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 $n$).
Categories: math.PR