A martingale approach for the elephant random walk

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 $0\le p<3\mathrm{/}4$, the critical regime $p=3\mathrm{/}4$, and the superdiffusive regime $3\mathrm{/}4<p\le 1$. Notwithstanding of this trichotomy, we provide some new results on the almost sure convergence and the asymptotic normality of the ERW.