Elephant random walks with multiple extractions and general reinforcement functions

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 $n$).