  The aim of this paper is to investigate the asymptotic behavior of the
so-called elephant random walk with stops (ERWS). In contrast with the standard
elephant random walk, the elephant is allowed to be lazy by staying on his own
position. We prove that the number of ones of the ERWS, properly normalized,
converges almost surely to a Mittag-Leffler distribution. It allows us to carry
out a sharp analysis on the asymptotic behavior of the ERWS. In the diffusive
and critical regimes, we establish the almost sure convergence of the ERWS. We
also show that it is necessary to self-normalized the position of the ERWS by
the random number of ones in order to prove the asymptotic normality. In the
superdiffusive regime, we establish the almost sure convergence of the ERWS,
properly normalized, to a nondegenerate random variable. Moreover, we also show
that the fluctuation of the ERWS around its limiting random variable is still
Gaussian.
