Elephant random walk with polynomially decaying steps

Abstract

In this paper, we introduce a variation of the elephant random walk whose steps are polynomially decaying. At each time $k$, the walker's step size is $k^{-\gamma}$ with $\gamma>0$. We investigate effects of the step size exponent $\gamma$ and the memory parameter $\alpha\in [-1,1]$ on the long-time behavior of the walker. For fixed $\alpha$, it admits phase transition from divergence to convergence (localization) at $\gamma_{c}(\alpha)=\max \{\alpha,1/2\}$. This means that large enough memory effect can shift the critical point for localization. Moreover, we obtain quantitative limit theorems which provide a detailed picture of the long-time behavior of the walker.