont à une inégalité de concentration convexe si E[φ(F )] E[φ(G)] (.3.1) pour toute fonction convexe φ : R R. Par un argument classique, l application de (.3.1) à φ(x) = exp(λx), λ >, implique la borne de déviation P (F x) inf λ> E[eλ(F x) 1 {F x} ] inf λ> E[eλ(F x) ] inf λ> E[eλ(G x) ], (.3.2) x >, ainsi la probabilité de déviation pour F peut être estimée via la transformée de Laplace de G, voir [7] (1981) et [74] (1998) pour plus de résultats sur ce sujet. Soit (Ω, F, P ) un espace probabilisé muni d une filtration croissante (F t ) t R+ et une filtration décroissante (Ft ) t R+. Considérons (M t ) t R+ une martingale progressive par rapport à F t et (Mt ) t R+ une martingale rétrograde par rapport à Ft. Nous supposons toujours que (M t ) t R+ est càdlàg, et que (Mt ) t R+ est càglàd. Désignons par (Mt c ) t R+ (resp. (Mt c ) t R+ ) la partie continue de (M t ) t R+ (resp. de (Mt ) t R+ ). Le processus (M t ) t R+ (resp. (Mt ) t R+ ) a la mesure de saut µ(dt, dx) = s> 1 { Ms }δ (s, Ms)(dt, dx), ( resp. µ (dt, dx) = s> 1 { M s } δ (s, M s ) (dt, dx)), où δ (s,x) désigne la mesure de Dirac en (s, x) R + R. Soit ν(dt, dx) (resp. ν (dt, dx)) la projection duale (F t ) t R+ -prévisible ((Ft ) t R+ -prévisible) de µ(dt, dx) (resp. de µ (dt, dx)), i.e. t f(s, x)(µ(ds, dx) ν(ds, dx)) (resp. t g(s, x)(µ (ds, dx) ν (ds, dx))) est une martingale locale F t -progressive (F t -rétrograde) pour tout processus f (resp. g) F t -prévisible (resp. F t -prévisible) et suffisamment intégrable. Ils ont les formes suivantes: et ν(du, dx) = ν u (dx)du et ν (du, dx) = ν u(dx)du, (.3.3) d M c, M c t = H t 2 dt, et d M c, M c t = H t 2 dt, (.3.4) où (H t ) t R+, (H t ) t R+, sont respectivement prévisibles par rapport à (F t ) t R+ et à (F t ) t R+.

.3. CHAPITRE 3: INÉGALITÉS DE CONCENTRATION CONVEXE 23 Formule généralisée d Itô Nous généralisons la formule d Itô aux martingales progressives/rétrogrades. Théorème.3.1. Soient (M t ) t R+ et (Mt ) t R+ les processus au-dessus. Supposons que M t (resp. Mt ) est Ft -adapté (resp. F t -adapté). Alors quelle que soit la fonction f C 2 (R 2, R), nous avons f(m t, M t ) f(m, M ) = t f (M u, M x u)dm u + 1 + 1 2 + <u t t t 2 f (M x 2 u, Mu)d M c, M c u 1 ( f(m u, M u) f(m u, M u) M u f x 1 (M u, M u) f (M u, M u x +)d Mu 1 2 f (M 2 2 x 2 u, Mu)d M c, M c u 2 ( ) f(m u, Mu) f(m u, M u +) M u f (M u, M u x +), 2 u<t où d désigne la différentielle rétrograde d Itô. La section suivante est une application de la formule généralisée d Itô. t ) Inégalité de concentration convexe pour des martingales Appliquant la formule à f(x, y) = φ(x + y), nous prouvons que la fonction E[φ(M t + M t )] est décroissante. Voici les théorèmes: Théorème.3.2. Soient ν u (dx) = xν u (dx), ν u(dx) = xν u(dx), u R +, et supposons que : i) Les supports de ν, ν sont dans R +, ii) ν u ([x, )) ν u([x, )) <, x, u R, et iii) H u H u, dp du p.s.

24 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX Donc nous avons: E[φ(M t + M t )] E[φ(M s + M s )], s t, (.3.5) pour toute fonctionnelle convexe φ sur R. Le théorème suivant est une version différente du même résultat sous des hypothèses L 2. Théorème.3.3. Soient ν u (dx) = x 2 ν u (dx) + H u 2 δ (dx), ν u(dx) = x 2 ν u(dx) + H u 2 δ (dx), u R +, et supposons que: ν u ([x, )) ν u([x, )) <, x R, u R +. Alors nous avons: E[φ(M t + M t )] E[φ(M s + M s )], s t, (.3.6) quelle que soit la fonctionnelle φ sur R telle que φ est convexe. Remarque.3.4. Dans le cas H t = H t et ν t = ν t, dp dt-p.s., nous pouvons avoir l identité E[φ(M t + M t )] = E[φ(M s + M s )], s t, (.3.7) pour toute fonction suffisamment intégrable φ : R R. En particulier, la relation (.3.7) s étend naturellement aux cas des accroissements indépendants: étant données (Z s ) s [,t], ( Z s ) s [,t] deux copies indépendantes d un processus de Lévy sans dérive, définissons une martingale rétrograde (Z s ) s [,t] comme Z s = Z t s, s [, t], donc par la convolution, E[φ(Z s + Z s )] = E[φ(Z t )] qui ne dépend pas de s [, t]. Remarque.3.5. Si φ est croissante, le Théorème.3.2, le Théorème.3.3, le Corollaire.3.8 et le Corollaire.3.7 s étendent aux semi-martingales ( ˆM t ) t R+, ( ˆM t ) t R+ représentées par ˆM t = M t + t α s ds et ˆM t = M t + t β s ds, (.3.8) avec (α t ) t R+, (β t ) t R+ respectivement F t et Ft -adaptés et α t β t, dp dt-p.s.

.3. CHAPITRE 3: INÉGALITÉS DE CONCENTRATION CONVEXE 25 Soit maintenant (F M t ) t R+ (resp. (F M t ) t R+ ) la filtration engendrée par (M t ) t R+ (resp. (M t ) t R+ ). Corollaire.3.6. Sous les hypothèses du Théorème.3.2, si de plus E[M t F M t ] =, t R +. Alors E[φ(M t )] E[φ(M s + M s )], s t. (.3.9) En particulier, si M = E[M t ] est déterministe (ou F M est la σ-algèbre triviale), le Corollaire.3.6 dit que M t E[M t ] est plus concentré que M : E[φ(M t E[M t ])] E[φ(M )], t. Le cas à sauts bornés Ayant ces théorèmes, maintenant nous étudions des cas concrets, premièrement le cas à sauts bornés. Supposons que ν (dt, dx) a la forme ν (dt, dx) = λ t δ k dt, (.3.1) où k est une certaine constante positive et (λ t ) t R+ est un processus positif Ft -prévisible. Posons λ 1,t = xν t (dx), (resp. λ 2 2,t = x 2 ν t (dx)), t R + le compensateur (resp. la variation quadratique) de la partie de saut de (M t ) t R+, sous l hypothèse t R +, P -p.s.. x ν t (dx) <, (resp. x 2 ν t (dx) < ), (.3.11) Corollaire.3.7. Supposons que le saut de (M t ) t R+ (resp. de (Mt ) t R+ ) satisfait (.3.11) (resp. (.3.1)), que (M t ) t R+ (resp. (Mt ) t R+ ) est Ft -adapté (F t -adapté). Donc nous avons : E[φ(M t + Mt )] E[φ(M s + Ms )], s t, (.3.12) pour toute fonction convexe φ : R R sous l une des trois conditions suivantes:

26 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX i) M t k, dp dt p.s., et H t H t, λ 1,t kλ t, dp dt p.s., ii) M t k, dp dt p.s.,et H t H t, λ 2 2,t k 2 λ t, dp dt p.s., iii) M t, dp dt p.s., et H t 2 + λ 2 2,t H t 2 + k 2 λ t, dp dt p.s., de plus φ convexe dans les cas ii) et iii). Le cas des processus ponctuels Ensuite, nous traitons des cas des processus ponctuels. En particulier, (M t ) t R+ et (Mt ) t R+ peuvent être et M t = M + M t = t t H s dw s + H s d W s + t t J s (dz s λ s ds), t R +, (.3.13) J s (d Z s λ sds), t R +, (.3.14) où (W t ) t R+ est un mouvement brownien standard, (Z t ) t R+ est un processus ponctuel d intensité (λ t ) t R+, (Wt ) t R+ est un mouvement brownien rétrograde standard, et (Zt ) t R+ est un processus ponctuel rétrograde d intensité (λ t ) t R+, et (H t ) t R+, (J t ) t R+ (resp. (Ht ) t R+, (Jt ) t R+ ) sont prévisibles par rapport à (F t ) t R+ (resp. (Ft ) t R+ ). Dans ce cas, nous avons ν(dt, dx) = λ t δ Jt (dx)dt et ν (dt, dx) = λ t δ J t (dx)dt. (.3.15) Corollaire.3.8. Supposons que (M t ) t R+, (M t ) t R+ ont les sauts définis dans (.3.15) et que (M t ) t R+ est F t -adapté, que (M t ) t R+ est F t -adapté. Alors nous avons: E[φ(M t + M t )] E[φ(M s + M s )], s t, (.3.16) pour toute fonctionnelle convexe φ sous l une des trois conditions suivantes :

.3. CHAPITRE 3: INÉGALITÉS DE CONCENTRATION CONVEXE 27 i) J t Jt, λ t dp dt p.s. et H t Ht, λ t J t λ t Jt, dp dt p.s., ii) J t J t, λ t dp dt p.s., et H t H t, λ t J t 2 λ t J t 2, dp dt p.s.. iii) J t J t, λ t dp dt p.s., et H t 2 + λ t J t 2 H t 2 + λ t J t 2, dp dt p.s.. Ici, dans les cas ii) et iii) nous supposons de plus que φ est convexe. Application aux processus ponctuels Soient (W t ) t R+ et (Z t ) t R+ un mouvement brownien standard et un processus ponctuel d intensité (λ t ) t R+, qui engendrent une filtration (Ft M ) t R+. Considérons F une variable aléatoire ayant la représentation F = E[F ] + H t dw t + J t (dz t λ t dt), (.3.17) où (H u ) u R+ est un processus prévisible de carré intégrable et (J t ) t R+ un processus soit prévisible de carré intégrable soit positive et intégrable. Soient Ñ(c) une variable aléatoire de Poisson concentré d intensité c > et W (β 2 ) une variable aléatoire gaussienne centrée de variance β 2. Le Théorème.3.9 est une conséquence du Corollaire.3.8 au-dessus. Théorème.3.9. Soit F = E[F ] + H t dw t + J t (dz t λ t dt). i) Supposons que J t k, dp dt-p.s., pour certain k >, et posons β1 2 = H t 2 dt et α 1 = J t λ t dt. Puis quelle que soit la fonctionnelle convexe φ sur R, nous avons [ ( )] E[φ(F E[F ])] E φ W (β1) 2 + kñ(α 1/k). (.3.18)

28 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX ii) Supposons que J t k, dp dt-p.s., pour certain k >, et posons β2 2 = H t 2 dt et α2 2 = J t 2 λ t dt. Ainsi nous avons [ ( )] E[φ(F E[F ])] E φ W (β2) 2 + kñ(α2 2/k 2 ), (.3.19) où φ : R R est une fonctionnelle convexe de classe C 1 telle que φ est convexe. iii) Supposons que J t, dp dt-p.s., et posons β3 2 = H t 2 dt + J t 2 λ t dt. Donc nous avons E[φ(F E[F ])] E [ φ(w (β3)) ] 2, (.3.2) pour toute fonction convexe φ : R R de classe C 1 telle que φ est convexe. Application aux mesures aléatoires de Poisson Puisqu une grande famille de processus ponctuels peut être représentée comme les intégrales par rapport aux mesures aléatoires de Poisson (voir e.g. [45], Section 4, Ch. XIV), il est naturel de poursuivre les conséquences du Théorème.3.2 par rapport aux mesures aléatoires de Poisson. Soit σ une mesure de Radon sur R d qui diffuse sur R d \ {}, telle que σ({}) = 1, et ( x 2 1)σ(dx) <. R d \{} Considérons une mesure aléatoire ω(dt, dx) ayant la forme ω(dt, dx) = i N δ (ti,x i )(dt, dx) identifiée à son support {(t i, x i )} i N (localement finie). Nous supposons que la mesure ω(dt, dx) est de Poisson d intensité dtσ(dx) sur R + R d \{}, et considérons un mouvement brownien standard (W t ) t R+, indépendante de ω(dt, dx), sous une mesure de probabilité P sur Ω. Soient F t = σ(w s, ω([, s] A) : s t, A B b (R d \ {})), t R +, où B b (R d \ {}) = {A B(R d \ {}) : σ(a) < }.

.3. CHAPITRE 3: INÉGALITÉS DE CONCENTRATION CONVEXE 29 Théorème.3.1. Soit F = E[F ] + H s dw s + où (H t ) t R+ L 2 (Ω R + ) et (J t,x ) (t,x) R+ R d et R d \{} J u,x (ω(du, dx) σ(dx)du), sont F t-prévisibles, de plus (J t,x ) (t,x) R+ R d L1 (Ω R + R d \ {}, dp dt dσ) (J t,x ) (t,x) R+ R d L2 (Ω R + R d \ {}, dp dt dσ) respectivement dans (i) et dans (ii iii) suivants. i) Supposons que J u,x k, dp σ(dx)du-p.s., pour certain k >, et posons β1 2 = H u 2 du, et α 1 (x) = J u,x du, σ(dx) p.s. Donc nous avons E[φ(F E[F ])] E pour toute fonctionnelle convexe φ sur R. [ ( ( φ W (β1) 2 + kñ R d \{} ))] α 1 (x) σ(dx), k ii) Supposons que J u,x k, dp σ(dx)du-p.s., pour certain k >, et posons β2 2 = H u 2 du, et α2(x) 2 = J u,x 2 du, σ(dx) p.s. Alors nous avons E[φ(F E[F ])] E [ ( ( φ W (β2) 2 + kñ R d \{} ))] α2(x) 2 σ(dx), k 2 où φ : R R est une fonction convexe dans C 1 telle que φ est convexe. iii) Supposons que J u,x, dp σ(dx)du-p.s., et posons β3 2 = H u 2 du + J u,x 2 duσ(dx) Donc nous avons R d \{} E[φ(F E[F ])] E [ φ(w (β 2 3)) ], pour toute fonction convexe φ : R R dans C 1 telle que φ convexe..

3 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX Fonctionnelles dans L 2 La formule de Clark-Ocone pour les processus de Lévy, cf. [7], [77], dit que toute fonctionnelle F L 2 (Ω) a la représentation F = E[F ] + E[D s, F F s ]dw s + R d \{} E[D s,x F F s ](ω(ds, dx) σ(dx)ds). (.3.21) Corollaire.3.11. Soit F L 2 (Ω) définie dans (.3.21), et supposons de plus que E[D R d \{} s,xf F s ] σ(dx)ds < p.s. dans (i) suivant et E[D R d \{} s,xf F s ] 2 σ(dx)ds < p.s. dans (ii) (iii) suivants. i) Supposons que E[D u,x F F u ] k, dp σ(dx)du-p.s., pour certain k >, et posons β1 2 = σ(dx)-p.s.. Donc nous avons (E[D u, F F u ]) 2 du, et α 1 (x) = [ ( ( E[φ(F E[F ])] E φ W (β1) 2 + kñ R d \{} pour toute fonction convexe φ : R R. E[D u,x F F u ]du, ))] α 1 (x) σ(dx), (.3.22) k ii) Supposons que E[D u,x F F u ] k, dp σ(dx)du-p.s., pour certain k >, et posons β2 2 = (E[D u, F F u ]) 2 du, et α2(x) 2 = σ(dx)-p.s. Ainsi nous avons [ ( ( E[φ(F E[F ])] E φ W (β2) 2 + kñ R d \{} (E[D u,x F F u ]) 2 du, ))] α2(x) 2 σ(dx), (.3.23) k 2 pour toute fonction convexe φ : R R de classe C 1 avec φ convexe. iii) Supposons E[D u,x F F u ], dp σ(dx)du-p.s., et posons β3 2 = (E[D u, F F u ]) 2 du + R d \{} (E[D u,x F F u ]) 2 duσ(dx).

.3. CHAPITRE 3: INÉGALITÉS DE CONCENTRATION CONVEXE 31 Alors nous avons E[φ(F E[F ])] E [ φ(w (β 2 3)) ], (.3.24) pour toute fonction convexe φ : R R de classe C 1 avec φ convexe. Dans le résultat suivant, qui toutefois impose des bornes uniformes presque partout sur DF, nous considérons des mesures aléatoires de Poisson sur R d \ {} au lieu de R + R d \ {}. Corollaire.3.12. i) Supposons que D x F β(x) k, dp σ(dx)-p.s., où β( ) : R d \ {} [, k] est déterministe et k >. Donc pour toute fonction convexe φ, nous avons [ ( ( E[φ(F E[F ])] E φ kñ R d \{} ))] β(x) k σ(dx). ii) Supposons que D x F β(x) k, dp σ(dx)-p.s., où β( ) : R [, k] est déterministe et k >. Alors pour toute fonction convexe φ de classe C 1 telle que φ convexe, nous avons [ ( ( E[φ(F E[F ])] E φ kñ R d \{} ))] β 2 (x) σ(dx). k 2 iii) Supposons que β(x) D x F, dp σ(dx)-p.s., où β( ) : R [, ) est déterministe. Donc quelle que soit la fonctionnelle convexe φ de classe C 1 telle que φ convexe, nous avons [ ( ( ))] E[φ(F E[F ])] E φ W β 2 (x)σ(dx). R d \{} Martingales normales Ensuite, nous poursuivons les résultats sous la structure de martingales normales. Soit (Z t ) t R + une martingale normale, i.e. (Z t ) t R + est une martingale telle que d Z, Z t = dt. Si (Z t ) t R + est dans L 4 et il a une représentation chaotique, il satisfait l équation de structure d[z, Z] t = dt + γ t dz t, t R +,

32 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX où (γ t ) t R+ est un processus prévisible de carré intégrable, cf. [35]. Rappelons que les cas γ s =, γ s = c R \ {}, γ s = βz s, β ( 2, ), correspondent respectivement au mouvement brownien, aux processus ponctuels de Poisson concentré d intensité 1/c 2 dont l amplitude de saut vaut c, et aux martingales de Azéma. Considérons la martingale M t = M + t où (R u ) u R+ L 2 (Ω R + ) est prévisible. Nous avons et µ(dt, dx) = d M c, M c t = 1 {γt=} R t 2 dt Z s δ (s,rsγ s)(dt, dx), ν(dt, dx) = 1 γ 2 t R u dz u, (.3.25) Z s Comme application du Corollaire.3.8, nous avons le résultat suivant. δ Rsγ s (dx)dt. Théorème.3.13. Soit (M t ) t R+ définie dans (.3.25), et soit (M t ) t R+ représenté comme M t = t H s d W s + t J s (d Z s λ sds), supposons de plus que (M t ) t R+ est une F t -martingale F t -adaptée et que (M t ) t R+ est une F t -martingale rétrograde F t -adaptée. Donc nous avons E[φ(M t + M t )] E[φ(M s + M s )], s t, pour toute fonction convexe φ : R R de classe C 1, sous l une des trois conditions suivantes: i) γ t R t J t, 1 {γt=} R t 2 H t 2, et ii) γ t R t J t, 1 {γt=} R t 2 H t 2, et R t 1 {γt } λ t Jt, dp dt p.s., γ t 1 {γt } R t 2 λ t J t 2, dp dt p.s., et φ est convexe,

.3. CHAPITRE 3: INÉGALITÉS DE CONCENTRATION CONVEXE 33 iii) γ t R t, R t 2 Ht 2, Jt =, dp dt - p.s., et φ est convexe. Par conséquent, nous avons le théorème suivant: Théorème.3.14. Soit F L 2 (Ω, F, P ) une fonctionnelle définie comme F = E[F ] + R t dz t. i) Supposons que γ t R t k, dp dt-p.s., pour certain k >, et posons β1 2 = Alors nous avons 1 {γs=} R s 2 ds and α 1 = [ ( )] E[φ(F E[F ])] E φ W (β1) 2 + kñ(α 1/k), R s 1 {γs } ds γ s. quelle que soit la fonction convexe φ : R R. ii) Supposons que γ u R u k, dp dt-p.s., pour certain k > et posons β2 2 = 1 {γs=} R s 2 ds et α2 2 = 1 {γs } R s 2 ds. Donc pour toute fonction convexe φ de classe C 1 telle que φ convexe, nous avons [ ( )] E[φ(F E[F ])] E φ W (β2) 2 + kñ(α2 2/k 2 ). iii) Supposons que γ u R u et posons β3 2 = R s 2 ds. Alors pour toute fonction convexe φ de classe C 1 telle que φ convexe, nous avons E[φ(F E[F ])] E[φ(Ŵ (β2 3))].

34 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX.4 Chapitre 4 : Domination convexe pour des intégrales browniennes et stables avec dépendance Le contenu du Chapitre 4 provient d une courte note, co-écrite avec Aldéric Joulin, dont l objectif est d établir un principe de domination convexe pour des intégrales stochastiques dirigées par un mouvement brownien et un processus stable symétrique non-nécessairement indépendants. Nous démontrons que sous certaines conditions de bornitude, une variable aléatoire admettant une représentation en termes d intégrales stochastiques browniennes et stables corrélées est dominée au sens convexe par la somme indépendante de variables aléatoires gaussienne et stable symétrique. L approche utilisée repose sur encore la formule généralisée d Itô récemment développé dans [54] et nous permet de découpler la paire d intégrales stochastiques dépendantes. Soit (W t ) t un mouvement brownien standard réel non- nécessairement indépendant d un processus symétrique stable (Z t ) t d indice α (1, 2) et de mesure de Lévy stable définie sur R \ {} par σ(dx) := c x α 1 dx, c >. En notant la filtration F W,Z t := σ (W s, Z s : s t), t, nous supposons dans la suite que ces deux processus sont des (F W,Z t ) t -martingales. En d autres termes, les accroissements du premier processus sont indépendants du passé du second, et réciproquement nous considérons une variable aléatoire F ayant la représentation F E[F ] = H t dw t + K t dz t, (.4.1) où les processus bornés (H t ) t P 2 et (K t ) t P α sont supposés (F W,Z t ) t -prévisibles. Nous supposons de plus que (K t ) t est de carré intégrable afin que la variable aléatoire + K t dz t soit bien définie comme intégrale stochastique par rapport à la décomposition de Lévy-Itô du processus stable (Z t ) t. Nous allons établir un principe de domination convexe pour la variable aléatoire centrée F E[F ]. Théorème.4.1. Il existe une variable aléatoire gaussienne W (β 1 ) de variance β 1 := H t 2 dt, indépendante d une variable aléatoire réelle symétrique stable Z(β 2 ) d indice α et de mesure de Lévy sur R \ {} donnée par σ(dx) = β 2 cdx x α+1, avec < c c et β 2 := K t α dt,

.5. CHAPITRE 5: MODÈLE DU GAZ CONTINU 35 telles que pour toute fonction convexe φ à croissance au plus polynomiale d ordre p (, α) à l infini, nous avons la relation de domination convexe [ ( E [φ(f EF )] E φ W (β1 ) + Z(β )] 2 ). (.4.2) La preuve du Théorème.4.1, que nous allons décrire brièvement, est divisée en plusieurs étapes. Tout d abord, nous conditionnons la variable aléatoire centrée F E[F ] afin d obtenir une martingale (X t ) t, à laquelle on associe une filtration (F t ) t. Ensuite, nous construisons de manière appropriée une martingale rétrograde (Xt ) t par rapport à une filtration décroissante (F t ) t, qui ne dépend pas de (X t ) t qu à travers des temps aléatoires donnés par les intégrales t H s 2 ds et t K s α ds. En particulier, la valeur finale X étant la somme des variables indépendantes W (β 1 ) et Z(β 2 ), on remarque que la somme corrélée X t + Xt, t >, est découplée à l instant t =. Après avoir identifié les caractéristiques locales de ces martingales, la formule d Itô pour des martingales progressive/rétrograde, cf. [54, Théorème 8.1], appliquée à la somme (X t + Xt ) t, donne l identité [ E [φ(x t + Xt )] E φ( W (β 1 ) + Z(β ] 2 )) [ t 1 ] = E (1 τ) 2 φ (X u + Xu α (c c) + τx) K u dτdxdu, t, x α+1 quantité qui est négative car la fonction φ est convexe et < c c. Enfin, comme la projection de la variable aléatoire Xt sur la tribu F W,Z t est nulle à chaque instant t > fixé, un argument de type Jensen complète la démonstration..5 Chapitre 5: Modèle du gaz continu Introduisons le modèle du gaz continu sur R + ou R d, notés ensemble par E dans cette section. Soit Ω l espace des mesures ponctuelles i δ x i (sommes finies ou dénombrables) avec x i différents dans E, où δ x désigne la mesure de Dirac en x. Soient F = F E où F A := σ(ω(b); B(borelian) A). Etant donnée l activité z >, soit P la loi du processus ponctuel de Poisson sur E d intensité z. L interaction paire φ : E (, + ] est une fonction paire et mesurable borelienne qui est stable ([82]), i.e., H(ω) := φ(t i t j ) Bn, ω = 1 i<j n n δ tj, n 1. (stabilité) (.5.1) j=1 Rappelons que (voir [82]) la condition de stabilité est suffisante et nécessaire de définir des mesures de Gibbs sur les domaines bornés Λ. Etant donné un domaine ouvert et non vide Λ E et ω Ω, soit ω Λ = t i Λ supp(ω) δ t i la restriction de la mesure ω à Λ, et

36 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX Ω Λ = {ω Λ ; ω Ω}. La mesure image P Λ de P par ω ω Λ est la loi du processus ponctuel de Poisson d intensité z sur Λ. La mesure de Gibbs dans Λ pour une condition frontière fixée η Ω sur Λ c est une mesure de probabilité sur (Ω Λ, F Λ ) donnée par où µ η Λ (dω) := (Zη Λ ) 1 exp [ βh η Λ (ω)]p Λ(dω Λ ) (.5.2) H η Λ (ω) = H(ω Λ) + Λ ω Λ (dx) φ(x y)η(dy) Λ c est le Hamiltonien (H(ω Λ ) étant définie dans (.5.1)), Z η Λ est la constante de normalisation et β > est la température inverse. Soit E(t, ω) l énergie relative d interaction entre une particule située à t et la configuration ω comme suivant: { φ(t u)ω(du) si φ(u t) ω(du) < ; E(t, ω) := = + sinon. Inégalités de concentration convexe Dans cette section nous prenons Λ = [, T ] pour une certaine constante positive T suffisamment grande. Pour tout t T, définissons et λ t = ze η Λ [e E(t,ω) F F t ] (.5.3) F t = t où f est une fonction réelle sur Λ. Introduisons deux hypothèses essentielles. f(s)ω(ds), (.5.4) Hypothèse 1 Hypothèse 2 α := T f(s)λ sds < α 2 := T f 2 (s)λ s ds < Remarque.5.1. Si φ est positive, la condition de stabilité est satisfaite avec B = et λ s z. Donc l Hypothèse 1 (resp. l Hypothèse 2) peut être vérifiée par l intégrabilité (resp. carré intégrabilité) de f, qui est une condition souvent utilisée. Sous les hypothèses au-dessus, le gaz continu est juste un exemple des théorèmes dans le Chapitre 3. Donc nous avons trois types d inégalités de concentration convexe pour la mesure de Gibbs.

.5. CHAPITRE 5: MODÈLE DU GAZ CONTINU 37 Théorème.5.2. (de type espérance) Sous l Hypothèse 1, si de plus f(s) k, s T pour certain k >. Donc pour toute fonction convexe ψ, nous avons E η Λ [ψ (F T E η Λ [F T ])] E[ψ ( kn α/k α ) ], (.5.5) où N t est un processus de Poisson d intensité 1. Théorème.5.3. (de type variance) Sous l Hypothèse 2 et si f(s) k sur [, T ] pour certain k >, nous avons E η Λ [ψ(f T E η Λ [F T ])] E[ψ(kN α 2 k 2 α2 )], (.5.6) k où ψ est une fonction convexe de classe C 1 avec φ convexe et N t est un processus de Poisson d intensité 1. Théorème.5.4. (de type brownien) Sous l Hypothèse 2, supposons que f est négative. Alors pour toute fonction convexe φ de classe C 1 avec φ convexe, nous avons E η Λ [ψ(f T E η Λ [F T ])] E[ψ(W α 2)], (.5.7) où W t est un mouvement brownien standard. Domination stochastique Prenons maintenant E = R d et supposons de plus que φ est non-négative. Définition.5.5. Une fonction g sur Ω Λ est dite croissante(resp. décroissante ), si pour tout u Λ, D u g(ω) := g(w + δ u 1 u/ suppω ) g(ω δ u 1 u suppω ) ( resp. ). En fait, ici D est l opérateur aux différences sur Ω Λ (voir [72]).

38 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX Théorème.5.6. La mesure de Gibbs définie dans (.5.2) est stochastiquement dominée par P Λ, i.e., pour toute fonction croissante ψ L 1 (Ω Λ, P Λ ), E η Λ [ψ] E[ψ]. (.5.8) Ce résultat est une conséquence de la proposition suivante avec g 1. Proposition.5.7. Soient f une fonction croissante et g une fonction décroissante positive, donc nous avons E η Λ [fg] Eη Λ [g]e[f]. La preuve du Proposition.5.7 est facilement obtenue par l inégalité FKG sur l espace de Poisson. Soit H x, x Λ une fonction positive sur Ω Λ. Corollaire.5.8. pour tout A Λ, posons F A (ω) = A H xω(dx). Donc pour toute fonction croissante ψ sur R, telle que ψ F A (ω) := ψ(f A (ω)) est intégrable par rapport à P Λ, nous avons E η Λ [ψ F ] E[ψ F ]. (.5.9) Remarque.5.9. a.) Prenant H x (ω) = f(x) une fonction positive sur Λ, (.5.9) est, pour toute fonction croissante ψ sur R, E η Λ [ψ(f A)] E[ψ(F A )], A Λ. (.5.1) C est intuitif puisque φ est positive, toutes particules se poussent. b.) Soit H x (ω) 1, donc F A = ω(a) désigne le nombre de particules dans A, qui est un processus de Poisson d intensité z sur (Ω Λ, F Λ, P Λ ). D autre coté, par le théorème de Nguyen-Zessin, on sais que E η Λ (ω(a)) = Eη Λ A ze E(x, ) dx z A = E[ω A ], (.5.11) où E(x, ω) := φ(x y)ω(dy). Donc (.5.11) est juste (.5.1) avec ψ 1. Λ La partie prochaine est relative aux inégalités fonctionnelles.

.6. CHAPITRE 6: INÉGALITÉ FKG SUR L ESPACE DE WIENER 39.6 Inégalité FKG sur l espace de Wiener par la représentation prévisible En 1971, C.M. Fortuin, P.W. Kasteleyn, J. Ginibre ont premièrement introduit l inégalité FKG dans leur article [36]. Puis cette type d inégalité a attiré beaucoup de mathématiciens. En 1991, Herbst et Pitt ([41]) ont utilisé de l équation de diffusion pour étudier des monotonicités stochastique et des corrélations positives. Chen et Wang (1993) ont poursuivi ce sujet et ont donné le théorème de comparaison. En 1992, Bakry et Michel ([4]) a prouvé des inégalités FKG sur { 1, +1} n. Hu ([44], 1997) a utilisé de l expansion chaotique d Itô-Wiener pour les équations de diffusion pour obtenir une corrélation positive pour la mesure gaussienne. Il a aussi obtenu des inégalités FKG. Les motivations directes de cet article sont les travaux de Wu ([95], 2) et de Barbato ([5], 25). Wu a magnifiquement prouvé l inégalité FKG sur l espace de Poisson par l analyse de Malliavin dans le Remarque 1.5. (voir [95]). Barbato a défini une relation ordre et a prouvé une inégalité FKG pour le mouvement brownien par la méthode d approcher graduellement. Nous suivons l idée de [95] pour étudier des inégalités sur l espace de Wiener classique. Soit (Ω, F, P, ) l espace de Wiener classique, i.e., Ω = C ([, 1]; R), avec le mouvement brownien canonique (W t ) t [,1] engendrant la filtration (F t ) t [,1], dont la relation d ordre partiel sur Ω est définie par: Définition.6.1. Soient ω 1, ω 2 Ω, nous disons que ω 1 ω 2 si pour tous t 1 < t 2 1, nous avons ω 1 (t 2 ) ω 1 (t 1 ) ω 2 (t 2 ) ω 2 (t 1 ). Définition.6.2. Une variable aléatoire réelle F sur (Ω, F, P, ) est dite croissante si F (ω 1 ) F (ω 2 ) pour tous ω 1, ω 2 Ω satisfaisant ω 1 ω 2. L inégalité FKG dit que si F et G sont deux fonctionnelles croissantes de carré intégrable, donc F et G sont non négativement corrélées: Cov (F, G). Par la formule de Clark-Ocone, toute fonctionnelle dans L 2 (Ω) a une représentation prévisible: F = E[F ] + 1 E[D t F F t ]dw t,

4 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX qui implique l identité de covariance [ 1 Cov(F, G) = E où D est le gradient de Malliavin exprimé comme Par (.6.1), nous avons la proposition: ] E[D t F F t ]E[D t G F t ]dt, (.6.1) DF, ḣ L 2 ([,1]) = d dε F (ω + ɛh) ε=. (.6.2) Proposition.6.3. Si deux fonctionnelles F et G dans L 2 (Ω) satisfont E[D t F F t ]E[D t G F t ], nous avons Cov (F, G). Heureusement (.6.2) nous donne la possibilité de prouver l inégalité FKG par l analyse de Malliavin. Mais c est évident seulement pour une famille de fonctions, qui est dense dans L 2 (Ω). Puis avec le semigroupe de Ornstein-Uhlenbeck, par la méthode d approcher, finalement nous avons l inégalité FKG sur L 2 (Ω): Théorème.6.4. Soient F, G deux fonctionnelles croissantes dans L 2 (Ω), nous avons Cov (F, G). Application aux équations différentielles stochastiques Maintenant nous appliquons ce théorème aux équations différentielles stochastiques. Considérons les équations différentielles stochastiques { dxt = b t (X t )dt + σ t (X t )dw t (.6.3) X = x, et { d Xt = b t ( X t )dt + σ t ( X t )dw t X = x, (.6.4) où b, b, σ, σ sont fonctions sur R + R satisfaisant les conditions suivantes, cf. [67], page 99:

.6. CHAPITRE 6: INÉGALITÉ FKG SUR L ESPACE DE WIENER 41 (i) σ t (x) σ t (y) + b t (x) b t (y) K x y, x, y R, t [, 1], (ii) t σ t () et t b t () sont bornées sur [, 1], pour certain K >. Par la Proposition 1.2.3 et le Théorème 2.2.1 de [67], nous avons f(x s ) Dom(D), s [, 1], et { s s ( D r f(x s ) = 1 [,s] (r)σ r (X r )f (X s ) exp α u dw u + β u 1 ) } 2 α2 u du, (.6.5) r r, s [, 1], où (α u ) u [,1] et (β u ) u [,1] sont des processus adaptés et uniformément bornés. Alors nous avons Théorème.6.5. Soient s, t [, 1] et supposons que σ, σ satisfont la condition r σ r (x) σ r (y), x, y R, r s t. Donc nous avons Cov(f(X s ), g( X t )), (.6.6) pour toutes fonctions lipschitziennes croissantes f, g. Remarque.6.6. En fait, le Lemme 8 de [5] a montré que les solutions (X t ) t [,1], ( X t ) t [,1] de (.6.3) et de (.6.4) sont des fonctionnelles croissantes quand σ(x), σ(x) sont différentiables et de dérivée lipschitzienne en une variable, et satisfont des bornes uniformes du type < ε σ(x) M < and < ε σ(x) M <, x R. Alors par le Théorème.6.4, ils satisfont l inégalité FKG comme le Théorème 7 de [5]. Mais ses conditions sont plus fortes que les nôtres. Cas discret Soit Ω = { 1, 1} N et considérons (X k ) k 1 une famille de variables aléatoires à valeurs { 1, 1} indépendantes de Bernoulli construites comme les projections canoniques sur Ω, sous une mesure de probabilité P telle que p n = P(X n = 1) et q n = P(X n = 1), n N.

42 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX Soient F 1 = {, Ω} et F n = σ(x,..., X n ), n N. Considérons l opérateur gradient linéaire D défini comme D k F (ω) = p k q k (F ((ω i 1 {i k} + 1 {i=k} ) i N ) F (ω i 1 {i k} 1 {i=k} ) i N ), (.6.7) k N. Rappelons la formule de Clark-Ocone dans le cas discret, cf. la Proposition 7 de [79]: F = E[F ] + E[D k F F k 1 ]Y k, (.6.8) où qk k= pk Y k = 1 {Xk =1} 1 {Xk = 1}, k N p k q k sont une suite renormalisée de variables aléatoires centrées indépendantes et identiquement distribuées de variance un. La formule de Clark-Ocone déduit l identité de covariance, cf. le Théorème 2 de [79]: [ ] Cov (F, G) = E E[D k F F k 1 ]E[D k G F k 1 ], k= qui implique un analogue en temps discret. Lemme.6.7. Supposons F, G L 2 (Ω) satisfaisant Donc nous avons: E[D k F F k 1 ] E[D k G F k 1 ], k N. Cov (F, G). D après la définition suivante, quelle que soit la fonctionnelle F croissante, on a D k F pour tout k N. Définition.6.8. Une variable aléatoire F : Ω R est dite croissante si pour tous ω 1, ω 2 Ω, nous avons ω 1 (k) ω 2 (k), k N, F (ω 1 ) F (ω 2 ). Evidemment nous obtenons aussi une inégalité FKG dans le cas discret: Proposition.6.9. Si F, G L 2 (Ω) sont deux fonctionnelles croissantes, donc F et G satisfont : Cov (F, G).

.7. CHAPITRE 7: PROCESSUS DE NAISSANCE ET DE MORT 43.7 Inégalités de trou spectral et de concentration convexe pour les processus de naissance et de mort Ce travail, effectué avec Wei Liu, est soumis au journal Annals de l IHP. Considérons un processus de naissance et de mort sur N = {n : n } de taux de naissance (b i ) i N et de taux de mort (a i ) i N, i.e., son générateur L est défini par, pour toute fonction réelle f sur N, Lf(i) = b i (f(i + 1) f(i)) + a i (f(i 1) f(i)) (.7.1) où b i > (i ), a i > (i > ), a =. Soient µ = 1, µ n = b b 1 b n 1 a 1 a 2 a n, n 1, donc µ est une mesure invariante de L. Supposons que C := + n= µ n < + et définissons la mesure de probabilité normalisée π de µ comme π n = µn, n. C Chen Mu-Fa et Wang Feng-yu ont premièrement utilisé la méthode probabiliste (couplage) à étudier la première valeur propre pour des variétés (voir [2]), puis en utilisant cette méthode, eux et leur groupe ont obtenu une série de résultats importants. Ils ont caractérisé le trou spectral pour des processus de diffusion, des processus de naissance et de mort, des chaînes de Markov et des opérateurs elliptiques ((voir [19])). Miclo ([64], 1999 ) a donné une condition nécessaire et suffisante pour avoir le trou spectral pour des processus de naissance et de nort par l inégalité de Hardy. Chen, dans ([?], 23), par la méthode analytique, a redonné une formulation variationnelle du trou spectral des processus de naissance et de mort. Wu, dans [1], a considéré des processus de diffusion de générateur L et caractérisé la borne de ( L) 1 sur certain espace des fonctionnelles lipschitziennes. Puis comme application, avec la théorie spectrale, il a obtenu la formule variation de Chen pour des processus de diffusion. Notre méthode repose sur la sienne. Etant donnée une fonction strictement croissante ρ : N R, soit d ρ (i, j) = ρ(i) ρ(j) la métrique sur N associée avec ρ. Nous disons qu une fonctionnelle f sur N est lipschitzienne par rapport à ρ si qui est équivalent à f Lip(ρ) := sup i j f Lip(ρ) = sup i f(j) f(i) ρ(j) ρ(i) f(i + 1) f(i) ρ(i + 1) ρ(i) < +. < +, (.7.2)

44 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX Désignons par (C Lip(ρ), Lip(ρ) ) l espace des fonctionnelles lipschitziennes par rapport à ρ. Par l ergodicité, l opérateur L : D Lip(ρ) (L) C Lip CLip(ρ) est injective, où C Lip(ρ) désigne l ensemble des fonctionnelles lipschitziennes dont espérances sont nulles. Par la définition, L a un trou spectral dans C Lip(ρ) si est une valeur propre isolable dans C Lip(ρ) ou équivalemment ( L) 1 : CLip(ρ) C Lip(ρ) est borné. Le résultat principal peut être raconté comme suivant: Théorème.7.1. Soient L, ρ, Lip(ρ) donnés comme ci-dessus et supposons que ρ L 1 (µ), donc nous avons I(ρ) := ( L) 1 Lip(ρ) = sup i 1 µ((ρ π(ρ))i [i,+ ] ) µ i a i (ρ(i) ρ(i 1)) Rappelons l équation de Poisson, i.e., Lf = g. Donc, (.7.3) ( L) 1 Lip(ρ) = sup f Lip(ρ). g Lip(ρ) =1 A l aide au Théorème 1 et au Lemme 4 du Chapitre 7 dans [89], nous avons i 1 k= f(i + 1) f(i) = µ kg(k), µ i a i qui déduit que ( L) 1 Lip(ρ) = i 1 k= sup f Lip(ρ) = sup sup µ kg(k) g Lip(ρ)=1 g Lip(ρ)=1 i 1 µ i a i (ρ(i) ρ(i 1)). Une inégalité importante i=k µ ig(i) i=k µ i(ρ i π(ρ)), où g Lip(ρ) = 1 et µ(g) =, complète la preuve du Théorème.7.1. Application au trou spectral Soit A l ensemble des fonctionnelles strictement croissantes ρ sur N et ρ L 2 (π). Comme une application du Théorème.7.1, nous avons le résultat suivant, qui caractérise le trou spectral de L dans L 2 (µ). Théorème.7.2. Soit λ le trou spectral de L dans L 2 (µ), donc nous avons où I(ρ) est définie en (.7.3). λ sup I(ρ) 1, (.7.4) ρ A

.7. CHAPITRE 7: PROCESSUS DE NAISSANCE ET DE MORT 45 Puisque L est un opérateur auto-adjoint et défini positif, il admet une décomposition spectrale L = λde λ, (,+ ) donc nous avons ( L) 1 = λ 1 de λ. (,+ ) Nous démontrons par le Théorème.7.1 que pour tout λ (, I(ρ) 1 ), E λ =. Désignons par λ 1 le trou spectral de L, donc λ 1 sup ρ A I(ρ) 1. Inǵalité de concentration convexe pour purement martingales à sauts Dans cette sous-section, nous obtenons une inégalité de concentration convexe pour des martingale générales de saut pur. Le résultat est le suivant: Théorème.7.3. Soit M t une martingale de saut pur avec M =, satisfaisant quelque soit t, M t K et < M > t < +. Alors pour toute fonction convexe φ de classe C 2 avec φ croissante, nous avons pour tout t, E[φ(M t )] E [ φ(kn <M>t /K 2 < M > t /K) ], où (N t ) t est un processus de Poisson standard. Puisque M t est une martingale de saut pur et M t K, elle a une représentation suivante t M t = z(ω(du, dz) λ u (dz)du), [ K,K]\{} où ω(du, dz) est la mesure aléatoire de saut de (M t ) t d intensité λ u (dz)du. Donc, la formule généralisée d Itô dans le Chapitre 3 implique ce théorème.

46 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX Inégalité de concentration convexe pour des fonctionnelles lipschitziennes Dans cette sub-section, nous supposons toujours que (N t ) t est un processus de Poisson d intensité 1. Commençons une conséquence directe du Théorème.7.3, la proposition suivante, Proposition.7.4. Soit f une fonction satisfaisant et Définissons K = sup f(k) f(k 1) <, k 1 Γ(f) = sup(a k (f(k 1) f(k)) 2 + b k (f(k + 1) f(k)) 2 ) <. k 1 M f t donc pour toute fonction φ C c, = f(x t ) f(x ) t Lf(X s )ds, [ ( E[φ(M f t )] E φ KN Γ(f) t/k 2 Γ(f) )] t. K Etant donnée g une fonction sur N avec µ(g) =, considérons S t = t g(x s)ds, où X s est le processus de naissance et de mort au-dessus. Hypothèse A: Hypothèse B: sup k 1 K := sup k 1 k 1 i= µ jg(j)/µ k a k < +. a k ( k 1 i= µ jg(j) µ k a k ) 2 ( k ) i= + b µ 2 jg(j) k µ k a k <. Remarque.7.5. Soit Lf = g, donc K = sup k 1 f(k) f(k 1) et l Hypothèse B est équivalent à Γ(f) = sup a k (f(k 1) f(k)) 2 + b k (f(k + 1) f(k)) 2 <. k 1

.7. CHAPITRE 7: PROCESSUS DE NAISSANCE ET DE MORT 47 Théorème.7.6. Supposons que g, S t satisfont les conditions au-dessus. Donc, sous les Hypothèse A et B, nous avons pour toute fonction convexe φ de classe C 2 avec φ croissante, E[φ(S t )] E[φ(KN Γ(f) t/k 2 Γ(f) t/k)]. Soit T une constante fixée, pour tout t T, définissons et M t = f(x t ) f(x ) M t = f(x ) f(x t ) t t Lf(X s )ds Lf(X s )ds, par la décomposition de martingale progressive/rétrograde de Lyons-Zheng (Voir [58, 94]), nous savons que (M t ) t est une F t -martingale progressive Càdlàg de saut pur et ( M t ) t est Càdlàg telle que ( M T M T t, t T ) est une G T t -martingale rétrograde de saut pur, où F t = σ{x s, s t} et G t = σ{x s, t s T }. De plus, M t et M t ont la même loi par rapport à P µ. Puisque 2S t = M t + M t, par la convexité de φ, nous avons E µ [φ(s t )] E µ [φ(m t )]. Evidemment, M t sup k 1 f(k 1) f(k) = k et par l Hypothèse B quelque soit t, < M > t t Γ(f) est fini. Donc le Théorème.7.6 est une conséquence du Théorème.7.3. Remarque.7.7. Appliquant le Théorème.7.6 avec φ(x) = e λx, nous avons une inégalité de déviation pour S t, i.e., P (S t E[S t ] x) inf exp{ λx + Γ(f) t(e λk λk 1)/K 2 } λ> { = exp Γ(f) } t h(kx/ Γ(f) K 2 t), où h(u) = (1 + u) log(1 + u) u. Prenant x = ty, nous avons aussi P ( 1 { t (S t E[S t ]) y) exp Γ(f) } t h(ky/ Γ(f) K 2 ), où h(u) = (1 + u) log(1 + u) u. Ensuite, nous donnons deux exemples typiques: la file d attente M/M/1 et le modèle M/M/.

48 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX La file d attente M/M/1 Soit (X t ) t un processus de la file M/M/1, c est-à-dire que le générateur L de X t satisfait que pour toute fonction f sur N, nous avons Lf(i) = λ(f(i + 1) f(i)) + ν1 i (f(i 1) f(i)). Donc une mesure invariante µ de L vérifie que pour tout k, µ k = σ k, où σ := λ. ν Nous supposons que σ < 1 telle que i= µ i <. La loi unique stationnaire π est la loi géométrique de paramètre σ donnée par ν k = σ k (1 σ), k N. Proposition.7.8. Soient X t le processus au-dessus et g une fonctionn sur N satisfaisant µ(g) = et K := 1 λ ν sup k 1 i= π ig(i + k) <. Définissons S t = t g(x s)ds, donc pour toute fonction convexe φ de classe C 2 avec φ croissante, nous avons E[φ(S t )] E[φ(KN (λ+ν)t (λ + ν)kt)]. Corollary.7.1. Pour tout i N, posons ρ(i) = i 1 k=1, où a > 1 est une constante. k a Donc quelle que soit la fonction g CLip(ρ), nous avons, [ ( ρ E[φ(S t )] E φ ν λ g Lip(ρ)N (λ+ν)t ρ )] ν λ (λ + ν) g Lip(ρ)t, où φ est une fonction de classe C 2 avec φ croissante et ρ := sup k 1 ρ(k). Le modèle M/M/ Soit (X t ) t un processus de M/M/ dont générateur L satisfaisant que pour toute fonction f on N, nous avons Lf(i) = λ(f(i + 1) f(i)) + νi(f(i 1) f(i)), où λ, ν sont deux constantes positives. Donc la mesure invariante µ est µ k = σk, k, k! où σ := λ. Ce processus est ergodique de distribution réversible stationnaire π, la mesure ν de Poisson sur N de paramètre σ, i.e., σ σk π k = e k!, k N. Pour ce modèle, nous avons la proposition suivante:

.7. CHAPITRE 7: PROCESSUS DE NAISSANCE ET DE MORT 49 k 1 Proposition.7.2. Etant donnée une fonction g satisfaisant sup k 1 k i= σi g(i)/i! νkσ k k! K, où K est une constante positive et π(g) =. Donc quelle que soit la fonction convexe φ de classe C 2 avec φ croissante, nous avons E[φ( t g(x s )ds)] E[KN (λ+ν)t (λ + ν)kt]. Corollary.7.3. Soit ρ(k) = k, k 1. Supposons g CLip(ρ), donc pour toute fonction φ C 2 avec φ croissante, nous avons [ ( )] σe σ E[φ(S t )] E φ ν g Lip(ρ)N (λ+ν)t (λ + ν) σeσ ν g Lip(ρ)t.

5 CHAPITRE. INTRODUCTION ET RÉSULTATS PRINCIPAUX

Partie I Principes de grandes déviations 51

Chapitre 1 A sufficient and necessary condition of LDP for exchangeable sequence in the τ-topology Cet article en collaboration avec Qiongxia, Song et Liming, Wu est publié dans la revue Statistic and Probability Letters, 77(3), 239-246, 27. Abstract For an exchangeable sequence of random variables (X n ) n N taking values in a Polish space, we obtain a necessary and sufficient condition for the large deviation principles with respect to the τ-topology of the occupation measure L n := (1/n) n 1 k= δ X k and of the process-level empirical measures. 1.1 Introduction and Main result Let (X k ) k be an exchangeable sequence of random variables taking values in a Polish space E, i.e., (X, X 1,, X n ) law = (X σ(), X σ(1),, X σ(n) ) for any n 1 and for any permutation {σ (1),, σ (n) } of {1,, n}. Consider its empirical (or occupation) measures L n := 1 n 1 δ Xk (δ being the Dirac measure at ), n 1, n k= 53

54 CHAPITRE 1. LDP FOR EXCHANGEABLE SEQUENCES which are random elements in the space M 1 (E) of probability measures on E, equipped with the σ-field B(M 1 (E)) = σ(ν ν(f) f bb), where bb is the space of all real valued, bounded and B-measurable functions. Its process-level empirical measures are R n := 1 n 1 δ (Xk,X n k+1, ), n 1, k= which are random elements in the space M 1 (E N ) of probability measures on E N, equipped with the σ-field B(M 1 (E N )) = σ(q Q(F ) F bf N, N ), where F N = σ(x k ; k N). By the well known de Finetti theorem, the law P of (X k ) k is a mixture of homogeneous product measures, i.e., P = ρ N dm(ρ) (1.1.1) M 1 (E) for some probability measure m on M 1 (E). So the large deviation principle about the exchangeable sequences (X n ) should be reduced to that of independent identically distributed sequences as the reader can imagine. However this reduction is not at all easy, as shown by known works [24, 26, 85, 91, 98]. The exchangeable sequences are widely used in Bayesian statistical and the bootstrap statistics, see the recent work by Chen [14], Trashorras [85] and Chaganty [13] etc. The reader is refereed to Aldous [1] for backgrounds and other applications. A lot of work has been done on the large deviation principle (LDP in short) of L n (and of R n ) for exchangeable sequences. For (X k ) taking values in a separable Banach space de Acosta [26] proved a good upper bound for the large deviation for (1/n) n k=1 X k with a convex rate function which is not the exact one. When E is finite, Wu [91] obtained the first complete large deviation principle for R n and he found that the rate function is not convex(so the Ellis-Gartner theorem is not valid). Successively the subject was investigated by Dinwoodie and Zabell [32] and Daras [24, 25], Trashorras [85]. For instance, Daras [25] obtained the large deviation principle of R n with respect to (w.r.t. in short) the weak convergence topology when E is compact. A direct motivation of this paper is the recent work of the third author [98], in which the large deviation principles of L n on M 1 (E) and of R n on M 1 (E N ) w.r.t. the weak convergence topology have been characterized by the compactness of the support of m in the weak convergence topology. In this paper, we will characterize the corresponding LDP w.r.t. the τ-topology. The τ-topology on M 1 (E) is the weakest topology on M 1 (E) such that for all f bb, ν ν(f) is continuous, where ν(f) := fdν. The projective limit τ-topology on E M 1 (E N ), denoted by τ p, is defined as the weakest topology such that for all F bf N, Q Q(F ) is continuous on M 1 (E N ).

1.1. INTRODUCTION AND MAIN RESULT 55 The main result of this chapter is the following theorem. We will adapt the terminology and notation of [3]. Theorem 1.1.1. Let S τ denote the topological support of the mixture measure m appeared in (1.1.1) w.r.t. the τ-topology. The following properties are equivalent: (a) As n goes to infinity, the law of L n satisfies the LDP on M 1 (E) w.r.t. the τ- topology, with some rate function I : M 1 (E) [, + ]. (Convention: the LDP implies necessarily that the rate function is good or τ-infcompact, i.e., [I L] is τ-compact, L R +.) (b) The law of R n satisfies the LDP on M 1 (E N ) w.r.t. the τ p - topology with the rate function given by J(Q) := inf ρ S τ h sp (Q ρ N ), Q M 1 (E N ). (1.1.2) where h sp (Q ρ N ) is the specific relative entropy of Q w.r.t. ρ N, i.e., 1 lim n h sp (Q ρ N ) := h(q n [,n 1] ρ n ) if Q M1(E s N ) + otherwise, (1.1.3) where M1(E s N ) is the space of those Q M 1 (E N ) which are stationary, Q [,n 1] is the law of the first n coordinates (X k ) k n 1 of E N under Q, and h(q [,n 1] ρ n ) is the relative entropy of Q [,n 1] w.r.t. ρ n (see (1.1.5) below for definition). (c) The τ-topological support S τ = supp τ (m) is τ-compact in M 1 (E). In this case the rate function in the LDP of part (a) is given by I(ν) = inf ρ Sτ h(ν ρ), ν M 1 (E). (1.1.4) where h(ν ρ) is the Kullback relative entropy of ν w.r.t. ρ, i.e., dν dν log E dρ h(ν ρ) := dρ, if νlρ dρ + otherwise (1.1.5)

56 CHAPITRE 1. LDP FOR EXCHANGEABLE SEQUENCES Compared to [98], the main new difficulty in proving Theorem 1.1.1 resides in the fact that the τ-topology on M 1 (E) is not metrizable and thus many technical ingredients in [98], are no longer valid. Moreover, since S τ is compact with respect to the weak convergence topology when it is τ-compact, we see that our LDP results in parts (a) and (b) of Theorem 1.1.1 are stronger than those in [98]. This improvement is meaningful because many observations in practice are often not continuous on E. 1.2 Preliminaries Throughout this paper M 1 (E) and M 1 (E N ) are equipped with the τ- topology and the τ p - topology, the σ-fields B(M 1 (E)) and B(M 1 (E N )) (given in the Introduction), respectively. We shall assume that the exchangeable sequence (X n ) n N is the system of coordinates on ( ) (Ω, F, P) := E N, B(E) N, ρ N dm(ρ) = ρ N dm(ρ) S τ M 1 (E) where S τ = supp τ (m). Since S τ is not necessarily B(M 1 (E))-measurable, we take m as the outer measure and we have m (S τ ) = 1. Then A m (S τ A) is a probability measure restricted on S τ B(M1 (E)), which will again be denoted by m for simplicity. 1.2.1 Several elementary lemmas The following lemma is crucial when the τ-topology is considered, which can be found in Rudin [7]. Lemma 1.2.1. If X is a compact topological space and if some sequence {f n } n N of continuous real-valued functions on X separates points on X, then X is metrizable. Lemma 1.2.2. Let S be a τ-compact subset of M 1 (E), then (S, τ) is metrizable. Proof. It is known that there exists some sequence (g n ) n N C b (E), such that µ, ν M 1 (E), µ = ν if and only if (iff in short) µ, g n = ν, g n, n. For all µ S, consider f n (µ) := µ, g n, where µ, f := fdµ. It is obvious that {f E n} n N separates the points on S. So by Lemma 1.2.1, S is metrizable. Lemma 1.2.3. Let M M 1 (E). M is relatively τ-compact iff {f n } n N bb decreasing pointwise to, sup ν M ν, f n as n.

1.2. PRELIMINARIES 57 Proof. 1) Necessity. Let g n (ν) := ν, f n, n, then g n on the τ-closure M τ is τ continuous, decreases to (as n ) for each ν M τ. Since M τ is τ-compact, Dini s monotone convergence theorem implies that the convergence is uniform over M τ. 2) Sufficiency. Let B(, 1) be the closed unit ball (centered at ) in the topological dual space (bb). It is compact w.r.t. the weak -topology σ ( (bb), bb ), which, restricted to M 1 (E) (bb), coincides with the τ-topology. Consequently to show the relative τ-compactness of M, it is sufficient to show that each element φ belong to the ( ) closure of M in (bb), σ((bb), bb) is a probability measure. Indeed, such a continuous linear form φ on bb is finitely additive, nonnegative and φ(1 E ) = 1. By our assumption, φ, f n sup ν M ν, f n as n goes to infinity for any sequence {f n } n N bb decreasing pointwise to zero. Then φ is σ-additive. This shows that φ is a probability measure. Lemma 1.2.4. (a) (ν, ρ) h(ν ρ) is jointly lower semi-continuous (l.s.c. in short) and convex on (M 1 (E), τ) (M 1 (E), τ). (b) If S τ is τ-compact in M 1 (E), the functional I given by (1.1.4) is τ-inf-compact on M 1 (E), i.e., [I L] is τ-compact in M 1 (E) for each L R +. Proof. (a). It follows from Donsker-Varadhan s variational formula, ( ) h(ν ρ) = sup ν, f log e f dρ f bb E (1.2.1) (b). 1) Relative τ-compactness of [I L], L R +. By Lemma 1.2.3, we only need to prove sup ν, f n ν [I L] for any sequence {f n } n N bb decreasing pointwise to zero over E. Indeed, for any

58 CHAPITRE 1. LDP FOR EXCHANGEABLE SEQUENCES N 1, we obtain by (1.2.1) sup ν, f n ν [I L] ν [I L] ρ S τ sup inf 1 sup ν [I L] 1 NL 1 N + sup ρ S τ log ( ) h(ν ρ) + log e NLfn dρ E NL ( inf h(ν ρ) + sup ρ S τ e NLfn dρ. E ρ S τ log E ) e NLfn dρ Observing that the functionals φ n (ρ) := log E enlfn dρ are τ-continuous on S τ, and φ n (ρ) for each ρ, we have, by the τ-compactness of S τ and Dini s monotone convergence theorem, sup ρ Sτ φ n (ρ). Therefore, lim sup n sup ν [I L] Since N 1 is arbitrary, the desired result follows. ν, f n 1/N. 2) Closeness of [I L], L R +, i.e., the l.s.c. of I. Since [I L] is relatively τ-compact, its τ-closure is metrizable by Lemma 1.2.2. It remains to prove that if ν n [I L] and ν n τ ν, then ν [I L]. For each n, by the l.s.c. of ρ h(ν n ρ) and the τ-compactness of S τ, there is some ρ n S τ such that I(ν n ) = inf ρ S τ h(ν n ρ) = h(ν n ρ n )( [, + ]). Again by the τ-compactness of S τ, we can find a subsequence (ρ nk ) k N tending to some ρ S τ. Thus by part (a), L lim inf k h(ν n k ρ nk ) h(ν ρ) I(ν). The following is a well known fact.

1.2. PRELIMINARIES 59 Lemma 1.2.5. Suppose (I n : E [, + ]) n is an increasing sequence of infcompact functions on a Hausdorff topological space E, then inf sup x E n I n (x) = sup inf I n(x). n x E Lemma 1.2.6. If S τ is τ-compact in M 1 (E), then the functional J given by (1.1.2) is τ-inf-compact on M 1 (E N ). Proof. We prove that J given by (1.1.2) is τ-inf-compact on M s 1(E N ), since otherwise J = +. For each fixed Q M s 1(E N ), f(n) := h(q [,n 1] ρ n ) is super-additive, i.e., f(n + 1) f(n) + f(1), then h sp (Q ρ N ) = sup n 1 By Lemma 1.2.5 and the τ-compactness of S τ, Hence J is l.s.c. on M 1 (E N ). 1 n h(q [,n 1] ρ n ) = lim k 1 2 k h(q [,2 k 1] ρ 2k ). 1 J(Q) = lim inf k ρ S τ 2 h(q k [,2 1] ρ k 2k ). In further, for any L and for each k fixed, {Q [,2 k 1] : J(Q) L} { Q [,2 k 1] : 1 } 2 inf h(q k [,2 1] ρ ρ S k 2k ) L. (1.2.2) τ By Lemma 1.2.4(b) the right hand side in (1.2.2) is τ-compact in M 1 (E 2k ), so is the left hand side in (1.2.2). Note that k is arbitrary, [Q : J(Q) L] is then compact in (M 1 (E N ), τ p ). Lemma 1.2.7. Let (f n, f) n N be a family of functions defined on a metric space (E, d), taking values in [, + ]. Suppose lim inf n f n (x n ) f(x) for all x E and all sequences x n x, then for any x E, where B(x, δ) := {y E; d(y, x) < δ}. lim lim inf inf f n (y) f(x), δ + n y B(x,δ)

6 CHAPITRE 1. LDP FOR EXCHANGEABLE SEQUENCES Its proof is elementary, so omitted. We now arrive at the crucial Lemma 1.2.8. Let ρ n τ ρ in M 1 (E) and P n := ρ N n. Then (a) P n (L n ) satisfies the LDP on (M 1 (E), τ) with the rate function ν h(ν ρ). (b) More generally P n (R n ) satisfies the LDP on (M 1 (E N ), τ p ) with the rate function Q h sp (Q ρ N ). Proof. We prove part (b), and part (a) follows immediately from (b) by contraction principle. For all Q M 1 (E N ), let Π N Q := Q [,N 1], and Rn N := Π N R n, i.e., Rn N = 1 n 1 δ (Xk,...,X n k+n 1 ). k= By the projective limit theorem ([3], [92]), it is sufficient to show ( P n (R N n ) ) n 1 satisfies the LDP on (M 1 (E N ), τ), N N. By [92], Chap. II, Theorem 3.2, we can get the LDP of (R N n ) n 1 on (M 1 (E N ), τ) by establishing: Claim 1) for every F : E N R d bounded and measurable, ( P n ( R N n, F ) ) n 1 satisfies the LDP with the rate function I F ρ (z), where I F ρ (z) = inf { h sp (Q ρ N ) : } F dπ N Q = z ; (1.2.3) Claim 2) ( P n (R N n ) ) n 1 is exponentially tight, i.e., L >, τ-compact K M 1 (E N ), such that for all open measurable G K, lim sup n 1 n log P n(r N n / G) L. We verify them in the next two steps: Step 1: Proof of the claim 1. Let Q F n := law of (F (X k,..., X k+n 1 )) k on (R d ) N under ρ N n

1.2. PRELIMINARIES 61 Q F := law of (F (X k,..., X k+n 1 )) k on (R d ) N under ρ N N Since ρn τ p ρ N, we have Q F n weakly Q F. By the well known Skorokhod theorem, there exist random sequences (Z n k ) k law = Q F n and (Z k ) k law = Q F, defined on some probability space ( Ω, F, P r), such that Z n k Z k, P r a.s.(n ), k. Since P r( 1 n 1 Z i ) satisfies the LDP on R d with the rate function Iρ F (z) (it is the n i= classical i.i.d. case), by the approximation theorem in Dembo-Zeitouni [3], for Claim 1), it is sufficient to prove: In fact, λ >, lim sup n k= ( 1 n log P r 1 n 1 n k= ( 1 n 1 ) P r (Zk n Z k ) > δ e λnδ Ee λ n ) (Zk n Z k ) > δ =, δ >. e λnδ Ee λ n 1 1 N 1 N e λnδ Zk n Z k k= N 1 [ N n ] ZiN+j n Z in+j j= i= j= [ n Ee Nλ N ] ZiN+j n Z in+j i= = 1 N 1 N e λnδ (Ee λn Zn j Zj ) [ n N ]+1 j= = e λnδ (Ee Nλ Zn Z ) [ n N ]+1 where the third inequality holds by Jensen s inequality and the fourth equality follows from the fact that (Z in+j ) i is i.i.d.. Hence lim sup n ( 1 1 n log P n 1 r n λδ + lim sup n k= [ n N ] + 1 n ) Zk n Z k > δ log Ee λn Zn Z.

62 CHAPITRE 1. LDP FOR EXCHANGEABLE SEQUENCES Since Z n a.s. Z and bounded, the last limsup equals to zero, and the desired result follows since λ is arbitrary. Step 2: Proof of claim 2. Consider the Cramér functional Λ(V ) := lim sup n 1 log EPn n n 1 V (X k,...,x k+n 1 ) ek=, V bf N 1. Applying Theorem 3.6 and Theorem 3.2 in [91] Chap. II, we now prove the following: for any sequence V m bf N 1 decreasing to zero pointwise on E N, then Λ(V m ) as m. Observe as before Consequently n 1 V m(x k,...,x k+n 1 ) E Pn ek= = E P n e Λ(V m ) lim sup n [ n N ] + 1 n 1 N N 1 [ N n ] V m(x in+j,,x in+j+n 1 ) j= i= N 1 j= E Pn e [ N n ] NV m(x in+j,,x in+j+n 1 ) i= = 1 N 1 [E Pn e NVm(X j,,x j+n 1 ) ] [ n N ]+1 N j= = [E Pn e NVm(X,,X N 1 ) ] [ n N ]+1. Hence by monotone convergence lim m Λ(V m ) =. log E Pn e NVm(X,...,X N 1 ) = 1 N log EP e NVm(X,,X N 1 ). 1.2.2 General lower bound of large deviation Proposition 1.2.9. For any τ-open subset G of M 1 (Ω) (resp. M 1 (E)), lim inf n lim inf n (without compactness of S τ.) 1 n log P(R n G) inf inf h sp (Q ρ N ) (resp. Q G ρ S τ 1 n log P(L n G) inf inf h(ν ρ) ). ν G ρ S τ

1.2. PRELIMINARIES 63 Proof. By the contraction principle, we only need to show the first inequality. In fact, a sufficient requirement for the first inequality is: For each Q M s 1(Ω) and its arbitrary neighborhood in M 1 (E N, τ p ) with the form N(Q ) := { Q : Q (F ) Q (F ) < ɛ } = { Q : Q (F ) B(Q (F ), ɛ) }, where F (bf N 1 ) d, we have lim inf n 1 n log log P(R n N(Q )) h sp (Q ρ N ), ρ S τ. Obviously, P(R n N(Q )) = P ( R n (F ) B(Q (F ), ɛ) ) ( ) n 1 = P F 1 Z i B(Q (F ), ɛ), n where (Z i ) i N is the system of coordinates in (R d ) N and P F is the law of { F (X k,, i= X k+n 1 ) } k N under P. Let QF ρ be the law of (F (X k,, X k+n 1 )) k N under ρ N and m F the image measure of m under the mapping: ρ Q F ρ. Putting S F := { } Q F ρ ρ S τ, we have P F = Q F ρ m(dρ) = M 1 (E) M 1 ((R d ) N ) Since Iρ F (Q (F )) h sp (Q ρ N ), it is enough to prove: ( ) 1 1 n 1 lim inf log PF Z i B(Q (F ), ɛ) n n n where I F ρ is given by (1.2.3). i= To that end, consider g n (Q) := 1 n log Q ( 1 n i= Qm F (dq). I F ρ (Q (F )) (1.2.4) n 1 ) Z i B(Q (F ), ɛ), Q S F. If Q n S F weakly Q = Q F ρ S F, then by the proof of Lemma 2.8, By Lemma 1.2.7 we have lim inf n g n(q n ) I F ρ (Q (F )). lim lim inf inf g n (Q ) I F δ n Q B(Q F ρ,δ) ρ (Q (F )),

64 CHAPITRE 1. LDP FOR EXCHANGEABLE SEQUENCES where B(Q F ρ, δ) is the ball centered at Q F ρ with radius δ in S F M 1 ((R d ) N ) equipped with a metric compatible with the weak convergence topology. Thus the desired result (1.2.4) follows since ρ Q F ρ is continuous from (S τ, τ) to S F equipped with the weak convergence topology, and then m F (B(Q F ρ, δ)) = m{ρ ; Q F ρ B(QF ρ, δ)} > for every ρ S τ. 1.3 Proof of the main result We shall prove the cycle (c) = (b) = (a) = (c). Proof of (c) = (b). For the LDP of P(R n ), the τ p -inf-compactness of the rate function J is proved in Lemma 1.2.6, the lower bound in Proposition 1.2.9. It remains to show the corresponding upper bound, i.e., for any closed F (M 1 (E N ), τ p ), Let lim sup n 1 n log P(R n F ) inf Q F inf h sp (Q ρ N ). ρ S τ g n (ρ) := 1 n log ρ N (R n F ), g(ρ) := inf Q F h sp(q ρ N ). By Lemma 1.2.8, lim sup n g n (ρ n ) g(ρ) for any sequence (ρ n ) ρ in (M 1 (E), τ). By the τ-compactness of S τ and its metrizability (Lemma 1.2.2) and Lemma 1.2.7 (applied to f n = g n ), we have lim sup n sup g n (ρ) sup g(ρ) = inf ρ S τ ρ S τ Q F inf h sp (Q ρ N ). ρ S τ It entails the desired estimate for 1 n log P(R n F ) sup ρ S τ g n (ρ). Proof of (b) = (a)+the last claim. This follows immediately from the contraction principle.

1.3. PROOF OF THE MAIN RESULT 65 Proof of (a) = (c). Now we assume that P(L n ) satisfies the LDP with some good rate function I on (M 1 (E), τ). By Varadhan s Laplace principle, for any f bb, 1 Λ(f) := lim n n log EP exp (nl n (f)) = sup ( ν, f I(ν)) ν M 1 (E) Let (f n ) n N be a sequence of functions in bb, decreasing pointwise to over E. By Lemma 1.2.5 and the τ-inf-compactness of I, we get by the equality above, Λ(f n ). On the other hand, by the i.i.d. property of (X k ) under ρ N and the τ-continuity of ρ e f dρ, By Lemma 1.2.3, S τ ( n 1 ) 1 Λ(f) = lim n n log m(dρ) exp f(x k ) dρ N S τ Ω k= ( ) n 1 = lim n n log m(dρ) e f dρ S τ E = log sup ρ S τ e f dρ sup ρ S τ fdρ. E E is relatively τ-compact iff sup ρ Sτ ρ(f n ) for every sequence (f n ) n N bb decreasing pointwise to over E, the preceding monotone continuity property of Λ implies then the relative τ-compactness of S τ in (M 1 (E), τ), which is the desired result.

66 CHAPITRE 1. LDP FOR EXCHANGEABLE SEQUENCES

Chapitre 2 Moderate deviation principle for Lipschitzian additive functionals of Markov chains Ce travail co-écrit avec Qiongxia Song est publié dans la revue journal Acta. Math. Sina. (Chinese series), 5(1), 33-42, 27. Abstract In this chapter, we consider the Lipschitzian additive functionals of Markov processes, and give a moderate deviation principle for them. We use the Kato s analytic perturbation theory and verify the C 2 -regularity condition for central limit theorem and moderate deviations. 2.1 Introduction The moderate deviation (MD) estimations, like large deviation (LD) estimations, arise from requirements of statistics. And it offers more accurate estimations than CLT (the central limit theorem). Let (X k ) k be a Markov process on (Ω, F, P). Set L n := 1 n n k=1 δ X k, where δ denotes the Dirac measure, consider the limit behavior of P ν (L n (f) m(f) A n ), (2.1.1) where m is the invariant measure of the Markov process (X n ) n, ν is the initial distribution of X, f is a Lipschitzian function, A n is a Borel subset of R. (2.1.1) represents the deviation property of L n (f) from m(f). When A n = A for all n, (2.1.1) is the LD, when A n = 1 n A for all n, (2.1.1) is the CLT. If A n = λ(n) n A, where, < λ(n), λ(n), λ(n) n, 67

68 CHAPITRE 2. MDP FOR LIPSCHITZIAN FUNCTIONALS (2.1.1) becomes the moderate deviation. In this case (2.1.1) can be written as it is an estimation between CLT and LD. ( ) n P ν λ(n) (L n(f) m(f)) A, In recent years a lot of works have been done in the MDP problem, and we have definite results in the i.i.d. case, see, cf. Chen,1991; Ledoux, 1992, for a systematical study on MDP of martingales, see, cf. Puhaliski, 1993 and H.Djellout, 21. Recently more and more works are realized in dependent cases. In Gao [37](1993), the MDP of the Doeblin recurrent Markov processes had been discussed, and lately he obtained a MDP result for martingales and mixing random processes in Gao [38](1996). Wu, in his paper [92](1995), gave the MDP for additive functionals of Markov processes with a spectral gap. He considered the functionals on the space of bb, C b (E) and L 2 (E). In his paper he introduced the C 2 -regularity and used the method of Kato s analytic perturbation theory to prove the C 2 -regularity and then moderate deviations principle. de Acosta, in his paper [27](1997), obtained the lower bound of MDP for empirical measures of Markov Chains. Wu, in his paper [96](21) got the complete MDP under exponential recurrence and H.Djellout, A.Guillin(22) relaxed exponential recurrence. 2.2 Motivation: to beat the irreducibility assumption The above results are all gotten from a fundamental hypothesis that the Markov process is irreducible (except several results in Wu [92]). But many Markov processes are not irreducible. We present a currently encountered example which is the so called Autoregressive Model: For each n Z +, Y n and W n are random variables on R satisfying inductively for n 1 Y n = a 1 Y n 1 + a 2 Y n 2 + + a k Y n k + W n, for some a 1, a 2,, a k R and initial values (Y,, Y n k+1 ). The sequence (W n ) n is an error sequence of i.i.d. random variable valued in R. In the usual treatment(see e.g. [63]) the law of W n is assumed to be absolutely continuous w.r.t. Lebesgue measure on R. In fact, such assumption is indispensable for providing the irreducibility of the Markov process. We want to relax such condition. In this chapter, we follow Wu [92] and use his method to the space of Lipschitz functions, and we get the corresponding MD result for such a class of functions on the Markov process (X n ) n without the hypothesis of irreducibility.

2.3. MAIN RESULT 69 2.3 Main result Let (E, d) be a connected Polish space which satisfies: d(x, y) = inf γ lim sup δ τ: τ δ n d(γ(t i ), γ(t i+1 )), i= where γ is a continuous curve connecting x and y (i.e., γ() = x, γ(1) = y) and τ is a partition. τ = { = t < t 1 < < t n = 1}, τ := max 1 i n γ(t i ) γ(t i 1 ). E is the Borel σ-field of (E, d). We denote by M 1 (E) the space of probability measures on (E, E) and by be the space of real bounded measurable functions on E. For a measurable function f and a measure µ, we write µ(f) = f, µ = fdµ. f : E R is a Lipschitzian E function if f(x) f(y) f Lip := lim sup d(x,y)> d(x, y) We define the gradient operator on E as: f(x) := lim sup y x < +. f(x) f(y). d(x, y) Let (Ω, F, (F k ) k, (X k ), (P x ) x E ) be a Markov chain with transition probability and P ν = E P xν(dx) for an initial measure ν M 1 (E). We shall work in the following assumption: (H 1 ) P Lip < + ; N > and r [, 1) such that P N f Lip r f Lip, f is Lipschitzian. As proved in the following Lemma 2.4.1, under (H 1 ), P possesses a unique invariant probability m such that L := m(d(x, x )) <. We consider the functional space (C Lip, ), the space of real Lipschitz functions on (E, d) and the norm is defined as = Lip + m( ). Moreover (C Lip, ) is a Banach space. We consider the moderate deviation behavior of L n (f) on the Markov process (X n ) n, where L n (f) is defined as: L n (f) = 1 n 1 f(x k ). (2.3.1) n k= The main result of this chapter is the following theorem: Theorem 2.3.1. Assume that the transition kernel P satisfy (H 1 ). Let (λ(n)) n be a positive sequence which satisfies: < λ(n), λ(n), λ(n) n.

7 CHAPITRE 2. MDP FOR LIPSCHITZIAN FUNCTIONALS If f : E R verifies: (C) f is Lipschitzian, f be and f(x)(1 + d(x, x)) c for some constant c, then (i) P ν ( n[l n (f) m(f)] ) converge weakly to N(, σ 2 (f)) uniformly over ν B L for any L L as n, where N(, σ 2 (f)) is normalized distribution and B L := {ν M 1 (E)/ν(d(x, )) L}; where σ 2 (f) is given by: (ii) σ 2 (f) = m(f m(f)) 2 + 2 m(p k f(f m(f))). (2.3.2) k=1 ( P ν ( ) n [L λ(n) n(f) m(f)] ) satisfies the LDP uniformly over ν B L for any L L with speed λ 2 (n) and rate function I f (r) = A R, ( ) n lim sup λ 2 (n) log sup P ν ( n ν B L λ(n) [L n(f) m(f)] A) r2. In other words, for any Borel subset 2σ 2 (f) inf I f (z); z A and lim inf n ( λ 2 (n) log sup P ν ( ν B L ) n λ(n) [L n(f) m(f)] A) inf z A o I f(z). 2.4 Proof of the main results 2.4.1 Some preparations Firstly we give an elementary lemma to guarantee the existence and uniqueness of the invariant measure w.r.t. the Markov chains satisfying (H 1 ): Lemma 2.4.1. Under (H 1 ), P possesses an unique invariant measure. Proof. Let M 1 1 (E) := {ν M 1 (E)/ν(d(x, )) < + }, and W 1 (ν, µ) be its Wasserstein metric, i.e., W 1 (µ, ν) = inf d(x, y)dπ(x, y),

2.4. PROOF OF THE MAIN RESULTS 71 where the infimum is taken over all probability measures π on the product space E E with marginal distribution ν and µ. Recall that (M 1 1 (E), W 1 ) is again a Polish space, then by Kantorovich-Rubinstein s Theorem, W 1 (ν, µ) := sup ν(f) µ(f). f Lip 1 Consider the mapping F : M 1 1 (E) M 1 1 (E), F (µ) := µp N, then by (H 1 ), W 1 (F (µ), F (ν)) = sup F (µ)(f) F (ν)(f) f Lip 1 = sup µ(p N f) ν(p N f) f Lip 1 sup µ(f) ν(f) f Lip r = r sup µ(f) ν(f) f Lip 1 = rw 1 (µ, ν). Therefore F is a strict contraction. So by the fixed point theorem there exists an unique m M 1 1 (E), such that, F (m) = m, i.e., mp k M 1 1 (E) by (H 1 ) and mp N = m. Since for each k, (mp k )P N = mp k+n = (mp N )P k = mp k, we get mp k = m by the uniqueness of invariant measure in M 1 1. In further, since W 1 ( P kn (x, ), P kn) r k W 1 (δ x, m) = r k d(x, y)m(dy), as k +, we get P kn m weakly for every x E. Thus m is the unique invariant measure of P N and then of P. Now we introduce the C 2 -regularity and a main result in Wu [92] here. Let (µ i ɛ, i A, ɛ > ) be a family of probability measures on R, where A is an index set. For ɛ > and i A, define Λ i ɛ(t) = ɛ log exp (tx/ɛ)dµ i ɛ := ɛ log Zɛ(t). i (2.4.1) R We assume always the following is satisfied: (H 2 ) there is Λ : R (, + ] so that for every t R fixed, Λ i ɛ(t) tends to Λ(t) uniformly on i A, as ɛ and Λ is finite on a neighborhood of zero in R.

72 CHAPITRE 2. MDP FOR LIPSCHITZIAN FUNCTIONALS Definition 2.4.2. (µ i ɛ, i A) ɛ is said to be right(resp. left) C 2 -regular uniformly for i A, if [Λ i ɛ] (t) Λ (t) uniformly for i A and t [, δ] (resp. t [ δ, ]), where Λ () is interpreted as the right second derivative Λ +() := lim t +(Λ (t) Λ +())/t[resp. Λ ()]. If it is simultaneously right and left C 2 -regular uniformly on i A, we say that it is C 2 -regular uniformly on i A. The following lemma say that the C 2 -regularity is a sufficient condition for MD and CLT, and its proof can be found in Wu [92]. Lemma 2.4.3. Let (µ i ɛ, i A) ɛ be a family of probability measures on R satisfying (H 2 ). If it is right (resp. left) C 2 -regular uniformly for i A, then for any (a(ɛ)) ɛ> verifying (i) a(ɛ) and a(ɛ) ɛ or (ii) a(ɛ) = ɛ, and for all t (resp. t ), lim sup ɛ i A ɛ a 2 (ɛ) log where m(i, ɛ) = R xdµi ɛ(x). In particular, let R exp[t(x m(i, ɛ))a(ɛ)/ɛ]dµ i ɛ(x) 1 2 Λ ()t 2 =, (2.4.2) ν i ɛ( ) := µ i ɛ ( x; ) x m(i, ɛ), (2.4.3) a(ɛ) then under the right(resp. left) C 2 -regularity assumption, ν i ɛ with a(ɛ) = ɛ tends uniformly for i A to the normal law N(, Λ +()) (resp. Λ ()) as ɛ weakly. Additionally, under the C 2 -regularity assumption, (ν i ɛ) with a(ɛ) verifying (2.4.2), satisfies the uniform LDP for i A with speed ɛ/a 2 (ɛ) and rate function I(x) = x 2 /(2Λ ()). To verify the C 2 -regularity, we shall apply the Kato s analytic perturbation theory as in Wu [92]. For f be, in discrete time case, define P f (x, dy) := e f(x) P (x, dy). (2.4.4) We have the following simplified Feynman-Kac formula in the present case: (P f ) n g(x) := E x g(x n ) exp { n 1 } f(x k ). (2.4.5) k=

2.4. PROOF OF THE MAIN RESULTS 73 Thus taking µ ν ɛ as the law of L n (f) under P ν for ɛ = 1/n, then the function Λ ν ɛ defined above equals to 1 n log ν, (P f ) n 1(x). Consequently, the C 2 regularity becomes a property of the perturbation operator P f of P. Next we recall Kato s analytic perturbation theorem: Lemma 2.4.4. Let B be a Banach space and (A(z)) z D be a holomorphic family of bounded operators on B, where D is an open domain in C, (i.e., z A(z)x is differentiable on D for every x B). Let z D, suppose that A(z ) has an isolated point λ in its spectrum and the corresponding eigen-projection J is one-dimensional. Let J 1 = I J be the eigen-projection associated with the remaining part of the spectrum of A(z ). Then there is an open neighborhood U of z in D such that z U, the spectrum Σ(z) of A(z) is separated into two parts: Σ(z) = Σ (z) Σ 1 (z), which satisfy: (i) Σ (z) = {λ(z)}, where λ(z) is an isolated eigenvalue of A(z) and λ(z) is holomorphic on U with λ(z ) = λ. (ii) If one denotes by J (z) associated with Σ (z) = λ(z), then the family of eigenprojections are also holomorphic and also one-dimensional. In order to apply this result we require two lemmas. Lemma 2.4.5. M := sup x E f(x) = f Lip. Proof. f Lip = sup d(x,y)> f(x) f(y) d(x,y) lim sup y x f(x) f(y) d(x,y), i.e., f Lip M. Now for the inverse inequality, fix some arbitrary x, x 1 E. For any ɛ >, there is a continuous curve γ : [, 1] E such that γ() = x, γ(1) = x 1 and lim sup n 1 d(γ(t i ), γ(t i+1 )) d(x, x 1 )(1 + ɛ). δ τ δ i= On E E, equipped with dist ((x, y), (x, y )) := d(x, x ) + d(y, y ), we consider A 1 := {(x, y) E E/ f(x) f(y) (M + ɛ)d(x, y)},

74 CHAPITRE 2. MDP FOR LIPSCHITZIAN FUNCTIONALS A 2 := {(γ(t), γ(t))/t [, 1]}. Then A 2 is compact and A c 1 A2 =, where A c 1 is the complement of A 1. For any (x, x) A 2, assuming there is {(x n, y n )} n 1 A c 1 such that (x n, y n ) (x, x), then by the definition of dist((x, y), (x, y )), we have x n x, y n x, moreover d(x n, y n ). Then M f(x n, y n ) lim sup dist((x n,y n),(x,x)) d(x n, y n ), which is contrary to the fact that (x n, y n ) A c 1. So we get dist((x, x), A c 1) >. Since A 2 is compact, 2δ := dist(a c 1, A 2 ) = inf (x,x) A2 dist((x, x), A c 1) > is attained. If d (γ(t), γ(s)) δ, we get f(γ(s)) f(γ(t)) (M+ɛ)d(γ(s), γ(t)) for dist((γ(s), γ(t)), A 2 ) dist ((γ(t), γ(s)), (γ(t), γ(t))) = d (γ(s), γ(t)) δ implies (γ(t), γ(s)) A 1. Hence for any partition τ = { = t < t 1 < < t n = 1} satisfying τ δ, n 1 f(x ) f(x 1 ) f(γ(t i )) f(γ(t i+1 )) i= n 1 (M + ɛ)d(γ(t i ), γ(t i+1 )) i= n 1 (M + ɛ) d(γ(t i ), γ(t i+1 )), i= then f(x ) f(x 1 ) (M + ɛ)(1 + ɛ)d(x, x 1 ). We get f Lip M since ɛ > is arbitrary. Lemma 2.4.6. Let f satisfy condition (C). Then g (C Lip, ), z P zf g is a bounded holomorphic mapping from C to (C Lip, ). Proof. Since P zf(x) g(x) = e zf(x) P g(x) is a holomorphic function for each x E, we need only show that P zf g is Lipschitz and z d dz (P zf g) = e zf fp g is continuous from C to (C Lip, ), i.e., to verify e zf P g Lip = sup x E (e zf P g)(x) < + and e znf fp g

2.4. PROOF OF THE MAIN RESULTS 75 e zf fp g Lip, if z n z. (e f P g)(x) = lim sup y x lim sup y x (e f P g)(x) (e f P g)(y) d(x, y) e f(x) (P g(x) P g(y)) d(x, y) = (e f P g )(x) + ( f e f P g )(x). + lim sup y x e f(x) e f(y) f(x) 1 P g(y) d(x, y) Since g (C Lip, ), P g is Lipschitz, then P g(x) K(1 + d(x, x )) for some constant K. By the assumption, we easily have f(x)p g(x) c. So e f fp g < + and e f P g < +. Then e zf P g (C Lip, ). 2.4.2 Proof of the results Proof. We only consider the case m(f) =, otherwise we consider f = f m(f) instead. Obviously P n g = P n g Lip + m(p n g) = P n g Lip r n g Lip = r n g. On the other hand P c = c, and 1 is an eigenvalue. In fact it is an isolated eigenvalue with algebraic multiplicity equal to one(i.e, the associated eigen-projection is of dimension one) by our assumption (H 1 ). By Lemma 2.4.6, we know z P zf g is a holomorphic function on some neighborhood of in C. We can use the theorem of Kato s analytic perturbation in previous section. Let G(z) be the complex number with the largest modulus in the spectrum of P zf regarded as an operator on (C Lip, ). By Kato s theorem above, since I J (z) is the eigen-projection on E 1 (z) along E (z), we have (P zf ) n [G(z)] n J (z) = (P zf ) n (I J (z)), On the other hand, because the largest modulus in the spectrum of P restricted to E 1 is strictly smaller than 1 by the hypothesis, by Kato s Theorem we can choose a ball U = B(, δ) := {z C/ z < δ} sufficiently small in C, such that (P zf ) n [G(z)] n J (z) c 1 e γn (2.4.6)

76 CHAPITRE 2. MDP FOR LIPSCHITZIAN FUNCTIONALS and G(z) > e γ/2, where γ >, (2.4.7) J (z)1(x) 1 < 1, z U. 4(L + 1) Given L L, we have for any ν B L, ν(f) = f(x )m(d(x )) + (f(x) f(x ))ν(dx)m(d(x )) m(f) + f Lip ν(d(x, )) f (1 + ν(d(x, ))) f (1 + L). Therefore we have by the above inequalities, ν, (P zf ) n 1 ν, G(z) n J (z)1 (P zf ) n 1 G(z) n J (z)1 (1 + ν(d(x, ))) (2.4.8) c 1 e γn (1 + L), and ν, J (z)1 1 (1 + ν(d(x, ))) J (z) 1 1 (1 + L) 4(1 + L) = 1 4. (2.4.9) Observe that 1 n log ν, (P zf ) n 1 1 n log Gn (z) I + II, (2.4.1) where I = 1 n log ν, (P zf ) n 1 1 n log ν, Gn (z)j (z)1, II = 1 n log ν, Gn (z)j (z)1 1 n log Gn (z).

2.4. PROOF OF THE MAIN RESULTS 77 Next we estimate I and II. In fact, I 1 n log ν, (P zf ) n 1 ν, G n (z)j (z)1 1 n log c 1e γn (1 + L) + ν, G n (z)j (z)1 ν, G n (z)j (z)1 1 ( n log 1 + c ) 1e γn (1 + L) 3 4 Gn (z) 1 ( n log 1 + 4(1 + L)c ) 1 e γ 2 n. 3 (2.4.11) The second inequality is implied by (2.4.8) and the fourth is by (2.4.7) and (2.4.9). II ν, Gn (z)j (z)1 G n (z) 1 ( n log 1 + ν, ) Gn (z)j (z)1 G n (z) G n (z) = 1 n log(1 + ν, J (z)1 1 ) (2.4.12) 1 n log 5 4. Combining (2.4.1), (2.4.11) with (2.4.12), we get sup 1 z U n log ν, (P zf ) n 1 log G(z), as n +. From the classic Montel and Hurwitz theorems in the complex analysis, we have sup sup dk ν B L z U dz ( 1 zf log ν, (P k ν ) n 1 log G(z)), as n +, k. n In particular, (µ i ɛ) i A is uniformly C 2 -regular, where (µ i ɛ) i A = P ν ( 1 n ɛ = 1 n, and i = ν B L. It remains to verify: (a) m(i, ɛ) = E ν 1 n n 1 i= f(x i) converge to m(f) uniformly on B L ; (b) d2 dz 2 1 n log m, (P zf ) n 1 σ 2 (f), as z, n. n 1 k= f(x k) ), By (H 1 ), c 2 := sup 1 k N 1 P k Lip < + and P N Lip r < 1. For all k, there exists l, such that ln k < (l + 1)N, then P k f Lip c 2 r l f Lip.

78 CHAPITRE 2. MDP FOR LIPSCHITZIAN FUNCTIONALS So E ν 1 n 1 1 n 1 f(x i ) m(f) = P k f(x)dν(x) f(x)dm(x) n n i= k= = 1 n 1 ( ) P k f(x)dν(x) P k f(x)dm(x) n k= n 1 1 P k f Lip W 1 (ν, m) n 1 n k= [ n 1 N ]+1 N 1 l= k= 1 n c 2N f Lip W 1 (ν, m) P ln+k f Lip W 1 (ν, m) [ n 1 N ]+1 l= 1 n c 1 2N f Lip W 1 (ν, m) 1 r. Obviously (a) is obtained. To verify (b), we first prove that σ 2 (f) converge. In fact this r l is true since f(x)p k f(x)m(dx) c 3 P k f(x)m(dx) = c 3 P k f(x) P k f(y)m(dy) m(dx) c 3 P k f(x) P k f(y) m(dy)m(dx) c 3 P k f Lip d(x, y)m(dy)m(dx), where c 3 := sup x E f(x). Note that at z =, ( d 2 1 dz 2 n log < m, (P zf ) n 1 >= d2 1 dz 2 n log Em e z n 1 i= f(xi) = 1 n 1 ) n V ar m f(x i ), i= and ( n 1 ) 1 n V ar m f(x i ) = V ar m (f) + 2 n i= i<j n 1 E m f(x i )f(x j ). Finally we show that 1 n i<j n 1 Em f(x i )f(x j ) k=1 m(fp k f), as n. In-

2.5. APPLICATIONS: TWO TYPICAL MODELS 79 deed, i<j n 1 n 2 = i= n 1 i k=1 E m f(x i )f(x j ) = n 1 m(fp k f) = n 1 = (n k)m(fp k f). k=1 k=1 n 2 n 1 i= j=i+1 n 1 k i= m(fp j i f) m(fp k f) Since k=1 m(fp k f) converge, by Kronecher Lemma, we get 1 n 1 n k=1 (n k)m(fp k f) k=1 m(fp k f), the desired result. 2.5 Applications: Two typical models 2.5.1 Linear States Space Model(LSSM) This model(see [63], Chap.1) is given as: X = (X k ) k is a stochastic process on R d which satisfies X k+1 = F X k + GW k+1, k. X is arbitrary and F is a d d matrix, G is a d p matrix. (W k ) k are i.i.d. random varibles on R p and independent of X. Corollary 2.5.1. Let r sp (F ) := max{ λ /λ C is an eigenvalue of F }. Assume r sp (F ) < 1, then the MDP holds for any Lipschitz function f satisfying the condition (C). Proof. For any r (r sp (F ), 1), there is a positive constant C 3, such that F n C 3 r n. Let P n be the transitional function of X, then for all x, y R d, P n f(x) P n f(y) = Ef(X n (x)) Ef(X n (y)) = Ef(F n x + F n 1 GW 1 + + F GW n 1 + GW n ) Ef(F n y + F n 1 GW 1 + + F GW n 1 + GW n ) f Lip F n (x) F n (y) f Lip F n x y C 3 r n f Lip x y.

8 CHAPITRE 2. MDP FOR LIPSCHITZIAN FUNCTIONALS By Theorem 2.3.1, we get the result easily. Remark 2.5.2. The invariant measure m is the law of X. 2.5.2 Scalar Nonlinear state Space Model(SNSSM) Now we study the following non-linear model in R d (d 1): X (x) = x R d, X n+1 (x) = F (X n (x), W n+1 ), n, where (W n ) n is a sequence of R k (k 1)-valued i.i.d. random variable; F : R d R k R d. Corollary 2.5.3. If F (x, w) F (y, w) r x y, o r < 1, denotes an arbitrary norm in R d. Then MDP holds for Lipschitz function f satisfying (C). Proof. Set F 1 (x, w 1 ) := F (x, w 1 ) and for k 2, F k (x; w 1,, w k ) = F (F k 1 (x; w 1,, w k 1 ), w k ) : R d (R m ) k R d Then F n (x; w 1,, w n ) F n (y, w 1,, w n ) = F (F n 1 (x; w 1,, w n 1 ), w n ) F (F n 1 (y; w 1,, w n 1 ), w n ) r F n 1 (x; w 1,, w n 1 ) F n 1 (y; w 1,, w n 1 ) r 2 F n 2 (x; w 1,, w n 2 ) F n 2 (y; w 1,, w n 2 ) r n 1 F (x, w 1 ) F (y, w 1 ) r n x y. By above inequality, we get, P n f(x) P n f(y) = Ef(X n (x)) Ef(X n (y)) = Ef(F n (x; w 1,, w n )) Ef(F n (y; w 1,, w n )) f(f n (x; w 1,, w n )) f(f n (y; w 1,, w n )) f Lip F n (x; w 1,, w n ) F n (y; w 1,, w n ) r n x y f Lip. So by Theorem 2.3.1, the result is obtained.

2.5. APPLICATIONS: TWO TYPICAL MODELS 81 Remark 2.5.4. Autoregressive Model is a particular case of the previous LSSM, where F = (a ij ) n n, with a 1j = a j, a j1 = 1 for j 2 and a ij = otherwise, and G = (1,,, ) t. Remark 2.5.5. Classical theory always assume the law of (W n ) n is absolutely continuous and also the integrability. Here we only make some hypothesis on F both in LSSM and in SNSSM.

82 CHAPITRE 2. MDP FOR LIPSCHITZIAN FUNCTIONALS

Partie II Inégalités de concentration convexe 83

Chapitre 3 Convex concentration inequality and forward/backward martingale stochastic calculus Cet article, écrit avec Thierry Klein et Nicolas Privault, est publié dans le journal Electronic Journal of Probability, Vol 11, 26, 486-512. Abstract Given (M t ) t R+ and (M t ) t R+ respectively a forward and a backward martingale with jumps and continuous parts, we prove that E[φ(M t + M t )] is non-increasing in t when φ is a convex function, provided the local characteristics of (M t ) t R+ and (M t ) t R+ satisfy some comparison inequalities. We deduce convex concentration inequalities and deviation bounds for random variables admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component. 3.1 Introduction Two random variables F and G satisfy a convex concentration inequality if E[φ(F )] E[φ(G)] (3.1.1) for all convex functions φ : R R. By a classical argument, the application of (7.4.1) to φ(x) = exp(λx), λ >, entails the deviation bound P (F x) inf λ> E[eλ(F x) 1 {F x} ] inf λ> E[eλ(F x) ] inf λ> E[eλ(G x) ], (3.1.2) x >, hence the deviation probabilities for F can be estimated via the Laplace transform of G, see [7], [1], [74] for more results on this topic. In particular, if G is Gaussian then Theorem 3.11 of [74] shows moreover that P (F x) e2 P (G x), x >. 2 85

86 CHAPITRE 3. CONVEX CONCENTRATION INEQUALITIES On the other hand, if F is more convex concentrated than G then E[F ] = E[G] as follows from taking successively φ(x) = x and φ(x) = x, and applying the convex concentration inequality to φ(x) = x log x we get Ent[F ] = E[F log F ] E[F ] log E[F ] = E[F log F ] E[G] log E[G] E[G log G] E[G] log E[G] = Ent[G], hence a logarithmic Sobolev inequality of the form Ent[G] E(G, G) implies Ent[F ] E(G, G). In this paper we obtain convex concentration inequalities for the sum M t + Mt, t R +, of a forward and a backward martingale with jumps and continuous parts. Namely we prove that M t + Mt is more concentrated than M s + Ms if t s, i.e. E[φ(M t + M t )] E[φ(M s + M s )], s t, for all convex functions φ : R R, provided the local characteristics of (M t ) t R+ and (Mt ) t R+ satisfy the comparison inequalities assumed in Theorem 3.3.2 below. If further E[Mt Ft M ] =, t R +, where (Ft M ) t R+ denotes the filtration generated by (M t ) t R+, then Jensen s inequality yields and if in addition we have M =, then E[φ(M t )] E[φ(M s + M s )], s t, E[φ(M T )] E[φ(M )], T. (3.1.3) In other terms, we will show that a random variable F is more concentrated than M : E[φ(F E[F ])] E[φ(M )], provided certain assumptions are made on the processes appearing in the predictable representation of F E[F ] = M T in terms of a point process and a Brownian motion. Consider for example a random variable F represented as F = E[F ] + H t dw t + J t (dz t λ t dt), where (Z t ) t R+ is a point process with compensator (λ t ) t R+, (W t ) t R+ is a standard Brownian motion, and (H t ) t R+, (J t ) t R+ are predictable square-integrable processes satisfying J t k, dp dt-a.e., and H t 2 dt β 2, and J t 2 λ t dt α 2, P a.s.

3.1. INTRODUCTION 87 By applying (3.1.3) or Theorem 3.4.1 ii) below to forward and backward martingales of the form M t = E[F ] + t H u dw u + t J u (dz u λ u du), t R +, and Mt = Ŵβ 2 ŴV 2 (t) + k( ˆN α 2 /k 2 ˆN U 2 (t)/k 2) (α2 U 2 (t))/k, t R +, where (Ŵt) t R+, ( ˆN t ) t R+, are a Brownian motion and a left-continuous standard Poisson process, β, α, k >, and (V (t)) t R+, (U(t)) t R+ are suitable random time changes, it will follow in particular that F is more concentrated than M = Ŵβ 2 + k ˆN α 2 /k 2 α2 /k, i.e. [ E[φ(F E[F ])] E φ(ŵβ 2 + k ˆN ] α 2 /k 2 α2 /k) (3.1.4) for all convex functions φ such that φ is convex. From (7.4.2) and (3.1.4) we get ( α 2 P (F E[F ] x) inf exp λ> k 2 (eλk λk 1) + β2 λ 2 2 ) λx, i.e. ( ) x P (F E[F ] x) exp k β2 λ (x) (2 kλ (x)) (x + α 2 /k)λ (x), 2k where λ (x) > is the unique solution of e kλ (x) + kλ (x)β 2 α 2 1 = kx α 2. When H t =, t R +, we can take β =, then λ (x) = k 1 log(1 + xk/α 2 ) and this implies the Poisson tail estimate ( ( ) ( y y P (F E[F ] y) exp k k + α2 log 1 + ky )), y >. (3.1.5) k 2 α 2 Such an inequality has been proved in [3], [95], using (modified) logarithmic Sobolev inequalities and the Herbst method when Z t = N t, t R +, is a Poisson process, under different hypotheses on the predictable representation of F via the Clark formula, cf. Section (3.6). When J t = λ t =, t R +, we recover classical Gaussian estimates which can be independently obtained from the expression of continuous martingales as time-changed Brownian motions. We proceed as follows. In Section 3.3 we present convex concentration inequalities for martingales. In Sections 3.4 and 3.5 these results are applied to derive convex concentration inequalities with respect to Gaussian and Poisson distributions. In Section 3.6 we

88 CHAPITRE 3. CONVEX CONCENTRATION INEQUALITIES consider the case of predictable representations obtained from the Clark formula. The proofs of the main results are formulated using forward/backward stochastic calculus and arguments of [53]. Section 3.7 deals with an application to normal martingales, and in the appendix (Section 3.8) we prove the forward-backward Itô type change of variable formula which is used in the proof of our convex concentration inequalities. See [11] for a reference where forward Itô calculus with respect to Brownian motion has been used for the proof of logarithmic Sobolev inequalities on path spaces. 3.2 Notation Let (Ω, F, P ) be a probability space equipped with an increasing filtration (F t ) t R+ and a decreasing filtration (Ft ) t R+. Consider (M t ) t R+ an F t -forward martingale and (Mt ) t R+ an Ft -backward martingale. We assume that (M t ) t R+ has right-continuous paths with left limits, and that (Mt ) t R+ has left-continuous paths with right limits. Denote respectively by (Mt c ) t R+ and (Mt c ) t R+ the continuous parts of (M t ) t R+ and (Mt ) t R+, and by M t = M t M t, Mt = Mt M t +, their forward and backward jumps. The processes (M t ) t R+ and (Mt ) t R+ have jump measures µ(dt, dx) = 1 { Ms }δ (s, Ms)(dt, dx), s> and µ (dt, dx) = s> 1 { M s }δ (s, M s )(dt, dx), where δ (s,x) denotes the Dirac measure at (s, x) R + R. Denote by ν(dt, dx) and ν (dt, dx) the (F t ) t R+ and (Ft ) t R+ -dual predictable projections of µ(dt, dx) and µ (dt, dx), i.e. t f(s, x)(µ(ds, dx) ν(ds, dx)) and t g(s, x)(µ (ds, dx) ν (ds, dx)) are respectively F t -forward and Ft -backward local martingales for all sufficiently integrable F t -predictable, resp. Ft -predictable, process f, resp. g. The quadratic variations ([M, M]) t R+, ([M, M ]) t R+ are defined as the limits in uniform convergence in probability n [M, M] t = lim M t n n i M t n i 1 2, and i=1 n 1 [M, M ] t = lim Mt M n n i t n 2, i+1 i=

3.3. CONVEX CONCENTRATION INEQUALITIES FOR MARTINGALES 89 for all refining sequences { = t n t n 1 t n k n = t}, n 1, of partitions of [, t] tending to the identity. We then let Mt d = M t Mt c, Mt d = Mt Mt c, [M d, M d ] t = M s 2, [M d, M d ] t = Ms 2, and <s t s<t M c, M c t = [M, M] t [M d, M d ] t, M c, M c t = [M, M ] t [M d, M d ] t, t R +. Note that ([M, M] t ) t R+, ( M, M t ) t R+, ([M, M ] t ) t R+ and ( M, M t ) t R+ are F t -adapted, but ([M, M ] t ) t R+ and ( M, M t ) t R+ are not Ft -adapted. The pairs (ν(dt, dx), M c, M c ) and (ν (dt, dx), M c, M c ) are called the local characteristics of (M t ) t R+, cf. [47] in the forward case. Denote by ( M d, M d t ) t R+, ( M d, M d t ) t R+ the conditional quadratic variations of (Mt d ) t R+, (Mt d ) t R+, with d M d, M d t = x 2 ν(dt, dx) and d M d, M d t = x 2 ν (dt, dx). R The conditional quadratic variations ( M, M t ) t R+, ( M, M t ) t R+ of (M t ) t R+ and (Mt ) t R+ satisfy M, M t = M c, M c t + M d, M d t, and M, M t = M c, M c t + M d, M d t, t R +. In the sequel, given η, resp. η, a forward, resp. backward, adapted and sufficiently integrable process, the notation t η udm u, resp. η t udm u, will respectively denote the right, resp. left, continuous version of the indefinite stochastic integral, i.e. we have R t η u dm u = t + η u dm u and t η udm u = t η udm u, t R +, dp a.e. 3.3 Convex concentration inequalities for martingales In the sequel we assume that (M t ) t R+ is an Ft -adapted, F t -forward martingale, (3.3.1) and (Mt ) t R+ is an F t -adapted, Ft -backward martingale, (3.3.2) whose characteristics have the form ν(du, dx) = ν u (dx)du and ν (du, dx) = νu(dx)du, (3.3.3)

9 CHAPITRE 3. CONVEX CONCENTRATION INEQUALITIES and d M c, M c t = H t 2 dt, and d M c, M c t = H t 2 dt, (3.3.4) where (H t ) t R+, (H t ) t R+, are respectively predictable with respect to (F t ) t R+ and (F t ) t R+. Hypotheses (3.3.1) and (3.3.2) may seem artificial but they are actually crucial to the proofs of our main results. Indeed, Theorem 3.3.2 and Theorem 3.3.3 are based on a forward/backward Itô type change of variable formula (Theorem 3.8.1 below) for (M t, M t ) t R+, in which (3.3.1) and (3.3.2) are needed in order to make sense of the integrals and t t s s + φ (M u + M u)dm u φ (M u + M u +)d M u. Note that in our main applications (see Sections 3.4, 3.5, 3.6 and 3.7), these hypotheses are fulfilled by construction of F t and F t. Recall the following Lemma. Lemma 3.3.1. Let m 1, m 2 be two measures on R such that m 1 ([x, )) m 2 ([x, )) <, x R. Then for all non-decreasing and m 1, m 2 -integrable function f on R we have f(x)m 1 (dx) f(x)m 2 (dx). If m 1, m 2 are probability measures then the above property corresponds to stochastic domination for random variables of respective laws m 1, m 2. Theorem 3.3.2. Let and assume that: i) The supports of ν, ν are in R +, ν u (dx) = xν u (dx), ν u(dx) = xν u(dx), u R +, ii) ν u ([x, )) ν u([x, )) <, x, u R, and iii) H u H u, dp du a.e. Then we have: E[φ(M t + Mt )] E[φ(M s + Ms )], s t, (3.3.5) for all convex functions φ : R R. Next is a different version of the same result, under L 2 hypotheses.

3.3. CONVEX CONCENTRATION INEQUALITIES FOR MARTINGALES 91 Theorem 3.3.3. Let ν u (dx) = x 2 ν u (dx) + H u 2 δ (dx), ν u(dx) = x 2 ν u(dx) + H u 2 δ (dx), u R +, and assume that: Then we have: ν u ([x, )) ν u([x, )) <, x R, u R +. (3.3.6) E[φ(M t + M t )] E[φ(M s + M s )], s t, (3.3.7) for all convex functions φ : R R such that φ is convex. Remark 3.3.4. Note that in both theorems, (M t ) t and (Mt ) t do not have to be independent. In the proof we may assume that φ is C 2 since a convex φ can be approximated by an increasing sequence of C 2 convex Lipschitzian functions, and the results can then be extended to the general case by an application of the monotone convergence theorem. In order to prove Theorem 3.3.2 and Theorem 3.3.3, we apply Itô s formula for forward/backward martingales (Theorem 3.8.1 in the Appendix Section 3.8), to f(x 1, x 2 ) = φ(x 1 + x 2 ): φ(m t + M t ) = φ(m s + M s ) t t + φ (M u + Mu)dM u + 1 φ (M u + M s 2 u)d M c, M c u + s + (φ(m u + Mu + M u ) φ(m u + Mu) M u φ (M u + Mu)) s<u t t t φ (M u + M u +)d Mu 1 φ (M u + M s 2 u)d M c, M c u s (φ(m u + M u + + Mu) φ(m u + M u +) Muφ (M u + M u +)), s u<t s t, where d and d denote the forward and backward Itô differential, respectively defined as the limits of the Riemann sums and n (M t n i M t n i 1 )φ (M t n i 1 + Mt ) n i 1 i=1 n 1 (Mt M n i t n )φ (M i+1 t n i+1 + Mt ) n i+1 i= for all refining sequences {s = t n t n 1 t n k n = t}, n 1, of partitions of [s, t] tending to the identity.

92 CHAPITRE 3. CONVEX CONCENTRATION INEQUALITIES Proof of Theorem3.3.2. Taking expectations on both sides of the above Itô formula we get E[φ(M t + Mt )] = E[φ(M s + Ms )] + 1 [ t ] 2 E φ (M u + Mu)d( M c, M c u M c, M c u ) s [ t ] +E (φ(m u + Mu + x) φ(m u + Mu) xφ (M u + Mu))ν u (dx)du s [ t ] E (φ(m u + Mu + x) φ(m u + Mu) xφ (M u + Mu))ν u(dx)du s = E[φ(M s + Ms )] + 1 [ t ] 2 E φ (M u + Mu)( H u 2 Hu 2 )du s [ t ] +E ϕ(x, M u + Mu)( ν u (dx) ν u(dx))du, where s ϕ(x, y) = φ(x + y) φ(y) xφ (y), x, y R. x The conclusion follows from the hypotheses and the fact that since φ is convex, the function x ϕ(x, y) is increasing in x R for all y R. Proof of Theorem 3.3.3. Using the following version of Taylor s formula 1 φ(y + x) = φ(y) + xφ (y) + x 2 (1 τ)φ (y + τx)dτ, x, y R, which is valid for all C 2 functions φ, we get E[φ(M t + Mt )] = E[φ(M s + Ms )] + 1 [ t ] 2 E φ (M u + Mu)( H u 2 Hu 2 )du s [ t 1 ] +E x 2 (1 τ)φ (M u + Mu + τx)dτν u (dx)du s [ t 1 ] E x 2 (1 τ)φ (M u + Mu + τx)dτνu(dx)du s = E[φ(M s + Ms )] [ 1 +E (1 τ) t s ] φ (M u + Mu + τx)( ν u (dx) ν u(dx))dudτ, and the conclusion follows from the hypothesis and the fact that φ is convex implies that φ is non-decreasing. Note that if φ is C 2 and φ is also convex, then it suffices to assume that ν u is more convex concentrated than ν u instead of hypothesis (3.3.6) in Theorem 3.3.3.

3.3. CONVEX CONCENTRATION INEQUALITIES FOR MARTINGALES 93 Remark 3.3.5. In case H t = H t and ν t = ν t, dp dt-a.e., from the proof of Theorem 3.3.2 and Theorem 3.3.3 we get the identity E[φ(M t + M t )] = E[φ(M s + M s )], s t, (3.3.8) for all sufficiently integrable functions φ : R R. In particular, Relation (3.3.8) extends its natural counterpart in the independent increment case: given (Z s ) s [,t], ( Z s ) s [,t] two independent copies of a Lévy process without drift, define the backward martingale (Z s ) s [,t] as Z s = Z t s, s [, t], then by convolution E[φ(Z s + Z s )] = E[φ(Z t )] does clearly not depend on s [, t]. Remark 3.3.6. If φ is non-decreasing, the proofs and statements of Theorem 3.3.2, Theorem 3.3.3, Corollary 3.3.9 and Corollary 3.3.8 extend to semi-martingales ( ˆM t ) t R+, ( ˆM t ) t R+ represented as ˆM t = M t + t α s ds and ˆM t = M t + t β s ds, (3.3.9) provided (α t ) t R+, (β t ) t R+, are respectively F t and F t -adapted with α t β t, dp dt-a.e. Let now (F M t ) t R+, resp. (F M t ) t R+, denote the forward, resp. backward, filtration generated by (M t ) t R+, resp. (M t ) t R+. Corollary 3.3.7. Under the hypothesis of Theorem 3.3.2, if further E[M t F M t ] =, t R +, then E[φ(M t )] E[φ(M s + M s )], s t. (3.3.1) Proof. From (3.3.5) we get E [φ (M s + Ms )] E [φ(m t + Mt )] = E [ E [ ]] φ(m t + Mt ) Ft M E [ φ ( M t + E[Mt Ft M ] )] s t, where we used Jensen s inequality. = E [φ(m t )], In particular, if M = E[M t ] is deterministic (or F M is the trivial σ-field), Corollary 3.3.7 shows that M t E[M t ] is more concentrated than M : E[φ(M t E[M t ])] E[φ(M )], t. The filtrations (F t ) t R+ and (Ft ) t R+ considered in Theorem 3.3.2 can be taken as F t = F M Ft M, Ft = F M F M t, t R +, provided (M t ) t R+ and (Mt ) t R+ are independent. In this case, if additionally we have MT =, then E[M t Ft M ] = E[Mt ] = E[MT ] =, t T, hence the hypothesis of Corollary 3.3.7 is also satisfied. However the independence of M, M t with M, M t, t R +, is not compatible (except in particular

94 CHAPITRE 3. CONVEX CONCENTRATION INEQUALITIES situations) with the assumptions imposed in Theorem 3.3.2. In applications to convex concentration inequalities between random variables (admitting a predictable representation) and Poisson or Gaussian random variables, the independence of (M t ) t R+ with (M t ) t R+ will not be required, see Sections 3.4 and 3.5. The case of bounded jumps Assume now that ν (dt, dx) has the form ν (dt, dx) = λ t δ k (dx)dt, (3.3.11) where k R + and (λ t ) t R+ is a positive F t -predictable process. Let λ 1,t = xν t (dx), λ 2 2,t = x 2 ν t (dx), t R +, denote respectively the compensator and quadratic variation of the jump part of (M t ) t R+, under the respective assumptions x ν t (dx) <, and x 2 ν t (dx) <, (3.3.12) t R +, P -a.s. Corollary 3.3.8. Assume that (M t ) t R+ and (M t ) t R+ have jump characteristics satisfying (3.3.11) and (3.3.12), that (M t ) t R+ is F t -adapted, and that (M t ) t R+ is F t -adapted. Then we have: E[φ(M t + M t )] E[φ(M s + M s )], s t, (3.3.13) for all convex functions φ : R R, provided any of the three following conditions is satisfied: i) M t k, dp dt a.e., and ii) M t k, dp dt a.e., and iii) M t, dp dt a.e., and with moreover φ convex in cases ii) and iii). H t H t, λ 1,t kλ t, dp dt a.e., H t H t, λ 2 2,t k 2 λ t, dp dt a.e., H t 2 + λ 2 2,t H t 2 + k 2 λ t, dp dt a.e.,

3.3. CONVEX CONCENTRATION INEQUALITIES FOR MARTINGALES 95 Proof. The conditions M t k, M t k, M t, are respectively equivalent to ν t ([, k] c ) =, ν t ((k, )) =, ν t ((, )) =, hence under condition (i), the result follows from Theorem 3.3.2, and under conditions (ii) (iii) it is an application of Theorem 3.3.3. For example we may take (M t ) t R+ and (M t ) t R+ of the form M t = M + where (W t ) t R+ t H s dw s + t is a standard Brownian motion, and ( Mt = Hs d Ws + k Zt t x(µ(ds, dx) ν s (dx)ds), t, (3.3.14) t ) λ sds, (3.3.15) where (W t ) t R+ is a backward Brownian motion and (Z t ) t R+ is a backward point process with intensity (λ t ) t R+. However in Section 3.5 we will consider an example for which the decomposition (3.3.15) does not hold. The case of point processes In particular, (M t ) t R+ and (M t ) t R+ can be taken as and M t = M + M t = t t H s dw s + H s d W s + t t J s (dz s λ s ds), t R +, (3.3.16) J s (d Z s λ sds), t R +, (3.3.17) where (W t ) t R+ is a standard Brownian motion, (Z t ) t R+ is a point process with intensity (λ t ) t R+, (Wt ) t R+ is a backward standard Brownian motion, and (Zt ) t R+ is a backward point process with intensity (λ t ) t R+, and (H t ) t R+, (J t ) t R+, resp. (Ht ) t R+, (Jt ) t R+ are predictable with respect to (F t ) t R+, resp. (Ft ) t R+. In this case, taking ν(dt, dx) = ν t (dx)dt = λ t δ Jt (dx)dt and ν (dt, dx) = ν t (dx)dt = λ t δ J t (dx)dt (3.3.18) in Theorem 3.3.3 yields the following corollary. Corollary 3.3.9. Let (M t ) t R+, (Mt ) t R+ have the jump characteristics (3.3.18) and assume that (M t ) t R+ is Ft -adapted and (Mt ) t R+ is F t -adapted. Then we have: E[φ(M t + M t )] E[φ(M s + M s )], s t, (3.3.19) for all convex functions φ : R R, provided any of the three following conditions are satisfied:

96 CHAPITRE 3. CONVEX CONCENTRATION INEQUALITIES i) J t J t, λ t dp dt a.e. and ii) J t J t, λ t dp dt a.e., and iii) J t J t, λ t dp dt a.e., and with moreover φ convex in cases ii) and iii). H t H t, λ t J t λ t J t, dp dt a.e., H t H t, λ t J t 2 λ t J t 2, dp dt a.e.. H t 2 + λ t J t 2 H t 2 + λ t J t 2, dp dt a.e.. Note that condition i) in Corollary 3.3.9 can be replaced with the stronger condition: i ) J t J t, λ t dp dt a.e. and H t H t, λ t λ t, dp dt a.e. 3.4 Application to point processes Let (W t ) t R+ and (Z t ) t R+ be a standard Brownian motion and a point process, generating a filtration (Ft M ) t R+. We will assume that (W t ) t R+ is also an Ft M -Brownian motion and that (Z t ) t R+ has compensator (λ t ) t R+ with respect to (Ft M ) t R+, which does not in general require the independence of (W t ) t R+ from (Z t ) t R+. Consider F a random variable with the representation F = E[F ] + H t dw t + J t (dz t λ t dt), (3.4.1) where (H u ) u R+ is a square-integrable F M t -predictable process and (J t ) t R+ is an F M t - predictable process which is either square-integrable or positive and integrable. Theorem 3.4.1 is a consequence of Corollary 3.3.9 above, and shows that the possible dependence of (W t ) t R+ from (Z t ) t R+ can be decoupled in terms of independent Gaussian and Poisson random variables. Note that inequality (3.4.2) below is weaker than (3.4.3) but it holds for a wider class of functions, i.e. for all convex functions instead of all convex functions having a convex derivative. Theorem 3.4.1. Let F have the representation: F = E[F ] + H t dw t + J t (dz t λ t dt), and let Ñ(c), W (β 2 ) be independent random variables with compensated Poisson law of intensity c > and centered Gaussian law with variance β 2, respectively.

3.4. APPLICATION TO POINT PROCESSES 97 i) Assume that J t k, dp dt-a.e., for some k >, and let β1 2 = H t 2 dt and α 1 = J t λ t dt. Then we have [ ( )] E[φ(F E[F ])] E φ W (β1) 2 + kñ(α 1/k), (3.4.2) for all convex functions φ : R R. ii) Assume that J t k, dp dt-a.e., for some k >, and let β2 2 = H t 2 dt and α2 2 = J t 2 λ t dt. Then we have [ ( )] E[φ(F E[F ])] E φ W (β2) 2 + kñ(α2 2/k 2 ), (3.4.3) for all convex functions φ : R R such that φ is convex. iii) Assume that J t, dp dt-a.e., and let β3 2 = H t 2 dt + J t 2 λ t dt. Then we have E[φ(F E[F ])] E [ φ(w (β3)) ] 2, (3.4.4) for all convex functions φ : R R such that φ is convex. Proof. Consider the F M t -martingale M t = E[F F M t ] E[F ] = t H s dw s + t J s (dz s λ s ds), t, and let ( ˆN s ) s R+, (Ŵs) s R+ respectively denote a left-continuous standard Poisson process and a standard Brownian motion which are assumed to be mutually independent, and also independent of (F M s ) s R+. i) ii) For p = 1, 2, let the filtrations (F t ) t R+ and (F t ) t R+ be defined by F t = F M σ(ŵβ 2 p ŴV 2 p (s), ˆN α p p /k p ˆN U p p (s)/k p : s t}, and F t = σ(ŵs, ˆNs : s ) F M t, t R +, and let M t = Ŵβ 2 p ŴV 2 p (t) + k( ˆN α p p /k p ˆN U p p (t)/k p) (αp p U p p (t))/k p 1, (3.4.5)

98 CHAPITRE 3. CONVEX CONCENTRATION INEQUALITIES where V 2 p (t) = t H s 2 ds and U p p (t) = t J p s λ s ds, P a.s., s. Then (Mt ) t R+ satisfies the hypothesis of Corollary 3.3.9 i) ii), as well as the condition E[Mt Ft M ] =, t R +, with Hs = H s, Js = k, λ s = Js p λ s /k p, dp ds-a.e., hence E[φ(M t )] E[φ(M )], and letting t go to infinity we obtain (3.4.2) and (3.4.3), respectively for p = 1 and p = 2. iii) Let M s = W β 2 3 W U 2 3 (s), (3.4.6) where U 2 3 (s) = s H u 2 du + s J u 2 λ u du, P a.s. Then (M t ) t R+ satisfies the hypothesis of Corollary 3.3.9 iii) with H s 2 = H s 2 + J s 2 λ s and λ s = J s =, dp ds-a.e., hence E[φ(M t )] E[φ(M )], and letting t go to infinity we obtain (3.4.4). Remark 3.4.2. The proof of Theorem 3.4.1 can also be obtained from Corollary 3.3.8. Proof. Let µ(dt, dx) = Z s δ (s,js)(dt, dx), ν t (dx) = λ t δ Jt (dx). i) ii) In both cases p = 1, 2, let (F t ) t R+, (Ft ) t R+ and (Mt ) t R+ be defined in (3.4.5), with Vp 2 (t) = t H s 2 ds and Up p (t) = t J s p ds, P -a.s., t. Then (Mt ) t R+ satisfies the hypothesis of Corollary 3.3.8 i) ii), with Hs = H s, νs = J s p /k p, dp ds-a.e. iii) Let (Ms ) s R+ be defined as in (3.4.6), and let U3 2 (s) = s H u 2 du + s J u 2 du, Hs 2 = H s 2 + J s 2 and νs =, dp ds-a.e. In the pure jump case, Theorem 3.4.1-ii) yields ( ( ) ( y y P (M T x) exp k k + α2 2 log 1 + ky )) k 2 α2 2 ( exp y ( 2k log 1 + ky )), α2 2 y >, with α 2 2 = M, M T, cf. Theorem 23.17 of [5], although some differences in the hypotheses make the results not directly comparable: here no lower bound is assumed on jump sizes, and the presence of a continuous component is treated in a different way. The results of this section and the next one apply directly to solutions of stochastic differential equations such as dx t = a(t, X t )dw t + b(t, X t )(dz t λ t dt), with H t = a(t, X t ), J t = b(t, X t ), t R +, for which the hypotheses can be formulated directly on the coefficients a(, ), b(, ) without explicit knowledge of the solution.

3.5. APPLICATION TO POISSON RANDOM MEASURES 99 3.5 Application to Poisson random measures Since a large family of point processes can be represented as stochastic integrals with respect to Poisson random measures (see e.g. [45], Section 4, Ch. XIV), it is natural to investigate the consequences of Theorem 3.3.2 in the setting of Poisson random measures. Let σ be a Radon measure on R d, diffuse on R d \ {}, such that σ({}) = 1, and ( x 2 1)σ(dx) <, R d \{} and consider a random measure ω(dt, dx) of the form ω(dt, dx) = i N δ (ti,x i )(dt, dx) identified to its (locally finite) support {(t i, x i )} i N. We assume that ω(dt, dx) is Poisson distributed with intensity dtσ(dx) on R + R d \ {}, and consider a standard Brownian motion (W t ) t R+, independent of ω(dt, dx), under a probability P on Ω. Let F t = σ(w s, ω([, s] A) : s t, A B b (R d \ {})), t R +, where B b (R d \ {}) = {A B(R d \ {}) : σ(a) < }. The stochastic integral of a square-integrable F t -predictable process u L 2 (Ω R + R d, dp dt dσ) is written as u(t, )dw t + u(t, x)(ω(dt, dx) σ(dx)dt), (3.5.1) and satisfies the Itô isometry [ ( E u(t, )dw t + = E [ R + R d \{} R d \{} ] [ u 2 (t, )dt + E ) 2 ] u(t, x)(ω(dt, dx) σ(dx)dt) u 2 (t, x)σ(dx)dt R + R d \{} [ ] = E u 2 (t, x)σ(dx)dt. (3.5.2) R + R d Recall that due to the Itô isometry, the predictable and adapted version of u can be used indifferently in the stochastic integral (3.5.1), cf. p. 199 of [29] for details. When u L 2 (R + R d, dt dσ), the characteristic function of I 1 (u) := u(t, )dw t + u(t, x)(ω(dt, dx) σ(dx)dt), is given by the Lévy-Khintchine formula E [ e ] ( ii 1(u) = exp 1 u 2 (t, )dt + 2 R + R d \{} R + R d \{} ) (e iu(t,x) 1 iu(t, x))σ(dx)dt. ]

1 CHAPITRE 3. CONVEX CONCENTRATION INEQUALITIES Theorem 3.5.1. Let F with the representation F = E[F ] + H s dw s + R d \{} J u,x (ω(du, dx) σ(dx)du), where (H t ) t R+ L 2 (Ω R + ), and (J t,x ) (t,x) R+ R d are F t-predictable with (J t,x ) (t,x) R+ R d L 1 (Ω R + R d \{}, dp dt dσ) and (J t,x ) (t,x) R+ R d L2 (Ω R + R d \{}, dp dt dσ) respectively in (i) and in (ii iii) below. i) Assume that J u,x k, dp σ(dx)du-a.e., for some k >, and let β1 2 = H u 2 du, and α 1 (x) = J u,x du, σ(dx) a.e. Then we have E[φ(F E[F ])] E for all convex functions φ : R R. [ ( ( φ W (β1) 2 + kñ R d \{} ))] α 1 (x) σ(dx), k ii) Assume that J u,x k, dp σ(dx)du-a.e., for some k >, and let β2 2 = H u 2 du, and α2(x) 2 = J u,x 2 du, σ(dx) a.e. Then we have E[φ(F E[F ])] E [ ( ( φ W (β2) 2 + kñ R d \{} for all convex functions φ : R R such that φ is convex. iii) Assume that J u,x, dp σ(dx)du-a.e., and let β3 2 = H u 2 du + Then we have R d \{} E[φ(F E[F ])] E [ φ(w (β 2 3)) ], for all convex functions φ : R R such that φ is convex. ))] α2(x) 2 σ(dx), k 2 J u,x 2 duσ(dx) Proof. The proof is similar to that of Theorem 3.4.1, replacing the use of Corollary 3.3.9 by that of Corollary 3.3.8. Let t t M t = M + H u dw u + J u,x (ω(du, dx) σ(dx)du), R d \{}.

3.5. APPLICATION TO POISSON RANDOM MEASURES 11 generating the filtration (Ft M ) t R+. Here, ν t (dx) denotes the image measure of σ(dx) by the mapping x J t,x, t, and µ(dt, dx) denotes the image measure of ω(dt, dx) by (s, y) (s, J s,y ), i.e. µ(dt, dx) = δ (s,js,y)(dt, dx). ω({(s,y)})=1 i) ii) For p = 1, 2, let the filtrations (F t ) t R+ and (F t ) t R+ be defined by and and let where V 2 p (t) = F t = F M σ(ŵβ 2 p ŴV 2 p (s) + ˆN α p p /k p ˆN U p p (s)/k p : s t}), F t = F M t σ(ŵs, ˆNs : s ), t R +, M t = Ŵβ 2 p ŴV 2 p (t) + k( ˆN α p p /k p ˆN U p p (t)/k p) (αp p U p p (t))/k p 1, t H s 2 ds and U p p (t) = t x p ν s (dx)ds, P a.s., t. Then (Mt ) t R+ satisfies the hypothesis of Theorem 3.3.2 i) ii), and also the condition E[Mt Ft M ] =, t R +, with Hs = H s, νs = xp ν s (dx), dp ds-a.e., hence E[φ(M t )] E[φ(M )]. Letting t go to infinity we obtain (3.4.2) and (3.4.3), respectively for p = 1, 2. iii) Let M s = W β 2 3 W U 2 3 (s), where U 2 3 (s) = Then (M t ) t R+ s H u 2 du + and ν s =, dp ds-a.e., hence s x 2 ν u (dx)du, P a.s., s. satisfies the hypotheses of Theorem 3.3.2 iii) with H s 2 = H s 2 + x 2 ν s (dx) E[φ(M t )] E[φ(M )], and letting t go to infinity we obtain (3.4.4). In Theorem 3.4.1, (Z t ) t R+ can be taken equal to the standard Poisson process (N t ) t R+, which also satisfies the hypotheses of Theorem 3.5.1 since it can be defined with d = 1 and σ(dx) = 1 [,1] (x)dx as N t = ω([, t] [, 1]), t. In other terms, being a point process, (N t ) t R+ is at the intersection of Corollary 3.3.8 and Corollary 3.3.9, as already noted in Remark 3.4.2.

12 CHAPITRE 3. CONVEX CONCENTRATION INEQUALITIES 3.6 Clark formula In this section we examine the consequence of results of Section 3.5 when the predictable representation of random variables is obtained via the Clark formula. We work on a product (Ω, P ) = (Ω W Ω X, P W P X ), where (Ω W, P W ) is the classical Wiener space on which is defined a standard Brownian motion (W t ) t R+ and { } Ω X = ω X (dt, dx) = i N δ (ti,x i )(dt, dx) : (t i, x i ) R + (R d \ {}), i N. The elements of Ω X are identified to their (by assumption locally finite) support {(t i, x i )} i N, and ω X ω X (dt, dx) is Poisson distributed under P X with intensity dtσ(dx) on R + R d \ {}. The multiple stochastic integral I n (h n ) of h n L 2 (R + R d, dtdσ) n can be defined by induction with I n (h n ) = n I n 1 (πt,h n n )dw t + n I n 1 (πt,xh n n )(ω X (dt, dx) σ(dx)dt), R + R d where (π n t,xh n )(t 1, x 1,..., t n 1, x n 1 ) := h n (t 1, x 1,..., t n 1, x n 1, t, x)1 [,t] (t 1 ) 1 [,t] (t n 1 ), t 1,..., t n 1, t R +, x 1,..., x n 1, x R d. The isometry property E [ I n (h n ) 2] = n! h n 2 L 2 (R + R d,dt σ) n follows by induction from (3.5.2). Let the linear, closable, finite difference operator D : L 2 (Ω, P ) L 2 (Ω R + R d, dp dt dσ) be defined as D t,x I n (f n ) = ni n 1 (f n (, t, x)), cf. e.g. [68], [78], with in particular D t, I n (f n ) = ni n 1 (f n (, t, )), σ(dx)dtdp a.e., dtdp a.e., Recall that the closure of D is also linear, and given F Dom(D), for σ(dx)dt-a.e. every (t, x) R + (R d \ {}) we have D t,x F (ω W, ω X ) = F (ω W, ω X {(t, x)}) F (ω W, ω X ), P (dω) a.s.,

3.6. CLARK FORMULA 13 cf. e.g. [68], [72], while D t, has the derivation property, and D t, f(i 1 (f (1) 1 ),..., I 1 (f (n) 1 )) = n k=1 f (i) 1 (t, ) k f(i 1 (f (1) 1 ),..., I 1 (f (n) 1 )), dtdp -a.e., f (1) 1,..., f (d) 1 L 2 (R + R d, dtdσ), f C b (Rn ), cf. e.g. [77]. The Clark formula for Lévy processes, cf. [7], [77], states that every F L 2 (Ω) has the representation F = E[F ] + E[D s, F F s ]dw s + R d \{} E[D s,x F F s ](ω X (ds, dx) σ(dx)ds). (3.6.1) (The formula originally holds for F in the domain of D but its extension to L 2 (Ω) is straightforward, cf. [77], Proposition 12). Theorem 3.5.1 immediately yields the following corollary when applied to any F L 2 (Ω) represented as in (6.2.3). Corollary 3.6.1. Let F L 2 (Ω) have the representation (6.2.3), and assume additionally that E[D R d \{} s,xf F s ] σ(dx)ds < a.s. in (i) below. i) Assume that E[D u,x F F u ] k, dp σ(dx)du-a.e., for some k >, and let β1 2 = σ(dx)-a.e. Then we have (E[D u, F F u ]) 2 du, and α 1 (x) = [ ( ( E[φ(F E[F ])] E φ W (β1) 2 + kñ R d \{} for all convex functions φ : R R. E[D u,x F F u ]du, ))] α 1 (x) σ(dx), (3.6.2) k ii) Assume that E[D u,x F F u ] k, dp σ(dx)du-a.e., for some k >, and let β2 2 = σ(dx)-a.e. Then we have (E[D u, F F u ]) 2 du, and α2(x) 2 = [ ( ( E[φ(F E[F ])] E φ W (β2) 2 + kñ R d \{} for all convex functions φ : R R such that φ is convex. (E[D u,x F F u ]) 2 du, ))] α2(x) 2 σ(dx), (3.6.3) k 2

14 CHAPITRE 3. CONVEX CONCENTRATION INEQUALITIES iii) Assume that E[D u,x F F u ], dp σ(dx)du-a.e., and let β3 2 = (E[D u, F F u ]) 2 du + (E[D u,x F F u ]) 2 duσ(dx) Then we have R d \{} E[φ(F E[F ])] E [ φ(w (β 2 3)) ], (3.6.4) for all convex functions φ : R R such that φ is convex. As mentioned in the introduction, from (3.6.4) we deduce the deviation inequality ) P (F E[F ] y) e2 2 P (W (β2 3) > y) ( e2 2 exp y2, y >, 2β 2 3. provided E[D u,x F F u ], dp σ(dx)du-a.e., and (E[D u, F F u ]) 2 du + (E[D u,x F F u ]) 2 duσ(dx) β3, 2 R d \{} P a.s. Similarly from (3.6.3) we get provided and P (F E[F ] y) exp R + R d \{} ( ( ) ( y y k k + α2 2 log 1 + ky )), y >, (3.6.5) k 2 α2 2 E[D t,x F F t ] k, dp σ(dx)dt a.e., (3.6.6) (E[D t,x F F t ]) 2 σ(dx)dt α 2 2, P a.s., for some k > and α 2 2 >. In [3] this latter estimate has been proved using (modified) logarithmic Sobolev inequalities and the Herbst method under the stronger condition and R + R d \{} D t,x F k, dp σ(dx)dt-a.e., (3.6.7) D t,x F 2 σ(dx)dt α 2 2, P a.s., (3.6.8) for some k > and α 2 2 >. In [95] it has been shown, using sharp logarithmic Sobolev inequalities, that the condition D t,x F k can be relaxed to which is nevertheless stronger than (3.6.6). D t,x F k, dp σ(dx)dt-a.e., (3.6.9) In the next result, which however imposes uniform almost sure bounds on DF, we consider Poisson random measures on R d \ {} instead of R + R d \ {}.

3.7. NORMAL MARTINGALES 15 Corollary 3.6.2. i) Assume that D x F β(x) k, dp σ(dx) a.e., where β( ) : R d \ {} [, k] is deterministic and k >. Then for all convex functions φ we have [ ( ( ))] β(x) E[φ(F E[F ])] E φ kñ k σ(dx). R d \{} ii) Assume that D x F β(x) k, dp σ(dx)-a.e., where β( ) : R [, k] and k > are deterministic. Then for all convex functions φ with a convex derivative φ we have [ ( ( ))] β 2 (x) E[φ(F E[F ])] E φ kñ σ(dx). k 2 R d \{} iii) Assume that β(x) D x F, dp σ(dx)-a.e., where β( ) : R [, ) is deterministic. Then for all convex functions φ with a convex derivative φ we have [ ( ( ))] E[φ(F E[F ])] E φ W β 2 (x)σ(dx). R d \{} Proof. Assume that ω X (dt, dx) has intensity 1 [,1] (s)σ(dx)ds on R + R d \ {}, we define the random measure ˆω on R d \ {} with intensity σ(dx) as ˆω X (A) = ω X ([, 1] A), A B b (R d \ {}). Then it remains to apply Corollary 3.6.1 to ˆF (ω W, ω X ) := F (ω W, ˆω X ). In Corollary 3.6.2, R d \ {} can be replaced by R d without additional difficulty. 3.7 Normal martingales In this section we interpret the above results in the framework of normal martingales. Let (Z t ) t R + be a normal martingale, i.e. (Z t ) t R + is a martingale such that d Z, Z t = dt. If (Z t ) t R + is in L 4 and has the chaotic representation property it satisfies the structure equation d[z, Z] t = dt + γ t dz t, t R +, where (γ t ) t R+ is a predictable square-integrable process, cf. [35]. Recall that the cases γ s =, γ s = c R \ {}, γ s = βz s, β ( 2, ), correspond respectively to Brownian motion, the compensated Poisson process with jump size c and intensity 1/c 2, and to the Azéma martingales. Consider the martingale M t = M + t where (R u ) u R+ L 2 (Ω R + ) is predictable. We have d M c, M c t = 1 {γt=} R t 2 dt R u dz u, (3.7.1)

16 CHAPITRE 3. CONVEX CONCENTRATION INEQUALITIES and µ(dt, dx) = and the Itô formula, cf. [35]: φ(m t ) = φ(m s ) + + 1 2 t s t s Z s δ (s,rsγ s)(dt, dx), ν(dt, dx) = 1 γ 2 t 1 {γu=}r u φ (M u )dz u + 1 {γu=} R u 2 φ (M u )du + t 1 {γu } s t 1 {γu } s Z s δ Rsγ s (dx)dt, φ(m u + γ u R u ) φ(m u ) dz u γ u φ(m u + γ u R u ) φ(m u ) γ u R u φ (M u ) γ u 2 du, φ C 2 (R). The multiple stochastic integrals with respect to (M t ) t R+ I n (f n ) = n! tn t2 f n (t 1,..., t n )dm t1 dm tn, are defined as for f n a symmetric function in L 2 (R n +). As an application of Corollary 3.3.9 we have the following result. Theorem 3.7.1. Let (M t ) t R+ have the representation (3.7.1), let (Mt ) t R+ be represented as M t = t H s d W s + t J s (d Z s λ sds), assume that (M t ) t R+ is an F t -adapted F t -martingale and that (M t ) t R+ is an F t -adapted F t -martingale. Then we have E[φ(M t + M t )] E[φ(M s + M s )], s t, for all convex functions φ : R R, provided any of the following three conditions is satisfied: i) γ t R t J t, 1 {γt=} R t 2 H t 2, and ii) γ t R t J t, 1 {γt=} R t 2 H t 2, and and φ is convex, iii) γ t R t, R t 2 H t 2, J t As above, if further E[M t F M t R t 1 {γt } λ t Jt, dp dt a.e., γ t 1 {γt } R t 2 λ t J t 2, dp dt a.e., =, dp dt - a.e., and φ is convex. ] =, t R +, we obtain E[φ(M t )] E[φ(M s + M s )], s t. As a consequence we have the following result which admits the same proof as Theorem 3.4.1.

3.7. NORMAL MARTINGALES 17 Theorem 3.7.2. Let F L 2 (Ω, F, P ) have the predictable representation F = E[F ] + R t dz t. i) Assume that γ t R t k, dp dt-a.e., for some k >, and let β1 2 = 1 {γs=} R s 2 ds and α 1 = R s 1 {γs } ds γ s. Then we have [ ( )] E[φ(F E[F ])] E φ W (β1) 2 + kñ(α 1/k), for all convex functions φ : R R. ii) Assume that γ u R u k, dp dt-a.e., for some k > and β2 2 = 1 {γs=} R s 2 ds and α2 2 = 1 {γs } R s 2 ds. Then for all convex functions φ with a convex derivative φ, we have [ ( )] E[φ(F E[F ])] E φ W (β2) 2 + kñ(α2 2/k 2 ). iii) Assume that γ u R u and let β3 2 = R s 2 ds. Then for all convex functions φ with a convex derivative φ, we have E[φ(F E[F ])] E[φ(Ŵ (β2 3))]. Let now D : L 2 (Ω, F, P ) L 2 (Ω [, T ], dp dt) denote the annihilation operator on multiple stochastic integrals defined as D t = I n (f n ) = ni n (f n (, t)), t R +. The Clark formula for normal martingales [71] provides a predictable representation for F Dom(D) L 2 (Ω, F, P ), which can be used in Theorem 3.7.2: F = E[F ] + where F t = σ(z s, s t). E[D t F F t ]dz t,

18 CHAPITRE 3. CONVEX CONCENTRATION INEQUALITIES 3.8 Appendix In this section we prove the Itô type change of variable formula for forward/backward martingales which has been used in the proofs of Theorem 3.3.2 and Theorem 3.3.3. Assume that (Ω, F, P ) is equipped with an increasing filtration (F t ) t R+ and a decreasing filtration (F t ) t R+. Theorem 3.8.1. Consider (M t ) t R+ an Ft -adapted, F t -forward martingale with rightcontinuous paths and left limits, and (Mt ) t R+ an F t -adapted, Ft -backward martingale with left-continuous paths and right limits. For all f C 2 (R 2, R) we have f(m t, Mt ) f(m, M ) t f = (M u, M x u)dm u + 1 + 1 2 + <u t t t 2 f (M x 2 u, Mu)d M c, M c u 1 ( f(m u, M u) f(m u, M u) M u f x 1 (M u, M u) f (M u, M u x +)d Mu 1 2 f (M 2 2 x 2 u, Mu)d M c, M c u 2 ( ) f(m u, Mu) f(m u, M u +) M u f (M u, M u x +), 2 u<t where d denotes the backward Itô differential and (Mt c ) t R+, (Mt c ) t R+ respectively denote the continuous parts of (M t ) t R+, (Mt ) t R+. Proof. We adapt the arguments of Theorem 32 of Chapter II in [8], using here the following version of Taylor s formula: t f(y 1, y 2 ) f(x 1, x 2 ) = f(y 1, y 2 ) f(y 1, x 2 ) + f(y 1, x 2 ) f(x 1, x 2 ) (3.8.1) = (y 1 x 1 ) f (x 1, x 2 ) + 1 x 1 2 (y 1 x 1 ) 2 2 f (x 1, x 2 ) x 2 1 +(y 2 x 2 ) f (y 1, y 2 ) 1 x 2 2 (y 2 x 2 ) 2 2 f (y x 2 1, y 2 ) 2 +R(x, y), where R(x, y) o( y x 2 ). Assume first that (M s ) s [,t] and (M s ) s [,t] take their values in a bounded interval, and let { = t n t n 1 t n k n = t}, n 1, be a refining sequence of partitions of [, t] tending to the identity. As in [8], for any ε >, consider A ε,t, B ε,t two random subsets of [, t] such that i) A ε,t is finite, P -a.s., ii) A ε,t B ε,t exhausts the jumps of (M s ) s [,t] and (M s ) s [,t], iii) s B ε,t M s 2 + M s 2 ε 2, )

3.8. APPENDIX 19 iv) for each 1 i n, exactly one of the two sets A ε,t (t n i 1, t n i ] or B ε,t (t n i 1, t n i ] is non-empty, P -a.s. We have f(m t, M t ) f(m, M ) = f(m t n i, Mt ) f(m n t n, M i i 1 t ) n i 1 A ε,t (t n i 1,tn i ] + f(m t n i, Mt ) f(m n t n, M i i 1 t ), n i 1 B ε,t (t n i 1,tn i ] and from Taylor s formula (3.8.1) we get f(m t, Mt ) f(m, M ) = f(m t n i, Mt ) f(m n t n, M i i t ) + f(m n t n, M i 1 i t ) f(m n t n, M i 1 i 1 t ) n i 1 A ε,t (t n i 1,tn i ] + f(m t n i, Mt ) f(m n t n, M i i t ) + f(m n t n, M i 1 i t ) f(m n t n, M i 1 i 1 t ) n i 1 B ε,t (t n i 1,tn i ] = f(m t n i, Mt ) f(m n t n, M i i t ) + f(m n t n, M i 1 i t ) f(m n t n, M i 1 i 1 t ) n i 1 A ε,t (t n i 1,tn i ] = + i=1 M t n i 1 ) f x 1 (M t n i 1, M t n i 1 ) + 1 2 M t n i M t n i 1 2 2 f (M x 2 t n i 1, Mt ) n i 1 1 (M t n i B ε,t (t n i 1,tn i ] + (Mt M n i t ) f (M n i 1 t n x i, Mt ) 1 n i B ε,t (t n 2 2 M t n i i 1,tn i ] + R(M t n i, Mt, M n t n, M i i 1 t ) n i 1 B ε,t (t n i 1,tn i ] f(m t n i, Mt ) f(m n t n, M i i t ) + f(m n t n, M i 1 i t ) f(m n t n, M i 1 i 1 t ) n i 1 A ε,t (t n i 1,tn i ] n + (M t n i M t n i 1 ) f (M t n x i 1, Mt ) + 1 n i 1 1 2 M t n M i t n i 1 2 2 f (M x 2 t n i 1, Mt ) n i 1 1 (M t n i A ε,t (t n i 1,tn i ] n + (Mt n i i=1 M t n i 1 ) f x 1 (M t n i 1, M t n i 1 ) + 1 2 M t n i M t n i 1 ) f x 2 (M t n i, M t n i ) 1 2 M t n i (Mt M n i t ) f (M n i 1 t n x i, Mt ) 1 n i A ε,t (t n 2 2 M t n i i 1,tn i ] + R(M t n i, Mt, M n t n, M i i 1 t ). n i 1 B ε,t (t n i 1,tn i ] Mt 2 f n 2 (M i 1 x 2 t n i, Mt ) n i 2 M t n i 1 2 2 f (M x 2 t n i 1, Mt ) n i 1 1 Mt 2 f n 2 (M i 1 x 2 t n i, Mt ) n i 2 Mt 2 f n 2 (M i 1 x 2 t n i, Mt ) n i 2

11 CHAPITRE 3. CONVEX CONCENTRATION INEQUALITIES By the same arguments as in [8] and from conditions (3.3.1) and (3.3.2), letting n tend to infinity we get f(m t, Mt ) f(m, M ) = ( f(m u, Mu) f(m u, M f u) M u (M u, M x u) 1 ) 1 2 M u 2 2 f (M x 2 u, Mu) u A ε,t 1 ( f(m u, M u +) f(m u, Mu) Mu f (M u, M u x +) + 1 ) 2 2 Mu 2 2 f (M u, M u +) + u A ε,t t t f x 1 (M u, M u)dm u + 1 2 f x 2 (M u, M u +)d M u 1 2 t 2 f (M u, Mu)d[M, M] u x 2 1 t 2 f (M x 2 u, M u +)d[m, M ] u. 2 Then letting ε tend to, the above sum converges to f(m t, Mt ) f(m, M ) t f = (M u, M x u)dm u + 1 + 1 2 + <u t t t + 2 f x 2 1 ( f(m u, M u) f(m u, M u) M u f t (M u, M u)d[m, M] u x 1 (M u, M u) 1 2 M u 2 2 f f (M u, M u x +)d Mu 1 2 f (M 2 2 x 2 u, M u +)d[m, M ] u 2 ( f(m u, Mu) f(m u, M u +) Mu f (M u, M u x +) + 1 2 2 Mu 2 2 f u<t which yields f(m t, M t ) f(m, M ) = + <u t t t + f x 1 (M u, M u)dm u + 1 2 t 2 f x 2 1 ( f(m u, M u) f(m u, M u) M u f x 1 (M u, M u) x 2 1 f (M u, M u x +)d Mu 1 2 f (M 2 2 x 2 u, Mu)d M c, M c u 2 ( ) f(m u, Mu) f(m u, M u +) Mu f (M u, M u x +), 2 u<t t x 2 2 x 2 2 ) (M u, Mu) (M u, M u +) ), (M u, Mu)d M c, M c u ) where the integral with respect to ( M, M t ) t R+ is defined as a Stieltjes integral with respect to a (not necessarily Ft -adapted) increasing process. In the general case, define the stopping times R m = inf{u [, t] : M u m}, and R m = sup{u [, t] : M u m}.

3.8. APPENDIX 111 The stopped process (M u Rm, M u R m ) u [,t] is bounded by 2m and since Itô s formula is valid for (X Rm u ) u [,t] for each m, it is also valid for (X u ) u R+. 2 f Note that the cross partial derivative (M u, M x 1 x u) does not appear in the formula 2 and there is no need to consider or define a bracket of the form d M, M t.

112 CHAPITRE 3. CONVEX CONCENTRATION INEQUALITIES

Chapitre 4 A convex domination principle for dependent Brownian and stable stochastic integrals Cette note co-écrit avec Aldéric Joulin est en cours de finalisation et sera prochainement soumis pour publication. Abstract Based on the forward-backward stochastic calculus developed in [54], we show in this note that under some boundedness assumptions, the sum of dependent Brownian and stable stochastic integrals is convex dominated by the independent sum of a Gaussian and a symmetric stable random variables. 4.1 Introduction A random variable X is said to be convex dominated by another random variable Y if we have E [φ(x)] E [φ(y )], (4.1.1) for any integrable convex function φ. Such a domination principle between random variables may be seen as a generalization of the classical moment inequalities developed in the theory of stochastic processes and entail many interesting results, among them maximal inequalities for semimartingales, Orlicz embeddings and deviation estimates, see for instance [28, 54]. Recently, such a problem has been considered by Klein in his PhD thesis [53]. More precisely, using some stochastic calculus techniques, he showed that under several boundedness assumptions, a stochastic integral driven by a point process is convex dominated by a centered Poisson random variable whose intensity depends on the characteristics of the previous stochastic integral. This result has been extended in [54] to the general framework of martingales with jumps. In particular, they considered the case of Poisson random measures and derived a convex domination principle contained 113

114 CHAPITRE 4. A CONVEX DOMINATION PRINCIPLE in their Theorem 5.1, which is stated as follows: a centered random variable admitting a representation in terms of dependent Brownian and compensated Poisson stochastic integrals, is convex dominated by the independent sum of centered Gaussian and Poisson random variables, provided some boundedness and moment assumptions are made on the integrated processes. However, the analysis does not concern for instance the case of martingales with unbounded jumps and infinite variance. Hence, the purpose of this note is to extend such a result by replacing the driving compensated Poisson process by a symmetric stable process whose sample paths are of infinite variation. Our approach relies on the Itô s formula for forward-backward martingales introduced in [54] and allows us to decouple the pair of Brownian and stable stochastic integrals. 4.2 Main result Consider on a probability space (Ω, F, P) a real standard Brownian motion (W t ) t which is non-necessarily independent of a symmetric stable process (Z t ) t of index α (1, 2) and with stable Lévy measure defined on R \ {} by σ(dx) := cdx, c >. (4.2.1) x α+1 Denoting the filtration F W,Z t := σ (W s, Z s : s t), t, we assume in the remainder of the paper that the Lévy processes (W t ) t and (Z t ) t are (F W,Z t ) t -martingales. In other words, the latter condition states that the increments of the first process are independent of the past of the second one, and reciprocally. Let F be a random variable having the representation F E[F ] = H t dw t + K t dz t, (4.2.2) where the bounded processes (H t ) t L (Ω, L 2 (, + )) and (K t ) t L (Ω, L α (, + )) are supposed to be (F W,Z t ) t -predictable. We assume moreover that (K t ) t L 2 (Ω (, + )) so that the variable K t dz t is well-defined as a stochastic integral with respect to the Lévy-Itô decomposition of the stable process (Z t ) t. Let S p,α be the set of convex functions on R with at most polynomial growth of order p (, α) at infinity, i.e. lim x + x p φ(x) < +. The main result of this note, whose proof is given in the next section, is the following Theorem 4.2.1. There exists a Gaussian random variable W β1 with variance β 1 := H t 2 dt, independent of a symmetric stable random variable Z β2 of index α and with Lévy measure given on R \ {} by σ(dx) := β 2 cdx x, α+1 with < c c and β + 2 := K t α dt,

4.3. PROOF OF THEOREM 4.2.1 115 such that for any φ S p,α, we have the convex domination inequality [ ( E [φ(f EF )] E φ Wβ1 + Z )] β2. (4.2.3) Note that the integrating processes (W t ) t and (Z t ) t in the predictable representation of the random variable F are correlated, whereas the Gaussian and the symmetric stable random variables in the right-hand-side of (4.2.3) are independent. Hence the convex domination inequality (4.2.3) allows us to decouple the dependent pair (W t, Z t ) t. 4.3 Proof of Theorem 4.2.1 Before proceeding to the proof of Theorem 4.2.1, let us introduce some notation and preliminary results on the forward-backward stochastic calculus recently developed in [54]. 4.3.1 Forward-backward stochastic calculus We endow the probability space (Ω, F, P) with an increasing filtration (F t ) t and a decreasing filtration (Ft ) t, and we consider a (F t ) t -martingale (X t ) t and a backward (Ft ) t -martingale (Xt ) t. For the sake of briefness, the (forward) processes considered in the remainder of this section are supposed to be right-continuous with left limits, whereas the backward martingales are naturally supposed to be left-continuous with right limits. We denote in the sequel by (Xt c ) t, (Xt c ) t, the continuous parts of the processes (X t ) t, (Xt ) t, respectively. Let X s := X s X s, and X s := X s X s+, be the respective forward and backward jump sizes at time s > and we denote by ν X, ν X, the (F t ) t -, (F t ) t -, dual predictable projections of the jumping measures µ X, µ X, of the martingales (X t ) t, (X t ) t, respectively. Define as the limits in probability the bracket processes [X, X] t := lim n + n i=1 X t n i X t n i 1 2 and [X, X ] t := lim n + n Xt n i i=1 X t n i+1 2, for all refining subdivisions (t n i ) i=1,...,n of the time interval [, t], and denote X c, X c t := [X, X] t X s 2 and X c, X c t := [X c, X c ] t Xs 2. <s t s<t Note that the bracket processes ([X, X] t ) t, ( X c, X c t ) t, ([X, X ] t ) t and ( X c, X c t ) t are (F t ) t -adapted but not (F t ) t -adapted. According to the terminology of [54], the pairs (ν X (dt, dx), X c, X c t ) and (ν X (dt, dx), X c, X c t )

116 CHAPITRE 4. A CONVEX DOMINATION PRINCIPLE are called the local characteristics of the processes (X t ) t and (X t ) t, respectively. Now, let us quote Theorem 8.1 in [54], which is an Itô s type formula for forward-backward martingales: Lemma 4.3.1. Let (X t ) t, (X t ) t, be a (F t ) t -adapted (F t ) t -martingale and a (F t ) t -adapted (F t ) t -backward martingale, respectively. Then for real-valued function f C 2 (R 2 ), we have the Itô s formula: f(x t, X t ) = f(x, X ) + t + <s t s<t t + f (X s, Xs )dx s + 1 x 1 2 t t f (X s, X x s+)d Xs 1 2 2 ( f(x s + X s, Xs ) f(x s, Xs f ) X s + 2 f (X x 2 s, Xs )d X c, X c s 1 2 f (X x 2 s, Xs+)d X c, X c s 2 ( f(x s, X s+ + X s ) f(x s, X s+) X s where d and d are the forward and backward Itô s differential, respectively. ) (X s, Xs ) x 1 f x 2 (X s, X s+) As noticed in [54], The crossing adaptedness assumption of Lemma 4.3.1 allows us to define properly the forward and the backward Itô s stochastic integrals. ), 4.3.2 Integrability of convex functions In order for the convex domination principle of Theorem 4.2.1 to make sense, we have first to establish the following integrability property of convex functions in S p,α. Lemma 4.3.2. If the random variable F has the representation (4.2.2), then for any φ S p,α, the random variable φ(f EF ) is integrable. Proof. Let φ S p,α. By the continuity of the convex function φ, we only have to check its integrability outside a compact interval, hence at infinity. Thus, it is sufficient to show that the centered random variable F E[F ] has a finite moment of order p (, α). Since the process (H t ) t L 2 (Ω (, + )), we only have to verify the latter condition for the random variable K t dz t, and up to a conditioning argument, for any random variable T K tdz t, where T > is a fixed time horizon. We have [ T p] E K t dz t = ( T P ( P sup s T ) K t dz t x 1/p dx s ) K t dz t x 1/p dx.

4.3. PROOF OF THEOREM 4.2.1 117 By the maximal inequality in [48], there exists a constant D α > depending on α such [ T p/2, that for any x > E K t dt] 2 we have ( P sup s T s K t dz t x 1/p ) [ T p/2, Hence, denoting x := E K t dt] 2 we obtain [ T p] E K t dz t D [ T α/2 α x E K α/p t dt] 2. [ T ] α/2 x + D α E K t 2 dx dt x x, α/p which is finite provided < p < α. The proof is complete. 4.3.3 Proof of Theorem 4.2.1 We are able to start the proof of the main Theorem 4.2.1, which is divided into several steps. First, we have to introduce a forward and a backward martingales (relying on the representation of the random variable F ) with respect to an increasing and a decreasing filtration, respectively. Then we identify their local characteristics and take expectation in the Itô s formula for forward-backward martingales. Finally, we get the result by a limiting argument. Let ( W t ) t be a standard Brownian motion independent of a symmetric stable process ( Z t ) t of index α and with Lévy measure given on R \ {} by σ(dx) := c x α 1 dx. We assume that both processes are independent of the filtration (F W,Z t ) t. Consider the ) t -martingale (X t ) t given by (F W,Z t X t := E[F EF F W,Z t ], t. By assumption, the symmetric stable process (Z t ) t is a martingale with respect to the filtration (F W,Z t ) t, and so is the stochastic integral ( t K sdz s ) t, since it is constructed as the L 1 -limit of square-integrable martingales. Using a similar argument for the Brownian part, one deduces that the martingale (X t ) t is identified as X t = t H s dw s + Define the enlarged filtration (F t ) t as t K s dz s, t. F t := F W,Z t σ( W s, Z s : s ), t, then the process (X t ) t is still a martingale with respect to (F t ) t. Denote the continuous increasing processes γ t := t H s 2 ds and τ t := t K s α ds, t,

118 CHAPITRE 4. A CONVEX DOMINATION PRINCIPLE and consider the time-changed process Xt = W β1 W γt + Z β2 Z τt, t. Finally, we endow the probability space (Ω, F, P) with the decreasing filtration given by F t := σ( W β1 W γs, Z β2 Z τs : s t) F W,Z t, t. Note that the correlated processes (X t ) t and (X t ) t are (F t ) t - and (F t ) t -adapted, respectively, and their dependence is given through (γ t ) t and (τ t ) t. On the one hand, the independent processes ( W t ) t and ( Z t ) t are both independent of the (F W,Z t ) t -measurable time-change processes (γ t ) t and (τ t ) t. On the other hand, these two processes are centered and have independent increments. Hence, the process (Xt ) t is a (Ft ) t -backward martingale. Now, let us identify the local characteristics of the martingales (X t ) t and (X t ) t. First, the continuous bracket processes are given by d X c, X c t = d X c, X c t = H t 2 dt, t, whereas if we set Y t := t K sdz s, t, then the following change of variables formula t (f(y s + K s x) f(y s ) K s xf (Y s )) dxds t = x α+1 (f(y s + y) f(y s ) yf (Y s )) K s α dyds, t, y α+1 available for any real-valued function f C 1 (R), allows us to identify the local characteristics of the jump parts as follows: ν X (dt, dx) = K t α dt σ(dx), and ν X (dt, dx) = K t α dt σ(dx). Assume without loss of generality that the convex function φ C 2 (R), since any convex function can be approximated by convex C 2 (R)-functions. Using the Itô s formula of Lemma 4.3.1 and taking then expectation, which is allowed by Lemma 4.3.2 since φ S p,α, we get for any s t: E [φ(x t + Xt )] E [φ(x s + Xs )] [ t ] [ t ] = E φ (X u + Xu)d X c, X c u E φ (X u + Xu)d X c, X c u s s [ t ] +E (φ(x u + Xu + x) φ(x u + Xu) xφ (X u + Xu)) ν X (du, dx) s [ t ] E (φ(x u + Xu + x) φ(x u + Xu) xφ (X u + Xu)) ν X (du, dx) s

4.3. PROOF OF THEOREM 4.2.1 119 [ t = E s 1 ] (1 τ) 2 φ (X u + Xu α (c c) + τx) K u dτdxdu. x α+1 By the convexity of φ, the second derivative φ is non-negative and according to the comparison assumption on the weights < c c, one deduces that the function t E [φ(x t + Xt )] is non-increasing on R +. Note [ that this ] function is constant in the case c = c. Since we have the null projection E Xt F W,Z t = for any t, we obtain by Jensen s inequality [ ( [ ])] E φ E F E[F ] F W,Z t = E[φ(X t )] E[φ(X t + X t )] E[φ(X)] [ ( = E φ Wβ1 + Z )] β2. Finally, letting t going to infinity in the left-hand-side above achieves the proof of the Theorem 4.2.1.

12 CHAPITRE 4. A CONVEX DOMINATION PRINCIPLE

Chapitre 5 Convex concentration inequality for continuous gas Cette note a été soumise pour publication au journal Acta Mathematica Scientia(chine) Abstract In this paper, we consider the continuous gas in a bounded Λ of R + or R d described by a Gibbsian probability measure µ η Λ associated with a pair interaction φ, the inverse temperature β, the activity z > and the boundary condition η. Define F = f(s)ω Λ (ds), applying the generalized Ito s formula for forward-backward martingale(see [54]), we give convex concentration inequalities for F with respect to Gibbs measure µ η Λ. By stochastic calculus, we also give a new simple approach of the stochastic domination for Gibbs measure. 5.1 Introduction In this chapter, we consider the continuous gas on R + or R d, which both denoted by E in this section and later we will state what E exactly is. Let Ω be the space of all point measures i δ x i (finite or countable) with x i different in E, which are moreover Random measures, where δ x denotes the Dirac measure at x. Let F A := σ(ω(b); B(Borelian) A) and F = F E. Given the activity z >, let P be the law of the Poisson point process on E with intensity z. The pair interaction φ : E (, + ] will be a Borel-measurable even function which is stable([82]), i.e., H(ω) := φ(t i t j ) Bn, ω = 1 i<j n n δ tj, n 1. (stability) (5.1.1) j=1 Recall that (see [82]) the stability condition is sufficient and necessary to define the Gibbs measures on bounded domains Λ. Given a bounded open and non-empty domain Λ E 121

122 CHAPITRE 5. THE MODEL OF CONTINUOUS GAS and ω Ω, let ω Λ = t i Λ supp(ω) δ t i be the restriction of the measure ω to Λ, and Ω Λ = {ω Λ ; ω Ω}. The image measure P Λ of P by ω ω Λ is the law of Poisson point process on Λ with intensity z. The Gibbs measure in Λ for a given boundary condition η Ω on Λ c is a probability measure on (Ω Λ, F Λ ) given by where µ η Λ (dω) := (Zη Λ ) 1 exp [ βh η Λ (ω)]p Λ(dω Λ ) (5.1.2) H η Λ (ω) = H(ω Λ) + Λ ω Λ (dx) φ(x y)η(dy) Λ c is the Hamiltonian (H(ω Λ ) being given in (5.1.1)), Z η Λ is the normalization constant and β > is the inverse temperature. Recently continuous gas has attracted much attention since F. Cesi in [12](21) skillfully showed the technology to solve the problem from finite volume to infinite volume. A lot of works have been given. Bertini, Cancrini and Cesi [8](22) proved the spectral gap for the continuous gas and pointed that log-sobolev inequality is not valid for this model. L. Wu [98](24) improved their work : precisely characterized the spectral gap for continuous gas and extended the result to hard core case. On the other hand, Klein, Ma and Privault [54](26) generalized the Ito s formula for forward/backward martingale and obtained convex concentration inequalities for Poisson random measure and Brown motion. Using their method, we find suitable backward martingale and give convex concentration inequalities for the Gibbs measure µ η Λ of continuous gas. We also give a new approach to prove the stochastic domination for µ η Λ just for our independent interest. Throughout this paper, for simplicity, we use E η Λ, E to denote respectively the expectation with respect to µ η Λ, P Λ. Since (Ω Λ, F Λ, P Λ ) is a Poisson space, for any F L 2 (Ω Λ, F Λ, P Λ ), in [54], we have given convex concentration inequalities for F E P Λ [F ]. But in this paper, define F = f(s)ω Λ (ds), we want to give some similar convex concentration inequalities for F E η Λ [F ] with respect to µη Λ although (Ω Λ, F Λ, µ η Λ ) is not a Poisson space. That is to say we will find a random variable G such that for any convex function ψ, E η Λ [ψ(f Eη Λ [F ])] E[ψ(G)]. And we also prove that µ η Λ and P Λ satisfy stochastic domination, i.e., for any nondecreasing function f, E η Λ [f] E[f]. 5.2 Main Results In this section we just take Λ as [, T ] for some positive sufficiently large T. Let E(t, ω) be the relative energy of interaction between a particle located at t and the configuration

5.2. MAIN RESULTS 123 ω as follows: For all t T, set and { φ(t u)ω(du) if φ(u t) ω(du) < ; E(t, ω) := = + otherwise. λ t = ze η Λ [e E(t,ω) F F t ]. (5.2.1) F t = t f(s)ω(ds), (5.2.2) where f is a real function on Λ. Before the statements of Theorems, we give two essential assumptions. Assumption 1 Assumption 2 α := T f(s)λ sds < α 2 := T f 2 (s)λ s ds < Remark 5.2.1. If φ is nonnegative, then stability condition is satisfied with B = and λ s z. So Assumption 1(Assumption 2) can be obtained by requiring the local integrability(square integrability) of f, which is a very common condition. Theorem 5.2.2. (Expectation type) If f(s) k, s T for some positive k and furthermore Assumption 1 is satisfied, we have for any convex function ψ, E η Λ [ψ (F t E η Λ [F t])] E[ψ ( kn α/k α ) ], (5.2.3) where N t is a Poisson process with intensity dt. Theorem 5.2.3. (Variance type) Provided with Assumption 2 and f(s) k on [, T ] for some positive k, we have for any C 2 convex function ψ with nondecreasing second derivative, E η Λ [ψ(f t E η Λ [F t])] E[ψ(kN α 2 k 2 where N t is a Poisson process with intensity dt. α2 )], (5.2.4) k

124 CHAPITRE 5. THE MODEL OF CONTINUOUS GAS Theorem 5.2.4. (Brownian type) Suppose that f is nonpositive and Assumption 2 is satisfied. Then for any C 2 convex function ψ with nondecreasing second derivative, we have where W t is a standard Brownian Motion. E η Λ [ψ(f t E η Λ [F t])] E[ψ(W α 2)], (5.2.5) Remark 5.2.5. In fact, in the three theorems above, we can use respectively α t = t λ sf(s)ds et α 2 t = t λ sf 2 (s)ds in place of α and α 2. 5.3 Proof of the theorems We begin this section with two elementary lemmas. Lemma 5.3.1. Given λ t in (5.2.1) and F t in (5.2.2), we have for any t T, t f(s)(ω(ds) λ s ds) is a F F t -martingale under the Gibbs measure µ η Λ. Proof. By Nguyen-Zessin Theorem, a probability measure µ on Ω is called a Gibbs measure if and only if it satisfies µ(dω) ω(ds)f (ω, s) = R + Ω Ω µ(dω) zds exp [ E(s, ω)]f (ω s, s) R + for any measurable function F : Ω Λ [, + ]. Therefore we obtain the result considering only on Ω Λ and taking F (ω, s) = f(s) for all ω Ω Λ The following Lemma is just an evident fact. Lemma 5.3.2. Suppose N t, Ñt are Poisson processes with intensity dt respectively under P Λ, µ η Λ and W t, W t are two standard Brownian Motions with respect to P Λ, µ η Λ. Then we have for any t, E[ψ(aN t at)] = E η Λ [ψ(añt at)], E[ψ(W t )] = E η Λ [ψ( W t )],

5.3. PROOF OF THE THEOREMS 125 where ψ is any real function on Ω Λ and a is a constant. By Lemma 5.3.2, we prove the Theorems replacing E[ψ(aN t at], E[ψ(B t )] respectively with E η Λ [ψ(añt at)], E η Λ [ψ( B t )]. In order to simplify our notations, we remain N t, B t to denote Poisson process, Brownian Motion with respect to µ η Λ. We assume that N t, B t are independent of F, otherwise replace them by their independent copies which are independent of F. Firstly we prove Theorem 5.2.2 and 5.2.3 together. Proof. For all t T, p = 1, 2 define α p (t) = t f p (s)λ s d(s). Set X t = X t = k t f(s)ω(ds) ( N α p k p t f(s)λ s ds, ) N αp (t) k(α p /k p α p (t)/k p ). k p An increasing filtration (F t ) t T and a decreasing filtration (F t ) t T are given as: F t = F X t F N and F t = σ{n u : t u T } F X T, where F X denotes the natural filtration generated by X. By Lemma 5.3.1, under µ η Λ, X t is a F -measurable F t martingale. Since under µ η Λ, N t is a Poisson Process and N t is independent of F, X t is a F-measurable F t -backward pure discontinuous martingale., i.e., E η Λ [X t F s ] = X s, t s T. Then by Ito s formula, for any s t we have E η Λ [ψ(x t + Xt )] = E η Λ [ψ(x s + Xs )] [ t ] + E η Λ (ψ(x s + Xs + f(s)) ψ(x s + Xs ) f(s)ψ (X s + Xs )) λ s ds s [ t ] E η Λ (ψ(x s + Xs + k) ψ(x s + Xs ) kψ (X s + Xs )) f(s)p λ s k p s ds [ t ] E η Λ [ψ(x s + Xs )] + E η Λ λ s ϕ p (X s + Xs, f(s))ds. For p = 1, ϕ 1 (x, y) = ψ(x + y) ψ(x) y (ψ(x + k) ψ(x)), k

126 CHAPITRE 5. THE MODEL OF CONTINUOUS GAS which is certainly less than since ψ is convex and f(s) k. For p = 2, by Taylor s formula ψ(x + y) ψ(x) yψ (x) = 1 y2 (1 τ)ψ (x + yτ)dτ, we have ϕ 2 (x, y) = 1 f 2 (s)(ψ (x + yτ) ψ (x + kτ))(1 τ)dτ. The fact that f(s) k and ψ is nondecreasing implies ϕ 2. Therefore we know E η Λ [φ(x t + X t )] is nonincreasing, which deduces that E η Λ [φ(x t + X t )] E η Λ [ψ(x )]. By Jensen s inequality, we have E η Λ [ψ(x )] E η Λ [ψ(x t + Xt )] = E η Λ [Eη Λ [ψ(x t + Xt )] Ft F ] E η Λ [ψ(x t + E η Λ [X t Ft F ])] = E η Λ [ψ(x t)]. The last equality is from the fact that N is independent of F and X t is a F t -backward martingale. It remains to prove Theorem 5.2.4. Proof. Take the same X t and α 2 (t) as in preceding proof. Set X t = W α 2 W α 2 (t) and define respectively the increasing filtration F t and the decreasing filtration F t as: F t = F X t F W, F t = F X T σ(w u : t u T ). Then under µ η Λ, X t is F -measurable F t -martingale and X t is F-measurable F t -backwardmartingale. With the same argument of Theorem 5.2.2 and 5.2.3, it is enough to verify E η Λ [ψ(x t + X t )] is a nonincreasing function of t. By Ito s formula, we have for any

5.4. STOCHASTIC DOMINATION 127 s t T, E η Λ [ψ(x t + Xt )] E η Λ [ψ(x s + Xs )] [ t ] = E η Λ (ψ(x u + Xu + f(s)) ψ(x u + Xu) f(s)ψ(x u + Xu))λ s ds 1 [ t ] 2 Eη Λ ψ (X u + Xu)f 2 (s)λ s ds [ 1 t ] = E η Λ (1 τ)(ψ (X u + Xu + f(s)τ) ψ (X u + Xu))f 2 (s)λ s dsdτ. The last inequality is established since f(s) and ψ is nondecreasing. Therefore E η Λ [ψ(x t + X t )] is nonincreasing with respect to t. 5.4 Stochastic domination In this section we take E as R d and suppose that φ is nonnegative. By the Corollary 3.4 of [99], letting n +, we can have the stochastic domination of µ η Λ and P Λ. But here, we give another simple approach just for our independent interest. Definition 5.4.1. We call a functional on Ω Λ is nondecreasing, if for any u Λ, D u f(ω Λ ) = f(ω Λ + δ u 1 u/ suppωλ ) f(ω Λ δ u 1 u ωλ ), where D is the difference operator. Theorem 5.4.2. The Gibbs measure given in (5.1.2) is stochastic dominated by P Λ, i.e., for any nondecreasing function ψ L 1 (Ω Λ, P Λ ), E η Λ [ψ] E[ψ]. (5.4.1) It is just a particular case of the following simple but useful proposition (5.4.3) with h 1. Proposition 5.4.3. For any nondecreasing functional g L 1 (Ω Λ, P Λ ) and non-increasing nonnegative functional h L 1 (Ω Λ, µ η Λ ), we have E η Λ [gh] E[g]Eη Λ [h]. (5.4.2)

128 CHAPITRE 5. THE MODEL OF CONTINUOUS GAS Proof. By the definition of difference operator, D u (e βhη Λ h)(ω) = e βh η Λ (ω+δu1 u / suppω) h(ω + δ u 1 u/ suppω ) e βhη Λ (ω δu1u suppω) h(ω δ u 1 u suppω ) = D u e βhη Λ (ω) h(ω + δ u 1 u/ suppω ) + e βhη Λ (ω 1u suppω) D u h(ω), since h is nonnegative and h, e βhη Λ are both non-increasing. By the FKG inequality on Poisson space due to Janson al. [46] (see [95], Remark 1.5 for a simple proof), we have E η Λ [gh] = (Zη Λ ) 1 E[(e βhη Λ h)g] (Z η Λ ) 1 E[e βhη Λ h]e[g] = E η Λ [h]e[g]. Let H x, x Λ be a nonnegative function on Ω Λ. Corollary 5.4.4. For any A Λ, set F A (ω) = A H xω(dx). Then for any nondecreasing function ψ on R, such that ψ F A (ω) := ψ(f A (ω)) is integrable with respect to P Λ, we have E η Λ [ψ F ] E[ψ F ]. (5.4.3) Proof. By Theorem (7.2.1), we just need to verify the function ψ F is nondecreasing on Ω Λ. For any u Λ, we have D u ψ F (ω) = ψ(f (ω + δ u 1 u/ suppω )) ψ(f (ω δ u 1 u suppω )) = ψ(f (ω) + H u (ω)1 u/ suppω ) ψ(f (ω) H u (ω)1 u suppω ), which is certainly greater than since ψ in nondecreasing and H is nonnegative. Remark 5.4.5. a.) If we take H x (ω) = f(x) a nonnegative function on Λ, (5.4.3) is, for any nondecreasing function ψ on R, E η Λ [ψ(f A)] E[ψ(F A )], A Λ. (5.4.4)

5.4. STOCHASTIC DOMINATION 129 It is very intuitisnic since φ is nonnegative, all particles push the others. b.) If we take H x (ω) 1, F A = ω(a) denotes then the number of particles in A, which is a Poisson process with intensity zdx on (Ω Λ, F Λ, P Λ ). On other hand, by Nyugen-Zessin s Theorem, we know that E η Λ (ω(a)) = Eη Λ A ze E(x, ) dx z A = E[ω A ], (5.4.5) where E(x, ω) := φ(x y)ω(dy). Therefore (5.4.5) is just (5.4.4) with ψ 1. Λ

13 CHAPITRE 5. THE MODEL OF CONTINUOUS GAS

Partie III Inégalités fonctionnelles 131

Chapitre 6 FKG inequality on the Wiener space via predictable representation Cette note, co-écrit avec Nicolas Privault paraîtra prochainement dans Preceedings of the 25 Hammanet Conference on Stochastic Analysis and Probability. Abstract Using the Clark predictable representation formula, we give a proof of the FKG inequality on the Wiener space. Solutions of stochastic differential equations are treated as applications and we recover by a simple argument the covariance inequalities obtained for diffusions processes by several authors. 6.1 Introduction Let (Ω, F, P, ) be a probability space equipped with a partial order relation on Ω. An (everywhere defined) real-valued random variable F on (Ω, F, P, ) is said to be nondecreasing if F (ω 1 ) F (ω 2 ) for any ω 1, ω 2 Ω satisfying ω 1 ω 2. The FKG inequality [36] states that if F and G are two square-integrable random functionals which are non-decreasing for the order, then F and G are non-negatively correlated: Cov(F, G). It is well known that the FKG inequality holds for the standard ordering on Ω = R, since given X, Y : R R two non-decreasing functions on R we have: Cov(X, Y ) = 1 (X(x) X(y))(Y (x) Y (y))p(dx)p(dy) 2 R R 133

134 CHAPITRE 6. FKG INEQUALITY ON THE WIENER SPACE = 1 2 + 1 2. {y x} {x<y} (X(x) X(y))(Y (x) Y (y))p(dx)p(dy) (X(y) X(x))(Y (y) Y (x))p(dx)p(dy) The FKG inequality also holds on R n for the pointwise ordering, cf. e.g. Bakry and Michel [4]. On the Wiener space (Ω, F, P) with Brownian motion (W t ) t R+, Barbato [5] introduced a weak ordering on continuous functions and proved an FKG inequality for Wiener functionals, with application to diffusion processes. In this paper we recover the results of [5] under weaker hypotheses via a simple argument. Our approach is inspired by Remark 1.5 stated on the Poisson space in Wu [95], page 432, which can be carried over to the Wiener space by saying that the predictable representation of a random variable F as a an Itô integral, obtained via the Clark formula F = E[F ] + yields the covariance identity [ 1 Cov(F, G) = E 1 where D is the Malliavin gradient expressed as E[D t F F t ]dw t, ] E[D t F F t ]E[D t G F t ]dt, (6.1.1) DF, ḣ L 2 ([,1]) = d dε F (ω + ɛh) ε=. (6.1.2) From (6.1.2) we deduce that DF is non-negative when F is non-decreasing, which implies Cov(F, G) from (6.1.1). Applications are given to diffusion processes and in Theorem 6.3.6 we recover, under weaker hypotheses, the covariance inequality obtained in Theorem 3.2 of [44] and in Theorem 7 of [5]. We proceed as follows. Elements of analysis on the Wiener space and applications to covariance identities are recalled in Sections 6.2. The FKG inequality and covariance inequalities for diffusions are proved in Section 6.3. We also show that our method allows us to deal with the discrete case, cf. Section 6.4. 6.2 Analysis on the Wiener space In this section we recall some elements of stochastic analysis on the classical Wiener space (Ω, F, P) on Ω = C ([, 1]; R), with canonical Brownian motion (W t ) t [,1] generating the filtration (F t ) t [,1]. Our results extend without difficulty to the Wiener space

6.2. ANALYSIS ON THE WIENER SPACE 135 on C (R + ; R). Let H denote the Cameron-Martin space, i.e. the space of absolutely continuous functions with square-integrable derivative: { 1 } H = h : [, 1] R : ḣ(s) 2 ds <. Let I n (f n ), n 1, denote the iterated stochastic integral of f n in the space L 2 s([, 1] n ) of symmetric square-integrable functions in n variables on [, 1] n, defined as 1 tn t2 I n (f n ) = n! f n (t 1,..., t n )dw t1 dw tn, with the isometry formula E[I n (f n )I m (g m )] = n!1 {n=m} f n, g m L 2 ([,1] n ). Every F L 2 (Ω) admits a unique Wiener chaos expansion F = E[F ] + I n (f n ) with f n L 2 s([, 1] n ), n 1. Let (e k ) k 1 denote the dyadic basis of L 2 ([, 1]) given by n=1 e k = 2 n/2 1 [ k 2 n 2 n, k+1 2n 2 n ], 2n k 2 n+1 1, n N. Recall the following two equivalent definitions of the Malliavin gradient D and its domain Dom(D), cf. Lemma 1.2 of [66] and [69]: a) Finite dimensional approximations. Given F L 2 (Ω), let for all n N: G n = σ(i 1 (e 2 n),..., I 1 (e 2 n+1 1)), and F n = E[F G n ], and consider f n a square-integrable function with respect to the standard Gaussian measure on R 2n, such that F n = f n (I 1 (e 2 n),..., I 1 (e 2 n+1 1)). Then F Dom(D) if and only if f n belongs for all n 1 to the Sobolev space W 2,1 (R 2n ) with respect to the standard Gaussian measure on R 2n, and the sequence D t F n := 2 n i=1 e 2 n +i 1(t) f n x i (I 1 (e 2 n),..., I 1 (e 2 n+1 1)), t [, 1], converges in L 2 (Ω [, 1]). In this case we let DF := lim n DF n.

136 CHAPITRE 6. FKG INEQUALITY ON THE WIENER SPACE b) Chaos expansions. Let G L 2 (Ω) be given by G = E[G] + I n (g n ). n=1 Then G belongs to Dom(D) if and only if the series n!n g n 2 L 2 ([,1] n ) n=1 converges, and in this case, D t G = g 1 (t) + ni n 1 (g n (, t)), t [, 1]. n=1 In case (a) above the gradient DF n, ḣ L 2 ([,1]), h H, coincides with the directional derivative DF n, ḣ L 2 ([,1]) = d dε f n(i 1 (e 2 n) + ε e 2 n, ḣ L 2 ([,1]),..., I 1 (e 2 n+1 1) + ε e 2 n+1 1, ḣ L 2 ([,1])) ε= = d dε F n(ω + ɛh) ε=, where the limit exists in L 2 (Ω). Similarly, the Ornstein-Uhlenbeck semi-group (P t ) t R+ definitions, cf. e.g. [67], [87], [9]: admits the following equivalent a) Integral representation. For any F L 2 (Ω) and t R +, let P t F (ω) = F (e t ω + 1 e 2t ω)dp( ω), P(dω) a.s. (6.2.1) Ω b) Chaos representation. For any F L 2 (Ω) with the chaos expansion F = E[F ] + I n (f n ), n=1 we have P t F = E[F ] + e nt I n (f n ), t R +. (6.2.2) n=1

6.2. ANALYSIS ON THE WIENER SPACE 137 The operator D satisfies the Clark formula, i.e. F = E[F ] + 1 E[D t F F t ]dw t, F Dom(D), (6.2.3) cf. e.g. [87]. By continuity of the operator mapping F L 2 (Ω) to the adapted and square-integrable process (u t ) t R+ appearing in predictable representation F = E[F ] + 1 u t dw t, (6.2.4) the Clark formula can be extended to any F L 2 (Ω) as in the following proposition. Proposition 6.2.1. The operator F (E[D t F F t ]) t [,1] extends as a continuous operator on L 2 (Ω). Proof. We use the bound [ 1 ] E (E[D t F F t ]) 2 dt = E[(F E[F ]) 2 ] = E[F 2 ] (E[F ]) 2 F 2 L 2 (Ω), (6.2.5) for F Dom(D). Moreover, by uniqueness of the predictable representation of F L 2 (Ω), an expression of the form F = c + 1 u t dw t where c R and (u t ) t R+ is adapted and square-integrable, implies u t = E[D t F F t ], dt dp-a.e. The Clark formula and the Itô isometry yield the following covariance identity, cf. Proposition 2.1 of [43]. Proposition 6.2.2. For any F, G L 2 (Ω) we have This identity can be written as [ 1 ] Cov(F, G) = E E[D t F F t ]E[D t G F t ]dt. (6.2.6) [ 1 ] Cov(F, G) = E E[D t F F t ]D t Gdt, (6.2.7) provided G Dom(D). The following lemma is an immediate consequence of (6.2.6).

138 CHAPITRE 6. FKG INEQUALITY ON THE WIENER SPACE Lemma 6.2.3. Let F, G L 2 (Ω) such that E[D t F F t ] E[D t G F t ], dt dp a.e. Then F and G are non-negatively correlated: Cov(F, G). If G Dom(D), resp. F, G Dom(D), the above condition can be replaced by resp. E[D t F F t ] and D t G, dt dp a.e., D t F and D t G, dt dp a.e.. As recalled in the introduction, if X is a real random variable and f, g are C 1 (R) functions with non-negative derivatives f, g, then f(x) and g(x) are non-negatively correlated. Lemma 6.2.3 provides an analog of this result on the Wiener space, replacing the ordinary derivative with the adapted process (E[D t F F t ]) t [,1]. 6.3 FKG inequality on the Wiener space We consider the order relation introduced in [5]. Definition 6.3.1. Given ω 1, ω 2 Ω, we say that ω 1 ω 2 if and only if we have ω 1 (t 2 ) ω 1 (t 1 ) ω 2 (t 2 ) ω 2 (t 1 ), t 1 t 2 1. The class of non-decreasing functionals with respect to is larger than that of nondecreasing functionals with respect to the pointwise order on Ω defined by ω 1 (t) ω 2 (t), t [, 1], ω 1, ω 2 Ω. Definition 6.3.2. A random variable F : Ω R is said to be non-decreasing if ω 1 ω 2 F (ω 1 ) F (ω 2 ), P(dω 1 ) P(dω 2 ) a.s. Note that unlike in [5], the above definition allows for almost-surely defined functionals. The next result is the FKG inequality on the Wiener space. It recovers Theorem 4 of [5] under weaker (i.e. almost-sure) hypotheses.

6.3. FKG INEQUALITY ON THE WIENER SPACE 139 Theorem 6.3.3. For any non-decreasing functionals F, G L 2 (Ω) we have Cov(F, G). The proof of this result is a direct consequence of Lemma 6.2.3 and Proposition 6.3.5 below. Lemma 6.3.4. For every non-decreasing F Dom(D) we have D t F, dt dp a.e. Proof. For n N, let π n denote the orthogonal projection from L 2 ([, 1]) onto the linear space generated by (e k ) 2 n k<2n+1. Consider h in the Cameron-Martin space H and let h n (t) = t [π n ḣ](s)ds, t [, 1], n N. Let Λ n denote the square-integrable and G n -measurable random variable ( 1 Λ n = exp [π n ḣ](s)dw s 1 1 ) [π n ḣ](s) 2 ds. 2 From the Cameron-Martin theorem, for all n N and G n -measurable bounded random variable G n we have, letting F n = E[F G n ]: hence E[F n ( + h n )G n ] = E[Λ n F n G n ( h n )] = E[Λ n E[F G n ]G n ( h n )] = E[E[Λ n F G n ( h n ) G n ]] = E[Λ n F G n ( h n )] = E[F ( + h n )G n ], F n (ω + h n ) = E[F ( + h n ) G n ](ω), P(dω) a.s. If ḣ is non-negative, then π nḣ is non-negative by construction hence ω ω + h n, ω Ω, and we have F (ω) F (ω + h n ), P(dω) a.s.,

14 CHAPITRE 6. FKG INEQUALITY ON THE WIENER SPACE since from the Cameron-Martin theorem, P({ω + h n : ω Ω}) = 1. Hence with the notation of Section 6.2, F n (ω + h) = f n (I 1 (e 2 n) + e 2 n, ḣ L 2 ([,1]),..., I 1 (e 2 n+1 1) + e 2 n+1 1, ḣ L 2 ([,1])) = f n (I 1 (e 2 n) + e 2 n, π n ḣ L 2 ([,1]),..., I 1 (e 2 n+1 1) + e 2 n+1 1, π n ḣ L 2 ([,1])) = F n (ω + h n ) = E[F ( + h n ) G n ](ω) E[F G n ](ω) = F n (ω), P(dω) a.s., i.e. for any ε 1 ε 2 and h H such that ḣ is non-negative we have F n (ω + ε 1 h) F n (ω + ε 2 h), and the smooth function ε F n (ω + εh) is non-decreasing in ε on R, P(dω)-a.s. As a consequence, DF n, ḣ L 2 ([,1]) = d dε F n(ω + ɛh) ε=, for all h H such that ḣ, hence DF n. Taking the limit of (DF n ) n N as n goes to infinity shows that DF. Next, we extend Lemma 6.3.4 to F L 2 (Ω). Proposition 6.3.5. For any non-decreasing functional F L 2 (Ω) we have E[D t F F t ], dt dp a.e. Proof. Assume that F L 2 (Ω) is non-decreasing. Then P 1/n F, n 1, is non-decreasing from (6.2.1), and belongs to Dom(D) from (6.2.2). From Lemma 6.3.4 we have D t P 1/n F, dt dp a.e.,

6.3. FKG INEQUALITY ON THE WIENER SPACE 141 hence E[D t P 1/n F F t ], dt dp a.e. Taking the limit as n goes to infinity yields E[D t F F t ], dt dp-a.e. from (6.2.5) and the fact that P 1/n F converges to F in L 2 (Ω) as n goes to infinity. Conversely it is not difficult to show that if u L 2 ([, 1]) is a non-negative deterministic function, then the Wiener integral 1 u tdw t is a non-decreasing functional. Note however that the stochastic integral of a non-negative square-integrable process may not necessarily be a non-decreasing functional. For example, consider u t = G1 [a,1], t [, 1], where G L 2 (Ω, F a ) is non-negative and decreasing, then is not non-decreasing. 1 u t dw t = G(B 1 B a ) Example: maximum of Brownian motion. By Proposition 2.1.3 of [67] the maximum M = sup t 1 W (t) of Brownian motion on [, 1] belongs to Dom(D) and satisfies DM = 1 [,τ], where τ is the a.s. unique point where M attains its maximum. Here, M is clearly an increasing functional. Example: diffusion processes. Consider the stochastic differential equations { dxt = b t (X t )dt + σ t (X t )dw t X = x, (6.3.1) and { d Xt = b t ( X t )dt + σ t ( X t )dw t X = x, (6.3.2) where b, b, σ, σ are functions on R + R satisfying the following global Lipschitz and boundedness conditions, cf. [67], page 99: (i) σ t (x) σ t (y) + b t (x) b t (y) K x y, x, y R, t [, 1], (ii) t σ t () and t b t () are bounded on [, 1],

142 CHAPITRE 6. FKG INEQUALITY ON THE WIENER SPACE for some K >. Lemma 8 of [5] shows that the solutions (X t ) t [,1], ( X t ) t [,1] of (6.3.1) and (6.3.2) are increasing functionals when σ(x), σ(x) are differentiable with Lipschitz derivative in one variable and satisfy uniform bounds of the form < ε σ(x) M < and < ε σ(x) M <, x R. Thus from Proposition 6.3.5 it satisfies the FKG inequality as in Theorem 7 of [5]. Here the same covariance inequality can be obtained without using the FKG inequality, and under weaker hypotheses. Theorem 6.3.6. Let s, t [, 1] and assume that σ, σ satisfy the condition σ r (x) σ r (y), x, y R, r s t. Then we have for all non-decreasing Lipschitz functions f, g. Cov(f(X s ), g( X t )), (6.3.3) Proof. From Proposition 1.2.3 and Theorem 2.2.1 of [67], we have f(x s ) Dom(D), s [, 1], and D r f(x s ) = 1 [,s] (r)σ r (X r )f (X s )e s r αudwu+ s r (β u 1 2 α2 u)du, (6.3.4) r, s [, 1], where (α u ) u [,1] and (β u ) u [,1] are uniformly bounded adapted processes. Hence we have E[D r X s F r ] = 1 [,s] (r)σ r (X r )E [f (X s )e s r αudwu+ s r, s [, 1]. Similarly we show that E[D r g( X t ) F r ] has the form E[D r g( X t ) F r ] = 1 [,t] (r) σ r ( X r )E [g ( X t )e t r αudwu+ t r, t [, 1], and we conclude the proof from Lemma 6.2.3. r (β u 1 2 α2 u)du r ( β u 1 2 α2 u)du ] F r, ] F r, Note that (6.3.3) has also been obtained for s = t and X = X in [44], Theorem 3.2, by semigroup methods. In this case it also follows by applying Corollary 1.4 of [41] in dimension one. The argument of [44] can in fact be extended to recover Theorem 6.3.6 as above. Also, (6.3.3) may also hold under local Lipschitz hypotheses on σ and σ, for example as a consequence of Corollary 4.2 of [2].

6.4. THE DISCRETE CASE 143 6.4 The discrete case Let Ω = { 1, 1} N and consider the family (X k ) k 1 of independent Bernoulli { 1, 1}- valued random variables constructed as the canonical projections on Ω, under a measure P such that p n = P(X n = 1) and q n = P(X n = 1), n N. Let F 1 = {, Ω} and F n = σ(x,..., X n ), n N. Consider the linear gradient operator D defined as D k F (ω) = p k q k (F ((ω i 1 {i k} + 1 {i=k} ) i N ) F (ω i 1 {i k} 1 {i=k} ) i N ), (6.4.1) k N. Recall the discrete Clark Formula, cf. Proposition 7 of [79]: F = E[F ] + E[D k F F k 1 ]Y k, (6.4.2) where Y k = 1 {Xk =1} k= qk pk 1 {Xk = 1}, k N, p k q k defines a normalized i.i.d. sequence of centered random variables with unit variance. The Clark formula entails the following covariance identity, cf. Theorem 2 of [79]: [ ] Cov(F, G) = E E[D k F F k 1 ]E[D k G F k 1 ], k= which yields a discrete time analog of Lemma 6.2.3. Lemma 6.4.1. Let F, G L 2 (Ω) such that E[D k F F k 1 ] E[D k G F k 1 ], k N. Then F and G are non-negatively correlated: Cov(F, G). According to the next definition, a non-decreasing functional F satisfies D k F for all k N. Definition 6.4.2. A random variable F : Ω R is said to be non-decreasing if for all ω 1, ω 2 Ω we have ω 1 (k) ω 2 (k), k N, F (ω 1 ) F (ω 2 ).

144 CHAPITRE 6. FKG INEQUALITY ON THE WIENER SPACE The following result is then immediate from (6.4.1) and Lemma 6.4.1, and shows that the FKG inequality holds on Ω. Proposition 6.4.3. If F, G L 2 (Ω) are non-decreasing then F and G are non-negatively correlated: Cov(F, G). Note however that the assumptions of Lemma 6.4.1 are actually weaker as they do not require F and G to be non-decreasing.

Chapitre 7 Spectral gap and convex concentration inequalities for birth-death processes Ce travail, effectué en collaboration avec Wei Liu, est soumis au Journal Annales de l Institut Henri Poincare (B). Abstract In this paper, we consider a birth-death process with generator L and reversible invariant probability π. Given an increasing function ρ and defining a Lipschitzian norm Lip(ρ) with respect to ρ, we have a representation of ( L) 1 Lip(ρ). As a typical application, with spectral theory, we revisit one variational formula of Chen for the lower bound of the spectral gap of L in L 2 (π). Moreover, by Lyons-Zheng s forward-backward martingale decomposition theorem, we obtain a convex concentration inequality for additive functionals of birth-death processes. Keywords: birth-death processes; spectral gap; Lipschitzian functional; Poisson equation; convex concentration inequality. 2 MR Subject Classification : 6E15, 6G27. 7.1 Introduction Consider a birth-death process (X t ) t on N = {, 1, 2, } with birth rates (b i ) i N and death rates (a i ) i N, i.e., its generator L is given by, for any function f on N Lf(i) = b i (f(i + 1) f(i)) + a i (f(i 1) f(i)), (7.1.1) 145

146 CHAPITRE 7. SPECTRAL GAP OF BIRTH-DEATH PROCESS where b i > (i ), a i > (i 1), a =. Throughout this paper, we assume that the process is positively recurrent, i.e., where µ n (µ i b i ) 1 = and C := n i n + n= µ = 1, µ n = b b 1 b n 1 a 1 a 2 a n, n 1. Denote π n = µn, n the reversible invariant probability. C µ n < +, In this paper, we have two main aims to verify: spectral gap and convex concentration inequalities for additive functionals of birth-death processes. Let λ 1 be the spectral gap of L, i.e., the lowest eigenvalue above zero of L. λ 1 1 is the best constant in the following Poincaré inequality, V ar π (f) ce π (f), where V ar π (f) := i π i(f(i) π(f)) 2 and E π (f) := i π ib i (f(i + 1) f(i)) 2 are respectively the covariance and Dirichlet form of f with respect to π. Let (P t ) t be the corresponding semigroup of the process, then λ 1 is the optimal constant in which characterizes the decay of P t to π. P t f π(f) L 2 (π) e ɛt f π(f) L 2 (π), Since Chen and Wang firstly used the probabilistic method (coupling) to obtain the first eigenvalue on manifold (see [2]), they and their working group have obtained fruitful results (the reader are suggested to [19] for a systematical knowledge). Chen (1996, [15]) proved two variational formulas for the lower bound of spectral gap for birth-death processes by coupling (the diffusion case is due to Chen and Wang [21], 1997). Then for both birth-death processes and diffusion processes, an analytic proof was given by Chen in [16] (1999) and the criteria were presented in [17] (2) based on the variational formulas. Independently, in 1999, the criteria was obtained by Miclo [64] by using discrete Hardy s inequalities. Next one more variational formula for the lower bound and two dual formulas for the upper bound as well as some approximating procedures for the spectral gap were presented in [18] (21). Wu ([92]) gave another approach to obtain the spectral gap for one dimensional diffusion processes. He considered diffusion processes with generator L and gave the bound of ( L) 1 on some space of additive Lipschizian functionals, and then with spectral theory obtained the spectral gap. As stated above, the spectral gap for birth-death processes has been well studied. But here we give another new approach, i.e., we use the idea of [92] while working for diffusion

7.2. REPRESENTATION OF ( L) 1 LIP (ρ) 147 processes to obtain the spectral gap. Precisely, we obtain an exact Lipschitzian norm for the operator ( L) 1 on some Lipschitzian space and with the spectral theory, we revisit one variational formula of Chen for the lower bound of the spectral gap. Moreover by Lyons-Zheng s forward-backward martingale decomposition, we prove a convex concentration inequality for additive functionals of birth-death processes. The remainder of the paper will be organized as follows. In section 2 we concentrate on the representation of ( L) 1 and in section 3 we work on the spectral gap for L. The last section is about a convex concentration inequality for the functionals of birth-death processes. 7.2 Representation of ( L) 1 Lip(ρ) Given an increasing function ρ : N R, let d ρ (i, j) = ρ(i) ρ(j) be the metric on N associated with ρ. We call a function f on N is Lipschitzian with respect to ρ if which is equivalent to say f Lip(ρ) := sup i j f Lip(ρ) = sup i f(j) f(i) ρ(j) ρ(i) f(i + 1) f(i) ρ(i + 1) ρ(i) < +. < +, (7.2.1) In this whole paper, we work only for the Lipschitzian functionals with zero mean so that Lip(ρ) becomes a true norm. Denoted by (C Lip(ρ), Lip(ρ) ) the space of all Lipschitzian functions with respect to ρ with zero mean. As showed below, given ρ L 1 (π), if g is in C Lip(ρ) and f, g satisfy the equation of Poisson Lf = g, we can prove that f is Lipschitzian. If moreover we restrict ourselves to C Lip(ρ), which implies is a single eigenvalue of L (this point is very important), then L is one-toone mapping from C Lip(ρ) to C Lip(ρ) (so does ( L) 1 ). By definition, L has a spectral gap in C Lip(ρ) if is an isolated eigenvalue in C Lip(ρ) or equivalently ( L) 1 : C Lip(ρ) C Lip(ρ) is bounded. Recall the usual Lipschitzian norm of ( L) 1 : ( L) 1 Lip(ρ) = sup ( L) 1 g Lip(ρ) = sup ( L) 1 g Lip(ρ). g Lip(ρ) 1 g Lip(ρ) =1 Then ( L) 1 Lip(ρ) has the following representation:

148 CHAPITRE 7. SPECTRAL GAP OF BIRTH-DEATH PROCESS Theorem 7.2.1. Let L, ρ, Lip(ρ) be as above consideration and assume that ρ L 1 (π), we have I(ρ) := ( L) 1 Lip(ρ) = sup i 1 k=i π k(ρ(k) π(ρ)) π i a i (ρ(i) ρ(i 1)). (7.2.2) Our approach is similar to that of [92] while working for diffusion processes. We begin the proof with two lemmas. Lemma 7.2.2. Given a function g on N with µ(g) =, consider the Poisson equation Lf = g. (7.2.3) For any i, the solution of the above equation (7.2.3) satisfies the following relation: i f(i + 1) f(i) = π jg(j). (7.2.4) π i+1 a i+1 Proof. By Theorem 1 and Lemma 4 of Chap 7 in [89], the solution of the equation (7.2.3) can be represented as i 1 f(i) = f() (Z(i) Z(j))µ j g(j), (7.2.5) j= where Z() =, Z(1) = 1 b, Z(n) = Z() + 1 b + n 1 i=1 f(i + 1) f(i) = a 1 a i b b 1 b i, n 2. Therefore i i 1 (Z(i + 1) Z(j))µ j g(j) + (Z(i) Z(j))µ j g(j) j= i 1 = (Z(i + 1) Z(i))µ j g(j) (Z(i + 1) Z(i))µ i g(i) j= = (Z(i + 1) Z(i)) i = π jg(j). π i+1 a i+1 i µ j g(j) Now we prove the crucial point of this section:

7.2. REPRESENTATION OF ( L) 1 LIP (ρ) 149 Lemma 7.2.3. Provided with g Lip(ρ) = 1 and π(g) =, we have π i g(i) π i (ρ(i) π(ρ)). (7.2.6) i k i k Proof. Set F (k) = + π i g(i) + i=k i=k π i (ρ(i) π(ρ)), which satisfies that F () = and lim k F (k) =. It suffices to show either F or there exists some K, k K, F is non-increasing and k > K, F is nondecreasing. Below we suppose that F is not identically zero. Simple calculus gives us, F (k + 1) F (k) = π k g(k) + π k (ρ(k) π(ρ)) = π k (ρ(k) g(k) π(ρ)). Define G(k) := Since g Lip(ρ) = 1, we have F (k + 1) F (k) π k = ρ(k) g(k) π(ρ). G(k + 1) G(k) = ρ(k + 1) ρ(k) (g(k + 1) g(k)). If G() >, then G(k) > for any k, which deduces that F is increasing. We have a contradiction. Thus G(). = lim k F (k) F (1) = G() >, Since G is nondecreasing, there is at most one time to change its sign. If G doesn t change its sign, it means G(k) for any k, then F is non-increasing. For any N N, which shows F, it is not our case. = lim k F (k) F (N) F () =, Hence G changes its sign, i.e., there exists some K, such that G(k) >, k > K and G(k), k K.

15 CHAPITRE 7. SPECTRAL GAP OF BIRTH-DEATH PROCESS That is to say, for any 1 k K, F (k) F (k 1) and F (k 1) F (k) when k > K, which completes the proof. Remark 7.2.4. Applying (7.2.6) to g, we get i k π j g(j) i k π i (ρ(i) π(ρ)). (7.2.7) Proof of Theorem 7.2.1. For any 1-Lipschitzian function g L 1 (π), by the equation (7.2.3), we have By the definition of Lip(ρ) and Lemma 7.2.2, we have Then f Lip(ρ) = sup i ( L) 1 (g π(g)) = f. (7.2.8) = sup i f(i + 1) f(i) ρ(i + 1) ρ(i) + j=i+1 π j(g(j) π(g)) (ρ(i + 1) ρ(i))π i+1 a i+1. ( L) 1 Lip(ρ) = + j=i+1 sup sup π j(g(j) π(g)) g Lip(ρ) =1 i (ρ(i + 1) ρ(i))π i+1 a i+1 = sup i sup i 1 1 π i+1 a i+1 (ρ(i + 1) ρ(i)) j=i π j(ρ(j) π(ρ)) π i a i (ρ(i) ρ(i 1)), sup g Lip(ρ) =1 + j=i+1 π j (g(j) π(g)) (7.2.9) the last inequality follows from (7.2.7). On the other hand, ρ π(ρ) is also 1-Lipschitz function, then ( L) 1 Lip(ρ) sup i 1 Combining (7.2.9) and (7.2.1), we have ( L) 1 Lip(ρ) = sup i 1 j=i π j(ρ(j) π(ρ)) π i a i (ρ(i) ρ(i 1)). (7.2.1) j=i π j(ρ j π(ρ)) π i a i (ρ(i) ρ(i 1)). (7.2.11)

7.3. APPLICATION TO SPECTRAL GAP 151 7.3 Application to spectral gap Now we revisit one variational formula of the spectral gap for the birth-death process (see [15, 16] for complete background). Let A be the set of all real increasing functions ρ on N such that ρ L 2 (π). As an application of Theorem 7.2.1, we have the following result, which characterizes the spectral gap of L in L 2 (π). Theorem 7.3.1. Let λ 1 be the spectral gap of L in L 2 (π), then we have where I(ρ) is the same as in Theorem 7.2.1. Proof of Theorem 7.3.1. Taking any ρ A, let λ 1 sup I(ρ) 1, (7.3.1) ρ A D = L 2 (π) C Lip(ρ), a dense subset of L 2 (π), where represents zero mean. Since for any Lipschitzian function f L 2 (π), we could replace it by f π(f) which belongs to D, it is sufficient to work only on D in the following. We assume below that I(ρ) is finite otherwise λ 1 is the trivial case. L is self-adjoint and negative definite, so it admits the spectral decomposition on L 2 (π) (see [11]), L = λde λ, (7.3.2) (,+ ) then ( L) 1 = λ 1 de λ. (7.3.3) (,+ ) We will show that for any λ (, I(ρ) 1 ), E λ f = holds for any f D. Theorem 7.2.1 and the finiteness of I(ρ) guarantee that for any n 1, ( L) n f is in C Lip(ρ) once f C Lip(ρ). Precisely for any f D, ( L) n f Lip(ρ) I(ρ) n f Lip(ρ). (7.3.4)

152 CHAPITRE 7. SPECTRAL GAP OF BIRTH-DEATH PROCESS On the other hand, for any g C Lip(ρ), we have ρ(i) ρ() = i (ρ(k) ρ(k 1)) k=1 i k=1 g 1 Lip(ρ) (g(k) g(k 1)) = g 1 Lip(ρ) (g(i) g()), which implies π i (ρ(i) ρ()) 2 g 2 Lip(ρ) i= π i (g(i) g()) 2 V ar π (g), (7.3.5) i= since ρ(i) ρ() is strictly positive and ρ is in L 2 (π). Therefore, for ( L) n f, we have ρ ρ() 2 L 2 (π) ( L) n f 2 Lip(ρ) V ar π (( L) n f) Combining above inequality with (7.3.4), we get = ( L) n f 2 L 2 (π) = λ 2n d < E λ f, f > π (,+ ) λ 2n < E λ f, f > π. < E λ f, f > π C(f)(λ I(ρ)) 2n, (7.3.6) where C(f) = ( ρ ρ() L 2 (π) f Lip(ρ) ) 2 a finite constant independent of n. Since n is arbitrary, letting n +, we have E λ f = for any f D. As discussed above, we have E λ f = for any f L 2 (π), which means E λ =. Equivalently λ 1 I(ρ) 1, then we have λ 1 sup I(ρ) 1. (7.3.7) ρ A

7.4. CONVEX CONCENTRATION INEQUALITY 153 Remark 7.3.2. Using analytic method, Chen in [15] obtained some variational formula for spectral gap, one of which is equivalent to say by our notations, λ 1 = sup I(ρ) 1, (7.3.8) C where C L 1 (π) is the set of all increasing function ρ such that i 1 π iρ(i) >. He proved that the eigenvector ρ of L with respect to λ 1 is increasing and thereby the equality in (7.3.8) is attained at ρ. Remark 7.3.3. By simple calculus, I(ρ ) 1 = λ 1. Therefore if moreover ρ is in L 2 (π), then with the increasingness of ρ, the sign of the equality in (7.3.7) is attained. That is to say, we have also an equality. Unfortunately, limited by the valid ity of the important inequality (7.3.5), we do lose the class consisting of ρ L 1 (π) \ L 2 (π) even if we are not willing to. Then, to some extent, from the point of variational formula of the spectral gap, Theorem 7.3.1 brings nothing new. Even though, the birth of Theorem 7.2.1 and Lemma 7.2.3 is interesting (maybe have other applications). Remark 7.3.4. The criteria for spectral gap was invented by Miclo in [64] (1999) by using Hardy s inequality. Then based on the variational formulas such as (7.3.8), new criteria was given by Chen in [17] (2), which was more precise than that of Miclo and dug all potentials. So here we don t repeat Chen s hard work to waste the readers precious time just for nothing new. 7.4 Convex concentration inequality for additional functionals: method of Lyons-Zheng forward-backward martingale decomposition In this section, (N t ) t is always a standard Poisson process which is not necessarily independent of (X t ) t (if need, take an independent copy of (N t ) t which is independent

154 CHAPITRE 7. SPECTRAL GAP OF BIRTH-DEATH PROCESS of (X t ) t ) and let C c := {φ : φ C 2, φ is convex and φ is non-decreasing}. Two random variables F and G satisfy a convex concentration inequality if E[φ(F )] E[φ(G)] (7.4.1) for all convex functions φ : R R. This concept was firstly introduced by Hoeffding in 1963 (see [42]) and was realized for general martingales by Klein al. in [54] (26). By a classical argument, the application of (7.4.1) to φ(x) = exp(λx), λ >, entails the deviation bound P (F x) inf λ> E[eλ(F x) 1 {F x} ] inf λ> E[eλ(F x) ] inf λ> E[eλ(G x) ], (7.4.2) x >, hence the deviation probabilities for F can be estimated via the Laplace transform of G, that is the reason why much works have been done on this subject. Firstly, we recall one result for pure jump martingales (see [54] for more information): Theorem 7.4.1. Let M t be a pure jump martingale on some space (E, E) satisfying for all t, M t K and < M > t < +, where < M > t = sup ω E < M > t (ω). Then for any function φ C c and any t, we have, E[φ(M t E[M t ])] E [ φ(kn <M>t /K 2 < M > t /K) ]. (7.4.3) Moreover, (7.4.3) still holds if < M > t is replaced by R(t) < M > t. The above theorem derives the following lemma Lemma 7.4.2. Let f be a function satisfying K = sup k 1 f(k) f(k 1) < and { } Γ(f) = max b (f(1) f()) 2, sup(a k (f(k 1) f(k)) 2 +b k (f(k+1) f(k)) 2 ) k 1 <. For the birth-death process considered in the previous section, define Then for any function φ C c, M t = f(x t ) f(x ) E[φ(M t )] E t Lf(X s )ds. [ ( φ KN Γ(f) t/k 2 Γ(f) )] t. K

7.4. CONVEX CONCENTRATION INEQUALITY 155 Proof. Obviously, (M t ) t is a pure jump martingale and by its definition we have M t K, since M t sup f(k + 1) f(k). k On the other hand, for any t, by Ito s formula, E[Mt 2 ] = E t Γ(f)(X s)ds, hence < M > t = t which completes the proof by Theorem 7.4.1. Γ(f)(X s )ds Γ(f) t, Remark 7.4.3. Taking ρ as ρ(i) = i, i N ( i.e., the classic metric on N ), suppose that f C Lip(ρ) and Γ(f) <. We have for any function φ C c, [ )] E[φ(M f t )] E φ ( f Lip(ρ) N Γ(f) Γ(f) t/ f 2Lip(ρ) t/ f Lip(ρ). Now given g a function on N, without lost of generality, suppose that π(g) =, we consider functionals S t = t g(x s)ds. In the following, we will prove concentration convex inequalities for S t and then as introduced at the beginning of this section some deviation inequalities are given. Given T a constant( could be large enough if need but fixed ). For any t T, let and M t = f(x t ) f(x ) M t = f(x ) f(x t ) t t Lf(X s )ds Lf(X s )ds, where f satisfies Lf = g. Obviously we have 2S t = M t + M t. Define F t = σ(x s, s t), G t = σ(x s, t s T ). By the theorem of Lyons-Zheng s forward-backward martingale decomposition (see [58, 91]), M t is an additive F t forward Càdlàg pure jump martingale and M t is additive Càdlàg such that ( M T M T t, t T ) is a G T t -backward pure jump martingale. Moreover, since (X t ) t is a symmetric process, M t and M t have the same distribution with respect to P π. For simplicity, we use E to denote the expectation with respect to P π. Therefore we have for any convex function φ, M t + M t E[φ(S t )] = E[φ( )] 2 E[φ( M t ) + φ( M t )] 2 = E[φ( M t )].

156 CHAPITRE 7. SPECTRAL GAP OF BIRTH-DEATH PROCESS If we impose some conditions on g such that the solution f of the equation Lf = g verifies the conditions in the previous Lemma 7.4.2, then we can have convex concentration inequality for S t, the following essential hypothesis, Hypothesis A: Hypothesis B: sup k 1 K := sup k 1 k 1 i= a k ( k 1 i= π jg(j) π k a k π i g(i) π k a k <, ) 2 ( k ) i= + b π 2 jg(j) k π k+1 a k+1 <. Remark 7.4.4. By Lemma 7.2.2, we have K = sup k 1 f(k) f(k 1) and Hypothesis B is equivalent to say { Γ(f) = max b (f(1) f()) 2, Consequently we have the following } sup a k (f(k 1) f(k)) 2 +b k (f(k+1) f(k)) 2 <. k 1 Theorem 7.4.5. With Hypothesis A and B, we have for any φ C c, E[φ(S t )] E[φ(KN Γ(f) t/k 2 Γ(f) t/k)]. Remark 7.4.6. Applying the theorem above to φ(x) = e λx (λ > ), we have a deviation inequality for S t, i.e., P (S t E[S t ] x) inf exp{ λx + Γ(f) t(e λk λk 1)/K 2 } λ> { = exp Γ(f) ( )} t Kx h, K 2 Γ(f) t where h(u) = (1 + u) log(1 + u) u. Moveover, taking x = ty, we also obtain ( ) { 1 P t (S t E[S t ]) y exp Γ(f) ( )} t Ky h. (7.4.4) K 2 Γ(f) Remark 7.4.7. (7.4.4) is comparable to that obtained recently by Joulin [48] under different conditions and whose proof relies on the Wasserstein s curvature. Even though with our method, our hypothesis are quite natural. But any way it seems too complicated to be verified. Now we give two typical examples of birth-death process to show how to satisfy these hypothesis. In order to avoid repeating too much, we just give corresponding concentration inequalities for these two examples and the deviation inequalities omitted could be derived like Remark 7.4.6.

7.4. CONVEX CONCENTRATION INEQUALITY 157 7.4.1 Two classic examples The M/M/1 queueing process The M/M/1 queueing process is a simple birth-death process whose generator is given by, for any function f on N, Lf(i) = λ(f(i + 1) f(i)) + ν1 i (f(i 1) f(i)), i N, where the positive numbers λ and ν correspond respectively to the input rate and service rate of the queue: the independent and identically distributed interarrival times and independent and identically distributed service times of the customers follow an exponential law with respective parameters λ and ν. We assume here that σ := λ < 1, then the ν reversible stationary measure π of this ergodic process is the geometric distribution with parameter σ, i.e., π k = (1 σ)σ k, k. For this simple example, we have Proposition 7.4.8. Let (X t ) be the M/M/1 queueing process as above. Suppose that a function g on N satisfies π(g) = and K := 1 ν λ sup k 1 i= π ig(i + k) <. Therefore for any function φ C c, we have E[φ(S t )] E[φ(KN (λ+ν)t (λ + ν)kt)]. (7.4.5) Proof. For (7.4.5), by Theorem 7.4.5, what we need to do is to verify the Hypothesis A and B. Indeed, k 1 sup k 1 π i g(i) π k a k = sup 1 k 1 ν = 1 ν sup k 1 = 1 ν sup k 1 k 1 σ i g(i)/σ k i= σ i g(i)/a k i=k σ i g(i + k) i= = 1 ν (1 σ) 1 sup k 1 = K <, + i= π i g(i + k)

158 CHAPITRE 7. SPECTRAL GAP OF BIRTH-DEATH PROCESS which means Hypothesis A. Consequently we have Γ(f) (λ + ν)k 2 < +, Hypothesis B. Corollary 7.4.9. Given a > 1, for all i N, let ρ(i) = i k=1 1/ka. Then for any g C Lip(ρ) and φ C c, we have, [ ( ρ E[φ(S t )] E φ ν λ g Lip(ρ)N (λ+ν)t ρ )] ν λ (λ + ν) g Lip(ρ)t, where ρ := sup k 1 ρ(k). Proof. By definition, π i g(i + k) = π i (g(i + k) g(i)) which deduces that i= sup k 1 i= π i (ρ(i + k) ρ(i)) g Lip(ρ) i= = (1 σ) g Lip(ρ) (1 σ) g Lip(ρ) = g Lip(ρ) k j=1 1 j a, i= σ i σ i i= k j=1 k j=1 π i g(i + k) g Lip(ρ) ρ. i= 1 (j + i) a So the condition in Proposition 7.4.8 is satisfied with K = 1 (ν λ) g Lip(ρ) ρ 1 j a The M/M/ queueing process The M/M/ model is a particular birth-death process whose generator L satisfying for any functional f on N, we have Lf(i) = λ(f(i + 1) f(i)) + νi(f(i 1) f(i)),

7.4. CONVEX CONCENTRATION INEQUALITY 159 where λ, ν are two positive numbers. Then this process is ergodic with reversible stationary distribution π, the Poisson measure on N with parameter σ, i.e., σ σk π k = e k!, k N. For this model, we have the following proposition: k 1 Proposition 7.4.1. Given a function g satisfying sup k 1 k i= σi g(i)/i! νkσ k k! K is a positive constant and π(g) =. Then for any φ C c, we have [ ( t )] [ ( )] E φ g(x s )ds E φ KN (λ+ν)t (λ + ν)kt. Proof. As for the M/M/1 model, we verify Hypothesis A and B. By conditions, sup k 1 sup k 1 k 1 π i g(i) i= π k a k = sup k 1 K k = K, k 1 i= σ i 1)! g(i) (k i! νσ k K, where which implies Hypothesis A and moreover f(k 1) f(k) K k. As a consequence, Γ(f) = max{λ(f(1) f()) 2, sup νk(f(k 1) f(k)) 2 + λ(f(k) f(k + 1)) 2 } k 1 max{λ(f(1) f()) 2, sup k 1 (λ + ν)k 2, the Hypothesis B. νk(f(k 1) f(k)) 2 + sup λ(f(k + 1) f(k)) 2 } k 1 Corollary 7.4.11. Taking ρ as ρ(i) = i. Given g C Lip(ρ), then for any function φ C c, we have [ ( (1 + σ + σe σ ) E[φ(S t )] E φ g Lip(ρ) N (λ+ν)t (λ + ν) (1 + σ + )] σeσ ) g Lip(ρ) t. ν ν Proof. Without loss of generality, we suppose that g Lip(ρ) = 1. Since for any i k, k! i!

16 CHAPITRE 7. SPECTRAL GAP OF BIRTH-DEATH PROCESS 1, we have (i k)! k 1 sup k k 1 i= σi g(i)/i! σ k νk/k! + i=k sup σi (ρ(i) π(ρ))/i! k 1 σ k ν k/k! sup k 1 1 ν 1 ν i= i= σ i i! σ i i! i + 1 i + k k (1 + σ + σe σ )/ν, then Proposition 7.4.1 is satisfied with K = (1 + σ + σe σ )/ν.

Bibliographie [1] Aldous D.J., Exchangeability and related topics. In École d été de probabilités de Saint-Flour, XIII-1983, Lecture NOtes in Math., 1117 1-198, Springer, Berlin, 1985. [2] Alós E. and Ewald C.O., A note on the Malliavin differentiability of the Heston volatility. Economics and Business Working Papers Series, No 88, Universitat Pompeu Fabra, 25. [3] Ané C. and Ledoux M., On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields, 116(4) 573 62, 2. [4] Bakry D. and Michel D., Sur les inégalités FKG. In Séminaire de Probabilités, XXVI, volume 1526 of Lecture Notes in Math., pages 17 188. Springer, Berlin, 1992. [5] Barbato D., FKG inequality for Brownian motion and stochastic differential equations. Electron. Comm. Probab., 1 7-16 (electronic), 25. [6] Bass R.F., Stochastic differential equations driven by symmetric stable processes, In Séminaire de Probabilités, XXXVI, 32 313, Lecture Notes in Math., 181, Springer, Berlin, 23. [7] Bentkus V., On Hoeffding s inequalities. Ann. Probab., 32(2) 165 1673, 24. [8] L. Bertini, N. Cancrini and F. Cesi, The spectral gap for a Glauber-type dynamics in a continuous gas, Ann. Inst. H. Poincaré-PR 38, 1(22), 91-18. [9] Bretagnolle J., Statistique de Kolmogorov-Smirnov pour un échantillon non équirépartie dans l aspects statistiques et aspects physiques des processus gaussiens. Paris: Editions du centre nationnal de la recherche scientifique, 1981. [1] Burkholder D.L., Strong differential subordination and stochastic integration. Ann. Probab., 22(2):995 125, 1994. [11] Capitaine M., Hsu E.P. and Ledoux M., Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces. Electron Comm. Probab., 2: 71-81(electric), 1997. 161

162 BIBLIOGRAPHIE [12] F. Cesi, Quasi-factorisation of entropy and log-sobolev inequalities for Gibbs random fields, Prob. Th. Rel. Fields, 12(21), 569-584. [13] Chaganty N.R., Bootstrap applications in Statistics, preprint. [14] Chen J.W., Inference on prior distribution. Preprint. [15] M. F. Chen, Estimation of spectral gap for Markov chains, Acta Math Sin. New ser., 1996, 12(4), 337-36. [16] M.F. Chen, Analytic proof of dual variational formula for the first eigenvalue in dimension one. Sci. Sin. (A). 1999, 42(8), 85-815. [17] M. F. Chen, Explicit bounds of the first eigenvalue. Sci. china (A). 2 43(1), 151-159. [18] M. F. Chen, Variational formulas and approximation theorems for the first eigenvalue. Sci. China (A), 21, 44(4), 49-418. [19] M. F. Chen, Eigenvalues, inequalities and Ergodic Theory, Springer, 25. [2] M. F. Chen and F.Y. Wang, Application of coupling method to the first eigenvalue on manifold, Sci. Sin. (A), 1993, 23(11), 113-114 ( Chinese Edition); 1994, 37(1), 1-14 (English Edition). [21] M. F. Chen and F.Y. Wang, Estimation of spectral gap for elliptic operators, Trans. Amer. Math. Soc. 1997, 349(3). 1239-1267. [22] Chen X., The moderate bounds of independent random variables in a Banach space, Chinese J. Appl. Probab. Statist., 7 24-32, 1991. [23] Cramér H., Sur un nouveau théoème limite de la théorie de probabilités., Actualités Scientifiques et industrielles, 736, 5-23. Colloque consacré à la théorie de probabilités. Vol.3, Hermann, Paris, 1938. [24] Daras T., Large deviations for the empirical measures of an exchangeable sequence, Statistics and Probab. Letters, 36 91-1, 1997. [25] Daras T., Large deviations for empirical process of a symmetric measure: a lower bound. Preprint 22. [26] De Acosta A., Upper bounds for large deviations of dependent random variables. Z. Wahr. verw. Gebiete 69, 551-565, 1985. [27] De Acosta A., Moderate deviations for emprical measures of Markov Chains:lower bounds, Annals of Probability., 25 259-284, 1997. [28] Dellacherie C. and Meyer P. A., Probabilités et potentiel. Chapitres V à VIII, Théorie des martingales, 1385, Hermann, Paris, 198.

BIBLIOGRAPHIE 163 [29] Dellacherie C., Maisonneuve B., and Meyer P. A., Probabilités et Potentiel, volume 5. Hermann, 1992. [3] Dembo A. and Zeitouni O., Large deviations Techniques and Applications. Second Edition. Springer, New York, 1998. [31] Donsker M. D., Varadhan S. R. S., Asymptotic evaluation of certain Markov process expectations for large time, I-IV, Comm. Pur. Appl. Math., 28(1975), 1-47 and 279-31; 29(1976), 389-461; 36(1983), 183-212. [32] Dinwoodie I.H. and Zabell S.L., Large deviations for exchangeable random vectors, Ann. Probab., 2 1147-1166, 1992. [33] Eaton M. L., Lectures on Tpoics in Probability Inequalities. CWI Trac, 35. [34] Ellis R.S., Large deviations and Statistics Mechanics, Springer, Berlin, 1985. [35] Émery M., On the Azéma martingales. In Séminaire de Probabilités XXIII, Lecture Notes in Mathematics, 1372 66-87, Springer Verlag, 199. [36] Fortuin C.M., Kasteleyn P.W., Ginibre J., Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22 89-13, 1971. [37] Gao F.Q., Characterization of moderate deviations for Markov process, In Advance in Mathematics. Bejing University., 1993. [38] Gao F.Q., Moderate deviations for martingales and mixing random processes, Stochastic Processes and their Applications., 61 263-275, 1996. [39] Giné E. and Marcus M.B., The central limit theorem for stochastic integrals with respect to Lévy processes. Ann. Probab., 11(1) 58-77, 1983. [4] Gross K.I., Richards D. ST. P., Total positivity, finite reflection groups and a formula of Harish-Chandra. J. Approx. Theory, 82 6-87, 1995. [41] Herbst I., Pitt L., Diffusion equation techniques in stochastic monotonicity and positive correlations. Probab. Theory Related Fields 87 275-312, 1991. [42] Hoeffding W., Probability inequalities for sums of bounded random variables. J. Amer. Stat. Assoc, 58 13-3, 1963. [43] Houdré C. and Privault N., Concentration and deviation inequalities in infinite dimensions via covariance representations. Bernoulli, 8(6) 697 72, 22. [44] Hu Y. Z., Itô-Wiener chaos expansion with exact residual and correlation, variance inequalities. J. Theoret. Probab., 1(4) 835-848, 1997. [45] Jacod J., Calcul stochastique et problèmes de martingales, volume 714 of Lecture Notes in Mathematics. Springer Verlag, 1979.

164 BIBLIOGRAPHIE [46] Janson S., Bounds of the distributions of extremal values of a scaning process. Stoch. Proc. Appl., 18 313-328, 1984. [47] Jacod J. and Mémin J., Caractéristiques locales et conditions de continuité absolue pour les semi-martingales. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 35(1) 1-37, 1976. [48] Joulin A., Concentration et fluctuations de processus stochastique avec sauts, Thèse, Université de La Rochelle, 26. [49] Kallenberg O., Some time change representations of stable integrals, via predictable transformations of local martingales. Stochastic Process. Appl., 4(2) 199-223, 1992. [5] Kallenberg O., Foundations of modern probability. Probability and its Applications. Springer-Verlag, New York, 22. [51] Karlin S. and Rinott Y., Classes of orderings of measures and related correlation inequalities I. Multivariate totally positive distributions, J. Multivariate Anal. 1 467-498, 198. [52] Kato T., Perturbation Theory for Linear Operators, 2nd ed. (2nd corrected printing), Springer, Berlin, 1984. [53] Klein Th. Inégalités de concentration, martingales et arbres aléatoires. Thèse, Université de Versailles-Saint-Quentin, 23. [54] Klein T., Ma Y. T., and Privault N., Convex concentration inequalities and forwardbackward stochastic calculus, Electronic Journal of Probab., 11 486-512, 26. [55] Ledoux M., Sur les déviations modérées des sommes de variables aléatoires vectorielles independantes de même loi, Ann. Inst. H. Poincaré., 28 267-28, 1992. [56] Liggett T., Iteracting particle systems, Springer, New York, 1985. [57] Liu W. and Ma Y. T., Spectral gap and convex concentration inequality for birth-death process, preprint. [58] Lyons T. J. and Zheng W. A., A crossing estimate for the canonical process on a Dirithlet space and a tightness result, Astérique, Vol 157-158 (Colloque P. Lévy), 1988, 249-271. [59] Ma Y. T., Convex concentration inequalities for continuous gas and stochastic domanation, preprint. [6] Ma Y. T. and Privault N., FKG inequality on the Wiener space via predictable representation, preprint.

BIBLIOGRAPHIE 165 [61] Ma Y. T., Song Q. X. et Wu L. M., Large deviation principle for exchangeble sequences with respect to τ topology: sufficient and necessary conditions, Statistics and Probability Letters, 77(3), 27, 239-246. [62] Ma Y. T. and Song Q. X., Moderate deviation for additive Lipschitz functionals of Markov process, Acta. Math. Sino.(Chinese Series), 5(1), 27, 1-1. [63] Meyn S. P. and Tweedie R. L., Markov Chains and Stochastic Stability, 3rd ed. (3rd corrected printing), Springer-Verlag, 1996. [64] L. Miclo, An exemple of application of discrete Hardy s inequalities, Markov Process and Related Fields, 5, 1999, No 3, 319-33. [65] Nguyen X. X., Zessin H., Integral and differential characterizations of the Gibbs process. Math Nachr., 88 15-115, 1979. [66] Nualart D., Markov fields and transformations of the Wiener measure. In T. Lindstrom, B. Oksendal, and A. S. Üstünel, editors, Proceedings of the Fourth Oslo-Silivri Workshop on Stochastic Analysis, Stochastics Monographs, 8 Oslo, 1993. Gordon and Breach. [67] Nualart D., The Malliavin Calculus and Related Topics. Probability and its Applications. Springer-Verlag, 1995. [68] Nualart D. and Vives J., Anticipative calculus for the Poisson process based on the Fock space. In Séminaire de Probabilités XXIV, volume 1426 of Lecture Notes in Math., 1426 154-165, Springer, Berlin, 199. [69] Ocone D., A guide to the stochastic calculus of variations. In H. Körezlioǧlu and A.S. Üstünel, editors, Stochastic Analysis and Related Topics, volume 1316 of Lecture Notes in Mathematics, Silivri, Springer-Verlag, 1988. [7] Oksendal B. and Proske F., White noise of Poisson random measures. Potential Anal., 21(4) 375-43, 24. [71] Ma J., Protter Ph., and San Martin J., Anticipating integrals for a class of martingales. Bernoulli, 4 81-114, 1998. [72] Picard J., Formule de dualité sur l espace de Poisson, Ann. Inst. H. Poincaré(Prob. Stat.), 32(4), 59-548(1996). [73] Pinelis I., Optimal Bounds for the distribution of martingales in Banach spaces. Ann. Prob., 22(4) 1679-176, 1994. [74] Pinelis I., Optimal tail comparison based on comparison of moments. In High dimensional probability (Oberwolfach, 1996), Progr. Probab., 43 297-314, Birkhäuser, Basel, 1998.

166 BIBLIOGRAPHIE [75] Preston C. J., A generalization of the FKG inequalities. Comm. Math. Phys., 36 233-241, 1974. [76] Preston C. J., Random Fields. Lecture notes in Math. 534 Springer, New York, 1976. [77] Privault N., An extension of stochastic calculus to certain non-markovian processes. Prépublication 49, Université d Evry, 1997. [78] Privault N., Equivalence of gradients on configuration spaces. Random Operators and Stochastic Equations, 7(3) 241 262, 1999. [79] Privault N. and Schoutens W., Discrete chaotic calculus and covariance identities. Stochastics and Stochastics Reports, 72:289 315, 22. Eurandom Report 6, 2. [8] Protter Ph. Stochastic integration and differential equations: A new approach. Springer-Verlag, Berlin, 199. [81] Rudin W., Functional Analysis, International Series in Pure and Applied Mathematics,second Edition, 1991. [82] Ruelle D., Statistical Mechanics: Rigorous Results, Benjamin, 1969. [83] Schilder M., Some asymptotic formulaes for Wiener integrals, Trans. Amer. Math. Soc., 125 63-85, 1966. [84] Shao Q. M., A comparison theorem on moment inequalities between negatively associated and independent random Variables. Journ of Theor. Prob., 13(2) 343-356, 2. [85] Trashorras J., Large deviations for a triangular array of exchangeble random variables. Ann. Inst. H. Poincaré Probab. Stat., 38(5) 649-68, 22. [86] Trashorras J., On finite exchangeble sequences, Prepint. [87] Üstünel A. S., An introduction to analysis on Wiener space, Lecture Notes in Mathematics, 161, Springer Verlag, 1995. [88] Varadhan S. R. S., Large deviations and applications, Ecole d Eté de Probabilités de Saint-Flour XV-XVII, 1985. [89] Wang Z. K. and Yang X. Q., Birth-Death processes and Markov chains, Academic Press of China, Beijing, 25(in Chinese). [9] Watanabe S., Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Institute of Fundamental Research, 1984. [91] Wu L. M., Large deviation for the empirical fields of symmetric measures, Chinese Annals of Mathematics,12B : 3, 1991.

BIBLIOGRAPHIE 167 [92] Wu L. M., Moderate deviations of dependent random variables related to CLT., The Annals of Probability, 23 42-445, 1995. [93] Wu L. M., An introduction to large deviations (in Chinese), pp225-336 in Several Topics in Stochastic Analysis (authors : J.A. Yan, S. Peng, S. Fang, L. Wu), Press of Sciences of China, Beijing, 1997. [94] Wu L. M., Forward-backward martingale decomposition and compactness results for additive functionals of stationary ergodic Markov process. Annals de l I.H.P., B, Tome 35(2), 1999, 121-141. [95] Wu L. M., A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields, 118(3) 427-438, 2. [96] Wu L. M., Large and moderate deviations and exponential convergence for stochastic damping Hamilton systems, Stochastic Processes and their applications, 91(2) 25-238, 21. [97] Wu L. M., Essential spectral radius for Markov semigroups(i):discrete time case. Probab. Th. Rel. Fields, (23). [98] Wu L. M., Large deviation principle for exchangeable sequences: necessary and sufficient condition, J. of Theor. Probab., 17(4) 967-978, 24. [99] Wu L. M., Estimate of spectral gap for contiunous gas, Ann. Inst. H. Poincaré(Prob. Stat.), 24, 4, 387-49. [1] Wu L. M., Preprint, 26. [11] Yosida K., Functional Analysis, Sixth Edition, Spring-Verlag, 1999.