Elephant random walk with polynomially decaying steps

TL;DR

This work extends the elephant random walk by imposing polynomially decaying steps $k^{-\gamma}$ and analyzes how the memory parameter $\alpha\in[-1,1]$ and decay exponent $\gamma>0$ shape long-time behavior. A Doob decomposition $S_n = M_n + A_n$ uncovers a martingale component whose limit theorems (CLT and LIL) are established across regimes, while the predictable part $A_n$ is tightly linked to $S_n$ and the memory process $T_n$. The authors identify a phase transition at $γ_c(α)=\max\{α,1/2\}$: for $γ>γ_c(α)$ the walk converges a.s. (and in $L^2$) to a finite limit $S_∞$, with semi-explicit expressions for the moments of $S_∞$, whereas for $γ\le γ_c(α)$ the process diverges or oscillates depending on $α$. Collectively, the results provide a detailed, quantitative map of the localization-delocalization transition and deliver precise limit theorems across multiple memory- and decay-parameter regimes.

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

Figures (1)

• Figure 1: The classification of the long-time behavior of $\{S_{n}\}$.

Theorems & Definitions (10)

• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Theorem A.1