Asymptotic Normality of Superdiffusive Step-Reinforced Random Walks

2021/01/04 by Marco Bertenghi, Bertenghi, Marco · 2 citations
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2101.00906

openalex publication_date 2021/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we establish for the superdiffusive regime p ∈ (1/2,1) that the fluctuations of a general step-reinforced random walk around a_n W, where (a_n)_{n ∈ ℕ} is a non-negative sequence of order n^p and W is a non-degenerate random variable, is Gaussian. This extends a known result by Kubota and Takei for the elephant random walk to the more general setting of step-reinforced random walks. Further, we provide an application of the asymptotic normality of S around a_n W to reinforced empirical processes as studied recently by Bertoin, which yields a refined Donsker's invariance principle.

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• Random walks with echoed steps I
• Elephant random walks on infinite Cayley trees

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