具有多项式衰减步长的遗忘型大象随机游走

中文摘要 AI 辅助

本文提出了一种具有多项式衰减步长的遗忘型大象随机游走。该游走保留了遗忘型大象随机游走的记忆丧失效应，但其步长随时间呈多项式衰减。我们研究了步长指数和记忆参数对游走长期行为的影响。确定了决定其相图的两个临界阈值。我们获得了游走在不同参数区域下的几乎必然收敛结果、重对数律、渐近正态性和均方位移速率。步长的多项式衰减导致出现亚扩散区域，而在无衰减时恢复经典扩散。此外，我们确定了游走几乎必然收敛到有限随机变量的局域化区域。

英文摘要

In this paper, we introduce an amnesic elephant random walk with polynomially decaying step sizes. The walk retains the loss of memory effect of amnesic ERW, however its step sizes decay polynomially over time. We study the effect of step-size exponent and memory parameter on the long-time behaviour of the walk. Two critical thresholds are identified that determine its phase diagram. We obtain almost sure convergence results, law of iterated logarithm, asymptotic normality and mean square displacement rate of the walk across different parameter regimes. The polynomial decay of step sizes gives rise to a subdiffusive regime while classical diffusion is recovered in the absence of decay. Also, we identify localization regimes in which the walk converges almost surely to a finite random variable.

发表机构

• Indian Institute of Technology Bhilai（印度理工学院比莱分校）

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