Asymptotic Normality of Superdiffusive Step-Reinforced Random Walks

Abstract

In this article we establish for the superdiffusive regime $p\in (1/2,1)$ that the fluctuations of a general step-reinforced random walk around $a_n\hat{W}$, where $(a_n{)}_{n\in \mathbb{N}}$ is a non-negative sequence of order $n^p$ and $\hat{W}$ 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 $\hat{S}$ around $a_n\hat{W}$ to reinforced empirical processes as studied recently by Bertoin, which yields a refined Donsker's invariance principle.

arXiv comment18 pages. This version contains several corrections and improved readability, comments are welcome

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• Recurrence and transience of multidimensional elephant random walks