Large deviations for non-irreducible Markov chains on Euclidean spaces

TL;DR

This paper proves a large deviations principle for empirical measures of Markov chains on Euclidean spaces without requiring irreducibility, broadening the applicability of such results.

Contribution

It establishes the large deviations principle for Markov chains on Euclidean spaces under mild conditions, even when irreducibility is not assumed.

Findings

The large deviations principle holds without irreducibility.

The rate function may be non-convex without irreducibility.

The proof is self-contained and uses subadditivity.

Abstract

We establish the weak large deviations principle for empirical measures of Markov chains on $\mathbb R^d$ under mild assumptions. In particular, no irreducibility is assumed and the initial measure may be arbitrary. The proof is entirely self-contained and relies on subadditivity. In the absence of irreducibility, examples show that the rate function is not convex in general.