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.

翻译：本文旨在研究具有停止的多维大象随机游走（MERWS）的渐近行为。与标准大象随机游走不同，该模型允许"大象"停留在当前位置。我们证明，经过适当归一化后，与MERWS相关的格拉姆矩阵几乎必然收敛于一个确定性矩阵（该矩阵与MERWS均匀移动的坐标轴相关）与一个Mittag-Leffler分布的乘积。这一结果使我们能够扩展先前针对一维带停止大象随机游走建立的所有结论。具体而言，在扩散态与临界态下，我们证明了MERWS的几乎必然收敛性。在超扩散态下，我们建立了经适当归一化的MERWS几乎必然收敛于一个非退化随机向量。此外，我们还研究了MERWS的自归一化渐近正态性。