Strong Stability And Probability Inequlities For Some Dependent Sequences

Limit theory is one of the branches of probability theory, and it is the important basis of other branches. In order to enhance application value of limit theory, researchers focus on weakening the restriction of independence in recent years. Limit theory of dependent suquences is applied widely in statistics, reliability theory, econometrics and other fields. Some moment inequalities, probability inequalities for dependent sequences are discussed, such as Kolmogorov-type inequality, Hajek-Renyi-type inequality, and so on. And limit property, such as strong stability, almost sure convergence, strong growth rate, and so on, are studied based on these inequalities.In Chapter1, research background of this paper is introduced briefly. Definitions of dependent sequences studied in this paper and notion of strong stability are given in this chapter. In Chapter2, moment inequality for (α,β) mixing suquence is studied. Strong limit theorems and strong stability for (α,β) mixing suquence are obtained based on this inequality. These are new results for (α,β) mixing suquence. In Chapter3, almost sure convergence and strong stability for p mixing sequence are studied, which extend the results for independent sequence without necessarily adding any extra conditions. The results obtained in this chapter are new results for ρ mixing sequence.In Chapter4, inequalities and strong limit theorems for NSD sequence are discussed. Inspired by Shao[14], we study moment inequality and Kolmogorov-type inequality for NSD sequence. As a consequence, Khintchine-Kolmogorov convergence theorem, three series theorem and Marcinkiewicz strong law of large numbers are obtained, extending the corresponding results for independent sequence and NA sequence. In addition, Hajek-Renyi-type inequalities for NSD sequence are given, which can be applied to prove the integrability of supremum. There are not too many results for NSD sequence in literatures We give new results for NSD sequence in this chapter.In Chapter5, moment inequality for pairwise NQD sequence is obtained, which corrects Lemma2in [34] and Lemma1.2in [41]. It is as follows:And moment inequality for Lr(r>1) mixingale is obtained:We revise an error of Lemma2in [40] and give accurate coefficient Furthermore, we study Hajek-Renyi-type inequalities, strong law of large numbers and strong growth rate for pairwise NQD sequence and Lr (r>1) mixingale. As a conse-quence, the integrability of supremum for pairwise NQD sequence and Lr (r>1) mixin-gale can be proved. Strong law of large numbers for Lr (r>1) mixingale extends andimproves Corollary2in [40].