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On the multidimensional elephant random walk with stops

Bernard Bercu

TL;DR

This work extends the elephant random walk with stops to multidimensional spaces by analyzing the MERWS through a multidimensional martingale construction. The authors show that the Gram matrix $\\Sigma_n$ scaled by $n^{1-r}$ converges a.s. to $(1/d)\\Sigma I_d$ with $\\Sigma\sim ML(1-r)$, enabling a unified treatment across diffusive, critical, and superdiffusive regimes. In the diffusive and critical regimes they establish almost-sure convergence, laws of the iterated logarithm, and self-normalized asymptotic normality; in the superdiffusive regime they prove almost-sure convergence to a nondegenerate limit $L$ and provide Gaussian fluctuations around $L$ with explicit variance, plus ML$(1-r)$-mixed asymptotics. The results reveal that the asymptotics are governed by Mittag-Leffler distributions and a deterministic axis structure, generalizing known 1D results to higher dimensions and highlighting a rich probabilistic structure of MERWS. These findings advance understanding of memory-driven walks and open avenues for further multidimensional stochastic process analysis with stops.

Abstract

The goal of this paper is to investigate the asymptotic behavior of the multidimensional elephant random walk with stops (MERWS). In contrast with the standard elephant random walk, the elephant is allowed to stay on his own position. We prove that the Gram matrix associated with the MERWS, properly normalized, converges almost surely to the product of a deterministic matrix, related to the axes on which the MERWS moves uniformly, and a Mittag-Leffler distribution. It allows us to extend all the results previously established for the one-dimensional elephant random walk with stops. More precisely, in the diffusive and critical regimes, we prove the almost sure convergence of the MERWS. In the superdiffusive regime, we establish the almost sure convergence of the MERWS, properly normalized, to a nondegenerate random vector. We also study the self-normalized asymptotic normality of the MERWS.

On the multidimensional elephant random walk with stops

TL;DR

This work extends the elephant random walk with stops to multidimensional spaces by analyzing the MERWS through a multidimensional martingale construction. The authors show that the Gram matrix scaled by converges a.s. to with , enabling a unified treatment across diffusive, critical, and superdiffusive regimes. In the diffusive and critical regimes they establish almost-sure convergence, laws of the iterated logarithm, and self-normalized asymptotic normality; in the superdiffusive regime they prove almost-sure convergence to a nondegenerate limit and provide Gaussian fluctuations around with explicit variance, plus ML-mixed asymptotics. The results reveal that the asymptotics are governed by Mittag-Leffler distributions and a deterministic axis structure, generalizing known 1D results to higher dimensions and highlighting a rich probabilistic structure of MERWS. These findings advance understanding of memory-driven walks and open avenues for further multidimensional stochastic process analysis with stops.

Abstract

The goal of this paper is to investigate the asymptotic behavior of the multidimensional elephant random walk with stops (MERWS). In contrast with the standard elephant random walk, the elephant is allowed to stay on his own position. We prove that the Gram matrix associated with the MERWS, properly normalized, converges almost surely to the product of a deterministic matrix, related to the axes on which the MERWS moves uniformly, and a Mittag-Leffler distribution. It allows us to extend all the results previously established for the one-dimensional elephant random walk with stops. More precisely, in the diffusive and critical regimes, we prove the almost sure convergence of the MERWS. In the superdiffusive regime, we establish the almost sure convergence of the MERWS, properly normalized, to a nondegenerate random vector. We also study the self-normalized asymptotic normality of the MERWS.
Paper Structure (15 sections, 10 theorems, 179 equations)

This paper contains 15 sections, 10 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. On the asymptotic behavior of the Gram matrix
  3. Main results
  4. The diffusive regime
  5. The critical regime
  6. The superdiffusive regime
  7. Our multidimensional martingale approach
  8. Almost sure convergence.
  9. Law of iterated logarithm.
  10. Asymptotic normality.
  11. Almost sure convergence.
  12. Law of iterated logarithm.
  13. Asymptotic normality.
  14. Almost sure convergence.
  15. Gaussian fluctuations.

Key Result

Lemma 2.1

Whatever the values of $p$, $q$ in $[0,1]$ and $r$ in $]0,1[$, we have where $\Sigma$ stands for the $\mathcal{M}\mathcal{L}(1-r)$ distribution. In particular, Moreover, this convergence holds in $\mathbb{L}^m$ for any integer $m\geq 1$,

Theorems & Definitions (17)

  • Definition 2.1
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • Theorem 3.3
  • Remark 3.2
  • Theorem 3.4
  • Theorem 3.5
  • ...and 7 more