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 either k may 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).
