A Berry-Esseen theorem and Edgeworth expansions for uniformly elliptic inhomogeneous Markov chains

TL;DR

This paper establishes a Berry-Esseen theorem and Edgeworth expansions for sums of functions of inhomogeneous Markov chains, providing optimal conditions and extending results to cases with various regularity assumptions on the functions involved.

Contribution

It proves the first Berry-Esseen theorem without extra assumptions and derives optimal Edgeworth expansions for inhomogeneous Markov chains under different regularity conditions.

Findings

Berry-Esseen theorem holds without additional assumptions.

Optimal order Edgeworth expansions for functions with decay rates.

Extensions to Markov chains on manifolds with Lipschitz and H"older functions.

Abstract

We prove a Berry-Esseen theorem and Edgeworth expansions for partial sums of the form $S_N=\sum_{n=1}^{N}f_n(X_n,X_{n+1})$, where $\{X_n\}$ is a uniformly elliptic inhomogeneous Markov chain and $\{f_n\}$ is a sequence of uniformly bounded functions. The Berry-Esseen theorem holds without additional assumptions, while expansions of order $1$ hold when $\{f_n\}$ is irreducible, which is an optimal condition. For higher order expansions, we then focus on two situations. The first is when the essential supremum of $f_n$ is of order $O(n^{-\be})$ for some $\be\in(0,1/2)$. In this case it turns out that expansions of any order $r<\frac1{1-2\be}$ hold, and this condition is optimal. The second case is uniformly elliptic chains on a compact Riemannian manifold. When $f_n$ are uniformly Lipschitz continuous we show that $S_N$ admits expansions of all orders. When $f_n$ are uniformly H\"older continuous with some exponent $\al\in(0,1)$, we show that $S_N$ admits expansions of all orders $r<\frac{1+\al}{1-\al}$. For H\"older continues functions with $\al<1$ our results are new also for uniformly elliptic homogeneous Markov chains and a single functional $f=f_n$. In fact, we show that the condition $r<\frac{1+\al}{1-\al}$ is optimal even in the homogeneous case.