Random walks on hyperbolic spaces: second order expansion of the rate function at the drift
[Marches aléatoires sur les espaces hyperboliques : dérivée seconde en la vitesse de fuite de la fonction de taux des grandes déviations]
Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 549-573.

Soit (X,d) un espace Gromov-hyperbolique, géodésique et séparable, oX un point base et μ une mesure de probabilité non élémentaire et à support dénombrable sur le groupe Isom(X) des isométries de X. Notons par z n la marche aléatoire sur X induite par μ. Sous l’hypothèse de moment exponentiel fini de μ, nous donnons un développement de Taylor d’ordre 2 de la fonction de taux des grandes déviations de la suite de variables aléatoires 1 nd(z n ,o) et exprimons la dérivée seconde en la vitesse de fuite en fonction de la variance dans le théorème central limite que vérifie la suite d(z n ,o). Cela répond par l’affirmative à une question posée dans [6]. La preuve s’appuie sur l’étude de la transformée de Laplace de d(z n ,o) 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.

Let (X,d) be a separable geodesic Gromov-hyperbolic space, oX a basepoint and μ a countably supported non-elementary probability measure on Isom(X). Denote by z n the random walk on X driven by the probability measure μ. Supposing that μ has a finite exponential moment, we give a second-order Taylor expansion of the large deviation rate function of the sequence 1 nd(z n ,o) and show that the corresponding coefficient is expressed by the variance in the central limit theorem satisfied by the sequence d(z n ,o). This provides a positive answer to a question raised in [6]. The proof relies on the study of the Laplace transform of d(z n ,o) 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.

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DOI : 10.5802/jep.225
Classification : 20F67, 60G50, 60G42, 60F10, 60F05
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

1 University Gustave Eiffel, Champs-sur-Marne 5 boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France
2 Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373 13453 Marseille, France
3 Institut für Mathematik, Universität Zürich 190, Winterthurerstrasse, 8057 Zürich, Switzerland
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Random walks on hyperbolic spaces: second order expansion of the rate function at the drift},
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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, Tome 10 (2023), pp. 549-573. doi : 10.5802/jep.225. https://jep.centre-mersenne.org/articles/10.5802/jep.225/

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