Cramér moderate deviations for the elephant random walk

We establish some limit theorems for the elephant random walk (ERW), including Berry–Esseen's bounds, Cramér moderate deviations and local limit theorems. These limit theorems can be regarded as refinements of the central limit theorem for the ERW. Moreover, by these limit theorems, we conclude that the convergence rate of normal approximations and the domain of attraction of normal distribution mainly depend on a memory parameter p which lies between 0 and 3/4. (paper)