Edgeworth expansions for independent bounded integer valued random variables

TL;DR

This paper develops asymptotic Edgeworth expansions for sums of bounded independent integer-valued variables, identifying conditions for their validity and exploring obstructions related to distributional stability and modular distribution properties.

Contribution

It provides the first comprehensive set of necessary and sufficient conditions for the validity of Edgeworth expansions for bounded integer-valued sums, including stability criteria.

Findings

Edgeworth expansions involve polynomial and trigonometric polynomial products.

Two main obstructions can prevent the validity of Edgeworth expansions.

A quantitative Prokhorov condition is sufficient and optimal for these expansions.

Abstract

We obtain asymptotic expansions for local probabilities of partial sums for uniformly bounded independent but not necessarily identically distributed integer-valued random variables. The expansions involve products of polynomials and trigonometric polynomials. Our results do not require any additional assumptions. As an application of our expansions we find necessary and sufficient conditions for the classical Edgeworth expansion. It turns out that there are two possible obstructions for the validity of the Edgeworth expansion of order $r$. First, the distance between the distribution of the underlying partial sums modulo some $h\in \bbN$ and the uniform distribution could fail to be $o(\sigma_N^{1-r})$, where $\sigma_N$ is the standard deviation of the partial sum. Second, this distribution could have the required closeness but this closeness is unstable, in the sense that it could be destroyed by removing finitely many terms. In the first case, the expansion of order $r$ fails. In the second case it may or may not hold depending on the behavior of the derivatives of the characteristic functions of the summands whose removal causes the break-up of the uniform distribution. We also show that a quantitative version of the classical Prokhorov condition (for the strong local central limit theorem) is sufficient for Edgeworth expansions, and moreover this condition is, in some sense, optimal.