Asymptotic Normality of Superdiffusive Step-Reinforced Random Walks

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