Large Deviations for Some Dependent Sequences

Let (Xi) be a martingale difference sequence and Sn=n∑i=1 Xi. Suppose (Xi) is bounded in Lp. In the case p2, Lesigne and Volny (Stochastic Process. Appl. 96 (2001) 143) obtained the estimation μ(Sn n) < cn-p/2, Yulin Li (Statist. Probab.Lett. 62 (2003) 317) generalized the result to the case when p ∈(1, 2) and obtained μ(Sn n) < cn1-P, these are optimal in a certain sense. In this article, the authors study the large deviation of Sn for some dependent sequences and obtain the same order optimal upper bounds for μ(Sn n) as those for martingale difference sequence.