Title: On the multi-dimensional elephant random walk,
Abstract: The purpose of this paper is to investigate the asymptotic behavior of the
multi-dimensional elephant random walk (MERW). It is a non-Markovian random
walk which has a complete memory of its entire history. A wide range of
literature is available on the one-dimensional ERW. Surprisingly, no references
are available on the MERW. The goal of this paper is to fill the gap by
extending the results on the one-dimensional ERW to the MERW. In the diffusive
and critical regimes, we establish the almost sure convergence, the law of
iterated logarithm and the quadratic strong law for the MERW. The asymptotic
normality of the MERW, properly normalized, is also provided. In the
superdiffusive regime, we prove the almost sure convergence as well as the mean
square convergence of the MERW. All our analysis relies on asymptotic results
for multi-dimensional martingales.