Optimal Rates in Local and Central Limit Theorems for the Elephant Random Walk

Abstract

We determine the optimal rates of convergence in the local and central limit theorems for the one-dimensional elephant random walk in the diffusive and critical regimes, $0\le p\le3/4$. For every fixed initial distribution, we treat both exact finite-time centering and standardization and the normalization $S_n/\sqrt n$ (or $S_n/\sqrt{n\log n}$ at criticality). We establish uniform Edgeworth expansions for local probabilities and distribution functions, and use these expansions to derive the optimal convergence rates, which exhibit several phase transitions depending on the symmetry of the initial distribution, the normalization, and the type of approximation. Our approach uses elephant polynomials for precise characteristic-function expansions and a random recursive tree representation for global Fourier bounds.