标题： 关于多维带停步的大象随机游走 显示英文标题

标题： On the multidimensional elephant random walk with stops

摘要： 本文的目的是研究多维带停步的随机游走（MERWS）的渐近行为。与标准的随机游走不同，这里的“大象”可以停留在其当前位置。我们证明了与MERWS相关的Gram矩阵，在适当标准化后，几乎必然收敛于一个确定性矩阵与Mittag-Leffler分布的乘积，该矩阵与MERWS沿其运动的均匀轴相关。这使得我们可以将之前针对一维带停步随机游走的所有结果推广到多维情况。更具体地说，在扩散和临界情形下，我们证明了MERWS的几乎必然收敛性；在超扩散情形下，我们建立了MERWS经适当标准化后的几乎必然收敛性，其极限是非退化的随机向量。此外，我们还研究了MERWS的自归一化渐近正态性。 显示英文摘要

摘要： The goal of this paper is to investigate the asymptotic behavior of the multidimensional elephant random walk with stops (MERWS). In contrast with the standard elephant random walk, the elephant is allowed to stay on his own position. We prove that the Gram matrix associated with the MERWS, properly normalized, converges almost surely to the product of a deterministic matrix, related to the axes on which the MERWS moves uniformly, and a Mittag-Leffler distribution. It allows us to extend all the results previously established for the one-dimensional elephant random walk with stops. More precisely, in the diffusive and critical regimes, we prove the almost sure convergence of the MERWS. In the superdiffusive regime, we establish the almost sure convergence of the MERWS, properly normalized, to a nondegenerate random vector. We also study the self-normalized asymptotic normality of the MERWS.