  This paper is devoted to a direct martingale approach for one type of
reinforced elephant random walk (RERW). The elephant random walk is a
non-markovian process which has a complete memory of its entire history. In the
diffusive and critical regimes, we establish the almost sure convergence, the
law of iterated logarithm and the quadratic strong law for the RERW. The
distributional convergences of the RERW to some Gaussian processes are also
provided. In the superdiffusive regime, we prove the distributional convergence
as well as the mean square convergence of the RERW. All our analysis relies on
asymptotic results for multi-dimensional martingales with matrix normalization.
