Non-uniform Edgeworth expansions for weakly dependent random variables and their applications

TL;DR

This paper develops non-uniform Edgeworth expansions for weakly dependent, non-stationary sequences of random variables, extending classical results beyond independent cases and applying them to distribution estimates, moments, and Wasserstein distances.

Contribution

It provides the first non-uniform Edgeworth expansions for weakly dependent, non-stationary sequences, with multiple applications including distribution approximation and Wasserstein distance bounds.

Findings

Established non-uniform Edgeworth expansions for various dependent sequences.

Derived Berry-Esseen theorems in Wasserstein metrics.

Provided moment and distribution function approximations.

Abstract

We obtain non-uniform Edgeworth expansions for several classes of weakly dependent (non-stationary) sequences of random variables, including uniformly elliptic inhomogeneous Markov chains, random and time-varying (partially) hyperbolic or expanding dynamical systems, products of random matrices and some classes of local statistics. To the best of our knowledge this is the first time such results are obtained beyond the case of independent summands, even for stationary sequences. As an application of the non uniform expansions we obtain average versions of Edgeworth exapnsions, which provide estimates of the underlying distribution function in $L^p(dx)$ by the standard normal distribution function and its higher order corrections. An additional application is to expansions of expectations $\bbE[h(S_n)]$ of functions $h$ of the underlying sequence $S_n$, whose derivatives grow at most polynomially fast. In particular we provide expansions of the moments of $S_n$ by means the variance of $S_n$. A third application is to Edgeworth expansions in the Wasserstein distance (transport distance). In particular we prove Berry-Esseen theorems in the Wasserstein metrics. This paper compliments \cite{NonU BE} where non-uniform Berry-Esseen theorems were obtained.