Let be a separable geodesic Gromov-hyperbolic space, a basepoint and a countably supported non-elementary probability measure on . Denote by the random walk on driven by the probability measure . Supposing that has a finite exponential moment, we give a second-order Taylor expansion of the large deviation rate function of the sequence and show that the corresponding coefficient is expressed by the variance in the central limit theorem satisfied by the sequence . This provides a positive answer to a question raised in [6]. The proof relies on the study of the Laplace transform of at the origin using a martingale decomposition first introduced by Benoist–Quint together with an exponential submartingale transform and large deviation estimates for the quadratic variation process of certain martingales.
Soit un espace Gromov-hyperbolique, géodésique et séparable, un point base et une mesure de probabilité non élémentaire et à support dénombrable sur le groupe des isométries de . Notons par la marche aléatoire sur induite par . Sous l’hypothèse de moment exponentiel fini de , nous donnons un développement de Taylor d’ordre de la fonction de taux des grandes déviations de la suite de variables aléatoires et exprimons la dérivée seconde en la vitesse de fuite en fonction de la variance dans le théorème central limite que vérifie la suite . Cela répond par l’affirmative à une question posée dans [6]. La preuve s’appuie sur l’étude de la transformée de Laplace de en zéro en utilisant une approximation par une martingale introduite pour la première fois par Benoist-Quint, combinée avec une transformée exponentielle de martingales et des estimées de grandes déviations pour le crochet de certaines martingales.
Accepted:
Published online:
Keywords: Random walks, hyperbolic spaces, martingales, large deviations, central limit theorem
Mot clés : Marches aléatoires, espaces hyperboliques, martingales, grandes déviations, théorème central limite
Richard Aoun 1; Pierre Mathieu 2; Cagri Sert 3
@article{JEP_2023__10__549_0, author = {Richard Aoun and Pierre Mathieu and Cagri Sert}, title = {Random walks on hyperbolic spaces: second order expansion of the rate function at the drift}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {549--573}, publisher = {\'Ecole polytechnique}, volume = {10}, year = {2023}, doi = {10.5802/jep.225}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.225/} }
TY - JOUR AU - Richard Aoun AU - Pierre Mathieu AU - Cagri Sert TI - Random walks on hyperbolic spaces: second order expansion of the rate function at the drift JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 549 EP - 573 VL - 10 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.225/ DO - 10.5802/jep.225 LA - en ID - JEP_2023__10__549_0 ER -
%0 Journal Article %A Richard Aoun %A Pierre Mathieu %A Cagri Sert %T Random walks on hyperbolic spaces: second order expansion of the rate function at the drift %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 549-573 %V 10 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.225/ %R 10.5802/jep.225 %G en %F JEP_2023__10__549_0
Richard Aoun; Pierre Mathieu; Cagri Sert. Random walks on hyperbolic spaces: second order expansion of the rate function at the drift. Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 549-573. doi : 10.5802/jep.225. https://jep.centre-mersenne.org/articles/10.5802/jep.225/
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