  The purpose of this paper is to establish, via a martingale approach, some
refinements on the asymptotic behavior of the one-dimensional elephant random
walk (ERW). The asymptotic behavior of the ERW mainly depends on a memory
parameter $p$ which lies between zero and one. This behavior is totally
different in the diffusive regime $0 \leq p <3/4$, the critical regime $p=3/4$,
and the superdiffusive regime $3/4<p \leq 1$. Notwithstanding of this
trichotomy, we provide some new results on the almost sure convergence and the
asymptotic normality of the ERW.
