On Wasserstein-1 distance in the central limit theorem for elephant random walk

TL;DR

This paper investigates the Wasserstein-1 distance in the central limit theorem for the elephant random walk, revealing different behaviors depending on the memory parameter p, which was previously unexplored.

Contribution

It provides the first analysis of Wasserstein-1 distance in the CLT for the elephant random walk, showing distinct cases based on the memory parameter p.

Findings

Wasserstein-1 distance behavior varies with p in three cases

Different asymptotic behaviors are identified for p<1/2, 1/2<p<3/4, and p=3/4

New insights into normal approximation for elephant random walk

Abstract

Recently, the elephant random walk has attracted a lot of attentions. A wide range of literature is available for the asymptotic behavior of the process, such as the central limit theorems, functional limit theorems and the law of iterated logarithm. However, there is not result concerning Wassertein-1 distance for the normal approximations.In this paper, we show that the Wassertein-1 distance in the central limit theorem is totally different when a memory parameter $p$ belongs to one of the three cases $0< p < 1/2,$ $1/2< p<3/4$ and $p=3/4.$