Cram\'{e}r moderate deviations for the elephant random walk

TL;DR

This paper develops refined limit theorems, including Cramér moderate deviations, for the one-dimensional elephant random walk, revealing how the memory parameter influences the distribution's domain of attraction.

Contribution

It introduces Cramér moderate deviation results for the elephant random walk, enhancing understanding of its asymptotic behavior beyond the central limit theorem.

Findings

Limit theorems refine the CLT for elephant random walk.

Domain of attraction depends on the memory parameter p.

Established Berry-Esseen bounds and local limit theorems.

Abstract

We establish some limit theorems for one-dimensional elephant random walk, including Berry-Esseen bounds, Cram\'{e}r moderate deviations and local limit theorems. These limit theorems can be regarded as refinements of the central limit theorems for the elephant random walk. Moreover, by these limit theorems, we conclude that the domain of attraction of normal distribution mainly depends on a memory parameter $p$ which lies between $0$ and $3/4.$