The Moment Inequalities For L~r(r>1) Mixingale And ψ-mixing Sequences And Their Applications

Since the last two centuries, the limit theorem is the central subject of the probability limit theory. Through the whole evolution, what made people deeply impressed is that the final settlement of the classical problems, which mainly depends on the improvement of the tools and the precision of the method, such as, Cr inequality, Minkowski's inequality, Holder's inequality, etc. The study of the moment inequality and the three theorem for random variables sequence is an important part of the probability limit theory, which has already been studied by many scholars. Particularly, a lot of perfect results have been obtained.p-mixing random variables andφ-mixing random variables were introduced by Kolmogorov and Rozanov(1960) and Dobrushin(1956) for non-stationary Markov chain, respectively. A number of writers have studied them and established a series of useful results,ρ-mixing random variables were introduced by Bradley in 1990 andφ-mixing random variables can be found in Wu s artial which was published in 2004. The pairwise NQD sequence was introduced by Lehmann(1966) and Wu Qunying studied it deeply in [8]. In the last few years, many writers have studied qualities of Lr-mixingale sequence, for example, Hall and Heyde(1980), Hansen(1991) and Pan, et al(2003). Let{Xi,i≥1} is a bounded difference martingale sequence in Lp space. Lesigne and Volny(2001) obtained estimator:μ(Sn> n)≤Cn-p/2 for p≥2. Li(2003) generalized the result, and got that/μ(Sn> n)≤Cn1-p for 11)mixingale, which has the concrete coefficients, and get the large deviations fo Lr(r>1)mixingale. These results have the same order optimal upper bounds for martingale difference sequence in Lesigne and Volny(2001) and Li(2003). In addition, we give its application to the linear process and obtain a large deviation for it.At last, inspired by the currents method of three series theorem, moment inequality forψ/-mixing sequence is obtained. By using this inequality, we get the three series theorem forψ/-mixing sequence.