Moderate deviations for stationary sequences of Hilbert-valued bounded random variables

In this paper, we derive the Moderate Deviation Principle for stationary sequences of bounded random variables with values in a Hilbert space. The conditions obtained are expressed in terms of martingale-type conditions. The main tools are martingale approximations and a new Hoeffding inequality for non-adapted sequences of Hilbert-valued random variables. Applications to Cramér–Von Mises statistics, functions of linear processes and stable Markov chains are given.