Large deviations for possibly reducible Markov chains on discrete state spaces

TL;DR

This paper establishes large deviation principles for discrete-state Markov chains without assuming irreducibility or exponential tightness, allowing for reducible chains and nonconvex rate functions.

Contribution

It provides an elementary, self-contained proof of level-2 and level-3 large deviation principles applicable to possibly reducible Markov chains.

Findings

Rate functions may be nonconvex due to reducibility.

The proof does not rely on irreducibility or exponential tightness.

Classical rate functions like Donsker-Varadhan may differ outside specific sets.

Abstract

We study the large deviations of Markov chains under the sole assumption that the state space is discrete. In particular, we do not require any of the usual irreducibility and exponential tightness assumptions. Using subadditive arguments, we provide an elementary and self-contained proof of the level-2 and level-3 large deviation principles. Due to the possible reducibility of the Markov chain, the rate functions may be nonconvex and may differ, outside a specific set, from the Donsker-Varadhan entropy and other classical rate functions.