  In this work, we discuss the smoothly amnesia-reinforced multidimensional
elephant random walk (MARW). The scaling limit of the MARW is shown to exist in
the diffusive, critical and superdiffusive regimes. We also establish the
almost sure convergence in all of the three regimes. The quadratic strong law
is displayed in the diffusive regime as well as in the critical regime. The
mean square convergence towards a non-Gaussian random variable is established
in the superdiffusive regime. Similar results for the barycenter process are
also derived. Finally, the last two Sections are devoted to a discussion of the
convergence velocity of the mean square displacement and the Cram{\'e}r
moderate deviations.
