Moderate deviations for stationary sequences of Hilbert valued bounded random variables

TL;DR

This paper establishes a moderate deviation principle for stationary Hilbert space-valued bounded random variables using martingale techniques and a new inequality, with applications to statistical tests and stochastic processes.

Contribution

It introduces a novel Hoeffding inequality for non-adapted Hilbert-valued sequences and derives conditions for moderate deviations based on martingale approximations.

Findings

Established moderate deviation principles under martingale-type conditions.

Developed a new Hoeffding inequality for Hilbert-valued sequences.

Applied results to Cramer-Von Mises statistics and Markov chains.

Abstract

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 adpated sequences of Hilbert-valued random variables. Applications to Cramer-Von Mises statistics, functions of linear processes and stable Markov chains are given.