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

We consider a minimal model of one-dimensional discrete-time 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 $n=1,2,\cdots$, a random time $U_n$ between $1$ and $n$ is chosen uniformly, and if the walker moved forward [resp. remained at rest] at time $U_n$, 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.