A stochastic algorithm approach for the elephant random walk with applications

TL;DR

This paper introduces a stochastic algorithm approach to analyze the elephant random walk (ERW), providing Gaussian approximations and limit theorems that deepen understanding of ERW's asymptotic behaviors across various regimes.

Contribution

It links the ERW and response-adaptive designs with recursive stochastic algorithms, deriving new limit theorems and precise laws of the iterated logarithm for ERW.

Findings

Gaussian approximation of ERW and multi-dimensional ERW with random step sizes

Central limit theorem and law of the iterated logarithm for ERW

Chung type laws of the iterated logarithm for ERW with random step sizes

Abstract

The randomized play-the-winner rule (RPW) is a response-adaptive design proposed by Wei and Durham (1978) for sequentially randomizing patients to treatments in a two-treatment clinical trial so that more patients are assigned to the better treatment as the clinical trial goes on. The elephant random walk (ERW) proposed by Schutz and Trimper (2004) is a non-Markovian discrete-time random walk on $\mathbb Z$ which has a link to a famous saying that elephants can always remember where they have been. The asymptotic behaviors of RPW rule and ERW have been studied in litterateurs independently, and their asymptotic behaviors are very similar. In this paper, we link RPW rule and ERW with the recursive stochastic algorithm. With the help of a recursive stochastic algorithm, we obtain the Gaussian approximation of the ERW and multi-dimensional varying-memory ERW with random step sizes. By the Gaussian approximation, the central limit theorem, precise law of the iterated logarithm, and almost sure central limit theorem are obtained for the multi-dimensional ERW, the multi-dimensional ERW with random step sizes, and their centers of mass for all the diffusive, critical, and superdiffusive regimes. Based on the Gaussian approximation and the small ball probabilities for a new kind of Gaussian process, the precise Chung type laws of the iterated logarithm of the multi-dimensional ERW with random step sizes and its mass of center are also obtained for both the diffusive regime and superdiffusive regime.