Non-uniform bounds and Edgeworth expansions in self-normalized limit theorems

TL;DR

This paper develops non-uniform bounds for Edgeworth expansions in self-normalized limit theorems, improving tail accuracy and extending results to entropic CLT and total variation CLT under minimal moment conditions.

Contribution

It introduces new non-uniform bounds for Edgeworth expansions in self-normalized sums, including cases with non-integer moments, enhancing accuracy and applicability.

Findings

Established non-uniform bounds for CLT expansions with minimal moments

Derived Edgeworth-type expansion in entropic CLT

Achieved CLT in total variation distance for self-normalized sums

Abstract

We study Edgeworth expansions in limit theorems for self-normalized sums. Non-uniform bounds for expansions in the central limit theorem are established while only imposing minimal moment conditions. Within this result, we address the case of non-integer moments leading to a reduced remainder. Furthermore, we provide non-uniform bounds for expansions in local limit theorems. The enhanced tail-accuracy of our non-uniform bounds allows for deriving an Edgeworth-type expansion in the entropic central limit theorem as well as a central limit theorem in total variation distance for self-normalized sums.