Journal of Applied Mathematics and Stochastic Analysis
Volume 2005 (2005), Issue 2, Pages 159-165

Davis-type theorems for martingale difference sequences

George Stoica

Department of Mathematical Sciences, University of New Brunswick, P.O. Box 5050, NB, Saint John E2L 4L5, Canada

Received 25 February 2004; Revised 10 August 2004

Copyright © 2005 George Stoica. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 moments hypothesis (1p<), and depending on the normalization factor, our results show that Davis' theorems either hold if and only if p>2 or fail for all p1. 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).