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,…,n}) a sample of k time-stamps, either with replacement or without, where either k may remain fixed as n grows, or k=k(n) may grow with n. Letting Un,1,Un,2,…,Un,k denote the time-stamps sampled, the step taken by the walker during the (n+1)-st time-stamp, denoted Xn+1, is a ±1-valued random variable whose distribution depends on the proportion of (+1)-valued steps out of XUn,1,XUn,2,…,XUn,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).