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On Limiting Behaviour of Moves in Multidimensional Elephant Random Walk with Stops

Shyan Ghosh, Manisha Dhillon, Kuldeep Kumar Kataria

TL;DR

The paper analyzes the number of moves in the multidimensional elephant random walk with stops (MERWS) by deriving conditional mean increments, obtaining a recursive expression for the expected move count, and constructing a multiplicative martingale. It then establishes asymptotic behavior through LLN and LIL results across regimes determined by the parameter $r$, revealing distinct almost-sure convergence and fluctuation patterns for the move count. The approach combines martingale techniques with asymptotic analysis of the associated sequences $a_n$ and $s_n$, yielding precise bounds and limit results for both moves and delays. Overall, the work extends memory-based random walk theory to MERWS and clarifies how memory and stopping interactions govern long-run behavior.

Abstract

In this paper, we discuss the number of moves in multidimensional elephant random walk with stops (MERWS). We obtain the conditional mean increments of the number of moves. Using this conditional result, a recursive relation for the expected number of moves of MERWS is derived and a multiplicative martingale is constructed. The asymptotic behaviour of the solution of this recursive relation is discussed. Later, we discuss several convergence results for the number of moves that include the law of large numbers and law of iterated logarithm.

On Limiting Behaviour of Moves in Multidimensional Elephant Random Walk with Stops

TL;DR

The paper analyzes the number of moves in the multidimensional elephant random walk with stops (MERWS) by deriving conditional mean increments, obtaining a recursive expression for the expected move count, and constructing a multiplicative martingale. It then establishes asymptotic behavior through LLN and LIL results across regimes determined by the parameter , revealing distinct almost-sure convergence and fluctuation patterns for the move count. The approach combines martingale techniques with asymptotic analysis of the associated sequences and , yielding precise bounds and limit results for both moves and delays. Overall, the work extends memory-based random walk theory to MERWS and clarifies how memory and stopping interactions govern long-run behavior.

Abstract

In this paper, we discuss the number of moves in multidimensional elephant random walk with stops (MERWS). We obtain the conditional mean increments of the number of moves. Using this conditional result, a recursive relation for the expected number of moves of MERWS is derived and a multiplicative martingale is constructed. The asymptotic behaviour of the solution of this recursive relation is discussed. Later, we discuss several convergence results for the number of moves that include the law of large numbers and law of iterated logarithm.
Paper Structure (10 sections, 5 theorems, 60 equations)

This paper contains 10 sections, 5 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized hypergeometric function
  4. $\mathbb{L}^p$ space
  5. Increasing process
  6. Laws of large numbers (LLN) for martingales
  7. Law of iterated logarithm (LIL)
  8. The number of moves in MERWS
  9. Laws of large numbers
  10. Laws of iterated logarithm

Key Result

Lemma 3.1

Let $a_1=1$ and Then, $\{M_n,\mathcal{F}_n\}_{n\ge 1}$ is a martingale, where $M_n=Z_n^*/a_n$ and $\mathcal{F}_n=\sigma(X_1,X_2,\dots,X_n)$.

Theorems & Definitions (12)

  • Lemma 3.1
  • Proposition 3.1
  • proof
  • proof
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.1
  • Lemma 3.3
  • proof
  • ...and 2 more