A central limit theorem for unbalanced step-reinforced random walks
Zhishui Hu, Liang Dong
TL;DR
This work analyzes unbalanced step-reinforced random walks, a unifying framework for the elephant random walk and its reinforced variants. By representing the process sum $T_n$ as a randomly weighted sum of i.i.d. increments $\{\xi_k\}$ via bond percolation on a random recursive tree, the authors develop a general theory for normal and stable central limit theorems of randomly weighted sums. They prove that, under $(2r-1)\alpha p<1$ and when $\xi_k$ lie in the domain of a symmetric $\alpha$-stable law with normalizing sequence $a_n$, one has $\frac{T_n}{a_n} \stackrel{d}{\rightarrow} (c(\alpha,p,r))^{1/\alpha} S$, where $S$ is symmetric $\alpha$-stable and $c(\alpha,p,r)$ is explicitly defined. In the Gaussian case $\alpha=2$, the constant simplifies to $c(2,p,r)=1/(1-(4r-2)p)$. The results extend and unify previous CLTs for ERW and reinforced walks by providing a cohesive framework for normal and stable limits through randomly weighted sums.
Abstract
In this paper, 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 first derive normal and stable central limit theorems for such randomly weighted sums, and then apply these results to obtain a unified central limit theorem for unbalanced step-reinforced random walks.