Elephant random walks with multiple extractions and general reinforcement functions

TL;DR

This paper studies a generalized elephant random walk model where the walker samples multiple past steps with a reinforcement function influencing future steps, analyzing its asymptotic behavior under various sampling and reinforcement scenarios.

Contribution

It introduces a new multi-sampling reinforcement model for elephant random walks and characterizes its asymptotic properties under different growth conditions.

Findings

Established conditions for strong and weak convergence.

Analyzed effects of sample size growth on walk behavior.

Provided insights into reinforcement function impacts.

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 $n$).