标题： 带多项式衰减步长的随机游走 显示英文标题

标题： Elephant random walk with polynomially decaying steps

摘要： 本文介绍了一种象随机行走的变化形式，其步长呈多项式衰减。在每个时间点 $k$，行进者的步长为 $k^{-\gamma}$，当 $\gamma>0$ 成立时。我们研究了步长指数 $\gamma$ 和记忆参数 $\alpha\in [-1,1]$ 对行进者长时间行为的影响。对于固定的 $\alpha$，它在 $\gamma_{c}(\alpha)=\max \{\alpha,1/2\}$ 处经历了从发散到收敛（定位）的相变。这意味着足够大的记忆效应可以改变定位的关键点。此外，我们获得了定量的极限定理，这些定理提供了行进者长时间行为的详细图景。 显示英文摘要

摘要： 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.