Large Deviations for Some Dependent Sequences

Let (Xi) be a martingale difference sequence and . Suppose (Xi) is bounded in Lp. In the case p≥ 2, 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.