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We consider a minimal model of one-dimensional discrete-time random walk withstep-reinforcement, introduced by Harbola, Kumar, and Lindenberg (2014): Thewalker 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 attime $n+1$ it can move forward with probability $p$ [resp. $q$], or withprobability $1-p$ [resp. $1-q$] it remains at its present position. For thecase $q>0$, 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 thewalker can exhibit all three forms of asymptotic behavior as $p$ is varied. Asa byproduct, we obtain limit theorems for the cluster size of the root inpercolation on uniform random recursive trees.

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