  Elephant random walk is a kind of one-dimensional discrete-time random walk
with infinite memory: For each step, with probability $\alpha$ the walker
adopts one of his/her previous steps uniformly chosen at random, and otherwise
he/she performs like a simple random walk (possibly with bias). It admits phase
transition from diffusive to superdiffusive behavior at the critical value
$\alpha_c=1/2$. For $\alpha \in (\alpha_c, 1)$, there is a scaling factor $a_n$
of order $n^{\alpha}$ such that the position $S_n$ of the walker at time $n$
scaled by $a_n$ converges to a nondegenerate random variable $W$, whose
distribution is not Gaussian. Our main result shows that the fluctuation of
$S_n$ around $W \cdot a_n$ is still Gaussian. We also give a description of
phase transition induced by bias decaying polynomially in time.
