Gaussian fluctuation for superdiffusive elephant random walks

Abstract

Elephant random walk is a kind of one-dimensional discrete-time random walk with infinite memory: For each step, with probability $\alpha$ the walker adopts one of his/her previous steps uniformly chosen at random, and otherwise he/she performs like a simple random walk (possibly with bias). It admits phase transition from diffusive to superdiffusive behavior at the critical value ${\alpha }_c=1/2$. For $\alpha \in ({\alpha }_c,1)$, there is a scaling factor $a_n$ of order $n^{\alpha }$ such that the position $S_n$ of the walker at time $n$ scaled by $a_n$ converges to a nondegenerate random variable $W$, whose distribution is not Gaussian. Our main result shows that the fluctuation of $S_n$ around $W\cdot a_n$ is still Gaussian. We also give a description of phase transition induced by bias decaying polynomially in time.

14 pages

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• New insights on the minimal random walk
• Rate of moment convergence in the central limit theorem for the elephant random walk
• Reinforced random walks under memory lapses
• Recurrence and transience of multidimensional elephant random walks
• The elephant random walk in the triangular array setting
• Limit theorems for a random walk with memory perturbed by a dynamical system
• Remarks on the limiting behaviors of generalized elephant random walks