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 $k^{-{\hat{I}}^3}$ with ${\hat{I}}^3>0$. We investigate effects of the step size exponent ${\hat{I}}^3$ and the memory parameter $\hat{I}\pm \in [-1,1]$ on the long-time behavior of the walker. For fixed $\hat{I}\pm$, it admits phase transition from divergence to convergence (localization) at ${\hat{I}}_c^3(\hat{I}\pm )=\max \{\hat{I}\pm ,1/2\}$. 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.

13 pages, 1 figure