A martingale minimax exponential inequality for Markov chains

TL;DR

This paper introduces a new martingale-based minimax exponential inequality that provides improved large deviation bounds for the empirical measure of Markov chains, applicable to both compact and non-compact spaces.

Contribution

It develops a novel inequality leveraging martingale and minimax techniques to control large deviations in Markov chains, extending existing results to non-compact spaces.

Findings

Provides a partial improvement over Donsker-Varadhan estimates for compact spaces.

Establishes bounds conditioned on visits to compact subsets in non-compact spaces.

Offers a unified approach for large deviations in Markov chains using martingale methods.

Abstract

We prove a new inequality controlling the large deviations of the empirical measure of a Markov chain. This inequality is based on the martingale used by Donsker and Varadhan and the minimax theorem. It holds for convex sets and it requires to take an infimum over the starting point. In the case of a compact space, this inequality is a partial improvement of the large deviations estimates of Donsker and Varadhan. In the case of a non compact space, we condition on the event that the process visits $n$ times a compact subset of the space and we still obtain a control on the exponential scale.