Rate of moment convergence in the central limit theorem for the elephant random walk

The one-dimensional elephant random walk is a typical model of discrete-time random walk with step-reinforcement, and is introduced by Sch\"{u}tz and Trimper (2004). It has a parameter $\alpha \in (-1,1)$: The case $\alpha=0$ corresponds to the simple symmetric random walk, and when $\alpha>0$ (resp. $\alpha<0$), the mean displacement of the walker at time $n$ grows (resp. vanishes) like $n^{\alpha}$. The walk admits a phase transition at $\alpha=1/2$ from the diffusive behavior to the superdiffusive behavior. In this paper, we study the rate of the moment convergence in the central limit theorem for the position of the walker when $-1 < \alpha \leq 1/2$. We find a crossover phenomenon in the rate of convergence of the $2m$-th moments with $m=2,3,\ldots$ inside the diffusive regime $-1<\alpha<1/2$.