Deviation inequalities for contractive infinite memory processes

Abstract

In this paper, we introduce a class of stochastic processes that encompasses many natural and widely used examples. A key feature of these processes is their infinite memory, which enables them to retain information from arbitrarily distant past states. Using the martingale decomposition method, we derive deviation and moment inequalities for separately Lipschitz functionals of such processes, under various moment conditions on certain dominating random variables. Our results extend those obtained for Markov chains by Dedecker and Fan [Stochastic Process. Appl., 2015], as well as recent results by Chazottes et al. [Ann. Appl. Probab., 2023] concerning specific infinite-memory models with sub-Gaussian concentration bounds. We also discuss an application to the stochastic gradient Langevin dynamics algorithm.