On multidimensional elephant random walk with stops and random step sizes

TL;DR

This paper analyzes a multidimensional elephant random walk with stops and random step sizes, establishing convergence results such as laws of large numbers and the central limit theorem using martingale techniques.

Contribution

It introduces new convergence results for a multidimensional elephant random walk model with stops and random step sizes, expanding understanding of its probabilistic behavior.

Findings

Law of large numbers established for the number of moves

Central limit theorem proved for the model

Almost sure convergence results obtained

Abstract

In this paper, we study the number of moves in a multidimensional elephant random walk with stops. We establish several convergence results for the number of moves, including the law of large numbers and the law of iterated logarithm. Using a martingale approach, we study the multidimensional elephant random walk with random step sizes. For this model, we obtain several almost sure convergence results for the number of moves, including the law of large numbers, the quadratic strong law, the law of iterated logarithm and the central limit theorem. Similar convergence results are derived for the multidimensional elephant random walk with random step sizes.