Non-Uniform Bounds in Local Limit Theorems in Case of Fractional Moments

TL;DR

This paper develops Edgeworth-type expansions with non-uniform error bounds for sums of i.i.d. random variables having fractional moments, extending classical results to cases with non-integer moment orders.

Contribution

It introduces new non-uniform bounds in local limit theorems for sums of i.i.d. variables with fractional moments, broadening the scope of classical limit theorems.

Findings

Established Edgeworth-type expansions with non-uniform error bounds

Extended local limit theorems to variables with fractional moments

Provided theoretical framework for non-integer moment orders

Abstract

Edgeworth-type expansions for convolutions of probability densities and powers of the characteristic functions with non-uniform error terms are established for i.i.d. random variables with finite (fractional) moments of order $s \geq 2$, where $s$ may be noninteger.