Gaussian fluctuation for superdiffusive elephant random walks

TL;DR

This paper studies the elephant random walk with infinite memory, revealing Gaussian fluctuations around a non-Gaussian limit in the superdiffusive phase, and explores phase transitions caused by decaying bias.

Contribution

It proves Gaussian fluctuation results for superdiffusive elephant random walks and characterizes phase transitions due to polynomially decaying bias.

Findings

Gaussian fluctuations around the non-Gaussian limit in the superdiffusive phase

Identification of phase transition induced by polynomially decaying bias

Scaling behavior of the walk's position in different regimes

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.