Berry-Esseen bounds for step-reinforced random walks

TL;DR

The paper addresses the problem of quantifying normal approximation errors for positively and negatively step-reinforced random walks with memory parameter $p$, under the assumption $\mathbb{E}(|X|^3)<\infty$. It represents the reinforced sums as randomly weighted sums and uses a conditional Berry-Esseen framework to compare against mixed normals, followed by a careful comparison to a standard normal to obtain sharp rates. The main results provide explicit Berry-Esseen bounds $d_K\left(\frac{\hat{S}_n-m_1 n}{\sigma_0\sqrt{b_n}}, \mathbf{Z}\right) \le C\delta_{1,n}$ for the positively reinforced case and $d_K\left(\frac{\check{S}_n-\check{b} n}{\check{\sigma}\sqrt{n}}, \mathbf{Z}\right) \le C\delta_{2,n}$ for the negatively reinforced case, with $b_n$ and the exponents $\delta_{1,n}, \delta_{2,n}$ expressed in terms of $p$. The proofs rely on a Berry-Esseen theorem for functionals of independent RVs and on percolation analyses for general graphs and random recursive trees to control auxiliary random structures, culminating in a detailed treatment of technical propositions. These results extend central limit phenomena for memory-influenced processes with quantitative rates, broadening the understanding of reinforced dynamics and their finite-sample normal approximation.

Abstract

We study both the positively and negatively step-reinforced random walks with parameter $p$. For a step distribution $μ$ 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 $μ$ 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.

Theorems & Definitions (23)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Theorem 3.1
• Remark 3.1
• proof : Proof of Theorem \ref{['thBE1']}