Stable limit theorems for unbalanced step-reinforced random walks

Abstract

We study a class of unbalanced step-reinforced random walks that unifies the elephant random walk, the positively step-reinforced random walk, and the negatively step-reinforced random walk. By establishing a connection with bond percolation on random recursive trees, these processes can be represented as randomly weighted sums of independent and identically distributed random variables. We derive Gaussian and non-Gaussian stable limit theorems for such randomly weighted sums, and then apply these results to obtain corresponding stable limit theorems for unbalanced step-reinforced random walks.