Berry-Esseen bounds for step-reinforced random walks

Abstract

We study both the positively and negatively step-reinforced random walks with parameter $p$. For a step distribution $\mu$ with finite second moment, the positively step-reinforced random walk with $p\in [1/2,1)$ and the negatively step-reinforced random walk with $p\in (0,1)$ converge to a normal distribution under suitable normalization. In this work, we obtain the rates of convergence to normality for both cases under the assumption that $\mu$ has a finite third moment. In the proofs, we establish a Berry-Esseen bound for general functionals of independent random variables, utilize the randomly weighted sum representations of step-reinforced random walks, and apply special comparison arguments to quantify the Kolmogorov distance between a mixed normal distribution and its corresponding normal distribution.