Davis‐type theorems for martingale difference sequences
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Abstract
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 ≤ 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).
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