Normal approximation, moderate and large deviations for the elephant random walk with stops and random step sizes

Abstract

We study the elephant random walk with stops and independent positive random step sizes. Building on fixed-movement estimates, we obtain quantitative normal approximations for an empirically normalized statistic by controlling its random denominator and the independent movement clock. A permutation comparison preserves the diffusive Kolmogorov rate, while variance matching permits third-order Zolotarev estimates. We also derive self-normalized Cramér-type moderate deviations. For regularly varying step-size tails, we obtain uniform precise large deviations and establish the necessity of the diffusive and critical threshold orders. Further applications of the clock representation yield large deviation principles for Weibull-type and light-tailed step sizes, and a precise light-tailed expansion in the balanced case.