Davis-type theorems for martingale difference sequences

We study Davis-type theorems on the optimal rate of convergence of moderate deviation probabilities. In the case of martingale difference sequences, under the finite pth mo-ments hypothesis (1 ≤ p <∞), and depending on the normalization factor, our results show that Davis ’ theorems either hold if and only if p> 2 or fail for all p ≥ 1. This is in sharp contrast with the classical case of i.i.d. centered sequences, where both Davis’ theorems hold under the finite second moment hypothesis (or less). 1