  Our goal is to investigate the asymptotic behavior of the center of mass of
the elephant random walk, which is a discrete-time random walk on integers with
a complete memory of its whole history. In the diffusive and critical regimes,
we establish the almost sure convergence, the law of iterated logarithm and the
quadratric strong law for the center of mass of the elephant random walk. The
asymptotic normality of the center of mass, properly normalized, is also
provided. Finally, we prove a strong limit theorem for the center of mass in
the superdiffusive regime. All our analysis relies on asymptotic results for
multi-dimensional martingales.
