Rates of convergence in the central limit theorem for the elephant random walk with random step sizes

TL;DR

This paper extends the elephant random walk model by allowing random step sizes and establishes new results on the law of the iterated logarithm, central limit theorem, and convergence rates in various metrics.

Contribution

It introduces a generalized model with random step sizes and provides novel rates of convergence in the CLT, even for the classical case.

Findings

Established the law of the iterated logarithm for the model.

Proved the central limit theorem with explicit convergence rates.

Derived convergence rates in Kolmogorov, Zolotarev, and Wasserstein distances.

Abstract

In this paper, we consider a generalization of the elephant random walk model. Compared to the usual elephant random walk, an interesting feature of this model is that the step sizes form a sequence of positive independent and identically distributed random variables instead of a fixed constant. For this model, we establish the law of the iterated logarithm, the central limit theorem, and we obtain rates of convergence in the central limit theorem with respect to the Kologmorov, Zolotarev and Wasserstein distances. We emphasize that, even in case of the usual elephant random walk, our results concerning the rates of convergence in the central limit theorem are new.