Limit theorems for the 'laziest' minimal random walk model of elephant type

Abstract

We consider a minimal model of one-dimensional discretetime random walk with step-reinforcement, introduced by Harbola, Kumar, and Lindenberg (2014): The walker can move forward (never backward), or remain at rest. For each n = 1, 2, • • • , a random time Un between 1 and n is chosen uniformly, and if the walker moved forward [resp. remained at rest] at time Un, then at time n + 1 it can move forward with probability p [resp. q], or with probability 1 − p [resp. 1 − q] it remains at its present position. For the case q > 0, several limit theorems are obtained by Coletti, Gava, and de Lima (2019). In this paper we prove limit theorems for the case q = 0, where the walker can exhibit all three forms of asymptotic behavior as p is varied. As a byproduct, we obtain limit theorems for the cluster size of the root in percolation on uniform random recursive trees.