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On the multidimensional elephant random walk with stops
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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.
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