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Elephant random walks with multiple extractions and general reinforcement functions

Moumanti Podder, Archi Roy

TL;DR

This work generalizes the elephant random walk by allowing the next step to depend on a sample of $k$ past steps, drawn with or without replacement, through a reinforcement function $f$ and memory parameter $p$. The authors cast the process as a stochastic approximation problem, proving almost-sure convergence of $S_n/n$ to a fixed point determined by the drift function (via $H$ for memory-with-replacement and its Bernstein-approximation $F_n$ for memory-without-replacement). They derive detailed weak convergence results that depend on the attraction strength $\tau = 1 - H'(x^*)$ (or its analogue in the without-replacement setting) and consider regimes where $k$ is fixed or grows with $n$, including growth rates that influence the scaling of fluctuations. The results unify and extend ERW-type limit theorems to settings with multiple past extractions and general reinforcement functions, offering rigorous decay rates, fixed-point criteria, and explicit asymptotic distributions with practical implications for memory-influenced, reinforcement-driven processes.

Abstract

We consider a generalized model of elephant random walks wherein the walker, during the $(n+1)$-st time-stamp, draws from the past (i.e. the set $\{1,2,\ldots,n\}$) a sample of $k$ time-stamps, either with replacement or without, where $k$ may either remain fixed as $n$ grows, or $k=k(n)$ may grow with $n$. Letting $\{U_{n,1}, U_{n,2}, \ldots, U_{n,k}\}$ denote the time-stamps sampled, the step taken by the walker during the $(n+1)$-st time-stamp, denoted $X_{n+1}$, is a $\pm 1$-valued random variable whose distribution depends on the proportion of $(+1)$-valued steps out of $X_{U_{n,1}},X_{U_{n,2}},\ldots,X_{U_{n,k}}$ via a reinforcement function $f$. In this paper, we investigate the asymptotic behaviour, i.e. strong and weak convergence, of this random walk model under suitable assumptions made on the function $f$ (as well as on the sequence $\{k(n)\}$ when the sample size varies with $n$).

Elephant random walks with multiple extractions and general reinforcement functions

TL;DR

This work generalizes the elephant random walk by allowing the next step to depend on a sample of past steps, drawn with or without replacement, through a reinforcement function and memory parameter . The authors cast the process as a stochastic approximation problem, proving almost-sure convergence of to a fixed point determined by the drift function (via for memory-with-replacement and its Bernstein-approximation for memory-without-replacement). They derive detailed weak convergence results that depend on the attraction strength (or its analogue in the without-replacement setting) and consider regimes where is fixed or grows with , including growth rates that influence the scaling of fluctuations. The results unify and extend ERW-type limit theorems to settings with multiple past extractions and general reinforcement functions, offering rigorous decay rates, fixed-point criteria, and explicit asymptotic distributions with practical implications for memory-influenced, reinforcement-driven processes.

Abstract

We consider a generalized model of elephant random walks wherein the walker, during the -st time-stamp, draws from the past (i.e. the set ) a sample of time-stamps, either with replacement or without, where may either remain fixed as grows, or may grow with . Letting denote the time-stamps sampled, the step taken by the walker during the -st time-stamp, denoted , is a -valued random variable whose distribution depends on the proportion of -valued steps out of via a reinforcement function . In this paper, we investigate the asymptotic behaviour, i.e. strong and weak convergence, of this random walk model under suitable assumptions made on the function (as well as on the sequence when the sample size varies with ).

Paper Structure

This paper contains 12 sections, 19 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Formal description of the model
  4. Model involving sampling with replacement
  5. Model involving sampling without replacement
  6. Review of literature pertinent to the work in this paper
  7. Main results of this paper
  8. When the sampling scheme is with replacement
  9. Where the sampling scheme is without replacement
  10. Proofs of the results from §\ref{['sec:main_results']}
  11. Proofs of the results from §\ref{['subsec:results_with']}
  12. Proofs of the results from §\ref{['subsec:without_results']}

Key Result

Theorem 4.1

Let $\{S_{n}\}$ be the stochastic process defined as in S_n, where the underlying model is as described in §subsec:with, with $p\in(0,1)\setminus\{1/2\}$ and the sample size, $k$, remaining constant with $n$. Recall the function $f$ introduced via X_n+1_distribution, and let the function $g:[0,1]\ri Let us further define the function $H:[-1,1]\rightarrow [-1,1]$ as If $H$ has a unique fixed point

Theorems & Definitions (35)

  • Theorem 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • Theorem 4.5
  • Theorem 4.6
  • Theorem 4.7
  • Corollary 4.8
  • Theorem 4.9
  • Theorem 4.10
  • ...and 25 more