Deviation inequalities for contractive infinite memory processes

In this paper, we introduce a class of processes that contains many natural examples. The interesting feature of such type processes lays on its infinite memory that allows it to record a quite ancient history. Then, using the martingale decomposition method, we establish some deviation and moment inequalities for separately Lipschitz functions of such a process, under various moment conditions on some dominating random variables. Our results generalize the Markov models of Dedecker and Fan [Stochastic Process. Appl., 2015] and a recent paper by Chazottes et al. [Ann. Appl. Probab., 2023] for the special case of a specific class of infinite memory models with discrete values. An application to stochastic gradient Langevin dynamic algorithm is also discussed.