  This paper is devoted to the asymptotic analysis of the reinforced elephant
random walk (RERW) using a martingale approach. 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.
