Series of moderate deviation probabilities for martingales

We study Davis-type theorems on the moderate deviation probabilities of martingale differences with finite pth moments (1 p <∞). We prove that Davis’ first result holds if p >4 and fails if 1 p <4, and Davis’ second result holds if p >2 and fails if 1 p <2.  2004 Elsevier Inc. All rights reserved.