Convergence Properties Of Mixing Random Sequences

Complete convergence property is an important convergence property of sequences of random variables, it was advanced by famous logistic statistician Xu Bao-lu and Robbins(American) in 1947. About independent sequences of random variables, it is investigated quitly perfectly. The most classical result is the famous Baum and Katz convergence theorem.The complete convergence property of the same distribution sequences of random variables, there are many results, too. This paper mainly studies the complete convergence of three abroad sequences of random variables. We obtain some important conclusions of complete convergence under different distribution.Chapter 1 studies the limit theorems of pairwise NQD.Pairwise NQD is a class of abroad sequences of random variables, which was researched by some predecessors, and they obtained many results about pairwise NQD.Such as: Matula achieved Kolmogorov strong law of large numbers of pairwise NQD.Wang Yue-bao obtained Baum and Katz complete convergence theorem.These results are almost identical circumstances with the independent , Wu Qun-ying studied weakly convergence , almost everywhere convergence of partial sums of pairwise NQD, and complete convergence of the same distribution pairwise NQD, and Jamison strongly convergence weight sum, almost obtained independent of the well-known circumstances Marcinkiewicz strong law of large numbers, the three series theorem, Jamison weighted of the strong convergence properties. The chapter is to give necessary and sufficient conditions for complete convergence of partial sum of pairwise NQD and the Law of large number.Chapter 2 studies the convergence natures of theρ|~ mixed sequences. The mixture is usually mixed withρmixed sequences certain similar, but not identical.ρ|~ mixed sequences is a wide range of mixed-dependent sequences ,its research is of great value.Such as : Wu Qun-ying studiedρ|~ mixed sequences of convergence property, got this basic inequality,obtained the same distribution of Baum and Katz complete convergence theorem, Marcinkiewicz strong law of large numbers, the three series theorem. Bradley studied the weak convergence theorems;Bryc and Smolenski and Yang Shan-chao studied theρ|~ mixed sequences of the strong convergence,and the complete convergence results ofρ|~ mixed sequences about different distribution has not yet seen.This chapter obtains a result ofρ|~ mixed sequences of the different distribution of necessary and sufficient conditions for complete convergence.Chapter 3 studies the convergence natures of theφ|~ mixed sequences. The mixture is usually mixed with ? mixed sequences certain similar, but not identical. Its research is of great value.Such as :Wu Qun-ying, Lin Liang studied the convergence properties ofφ|~ mixed sequences,obtained the same distribution of Baum and Katz complete convergence theorem and discussed some convergences of partial sums and weighted sequences.The complete convergence and strong convergence and Jamison weighted and strong convergence ofφ|~ mixed sequence was studied by Tang Guo-qiang and Wu Yan-chun. This chapter obtains a result ofφ|~ mixed sequences of the different distribution of necessary and sufficient conditions for complete convergence.