Edgeworth expansions for integer valued additive functionals of uniformly elliptic Markov chains

TL;DR

This paper derives explicit asymptotic Edgeworth expansions for the probabilities of partial sums of integer-valued additive functionals of uniformly elliptic Markov chains, with conditions for their validity based on conditional distribution uniformity.

Contribution

It provides explicit formulas for Edgeworth expansions of integer-valued additive functionals of Markov chains without extra assumptions, linking their accuracy to conditional distribution properties.

Findings

Explicit asymptotic expansions for probabilities of sums.

Edgeworth expansions hold under specific conditional distribution conditions.

Characterization of when standard Edgeworth expansions are valid.

Abstract

We obtain asymptotic expansions for probabilities $\mathbb{P}(S_N=k)$ of partial sums of uniformly bounded integer-valued functionals $S_N=\sum_{n=1}^N f_n(X_n)$ of uniformly elliptic inhomogeneous Markov chains. The expansions involve products of polynomials and trigonometric polynomials, and they hold without additional assumptions. As an application of the explicit formulas of the trigonometric polynomials, we show that for every $r\geq1\,$, $S_N$ obeys the standard Edgeworth expansions of order $r$ in a conditionally stable way if and only if for every $m$, and every $\ell$ the conditional distribution of $S_N$ given $X_{j_1},...,X_{j_\ell}$ mod $m$ is $o_\ell(\sigma_N^{1-r})$ close to uniform, uniformly in the choice of $j_1,...,j_\ell$, where $\sigma_N=\sqrt{\text{Var}(S_N)}.$