Cramér moderate deviations for the elephant random walk

Abstract

We establish some limit theorems for one-dimensional elephant random walk, including Berry-Esseen bounds, Cramér moderate deviations and local limit theorems. These limit theorems can be regarded as refinements of the central limit theorems for the elephant random walk. Moreover, by these limit theorems, we conclude that the domain of attraction of normal distribution mainly depends on a memory parameter $p$ which lies between $0$ and $3/4.$

References in corpus (5)

• Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk
• Elephant Random Walks and their connection to Pólya-type urns
• A martingale approach for the elephant random walk
• Central Limit Theorem for the Elephant Random Walk
• A strong invariance principle for the elephant random walk