Davis-type theorems for martingale difference sequences (Q2495092)

Summary: 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 \(p\)th moments hypothesis \((1\leq p < \infty)\), and depending on the normalization factor, our results show that Davis' theorems either hold if and only if \(p > 2\) or fail for all \(p \geq 1\). 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).