Convergence Properties Of Some Types Of Random Variable Sequences

In modern probability limit theory, one of the hot spots is to weaken the restrictionof independence, and to make it more practical application value, which yields theconcepts of dependent sequences. Limit theory of dependent sequences has a widerange of applications in probability and statistics, insurance, finance, econometrics,complex systems, reliability theory and other fields. In this dissertation, some limitconvergence properties for four kinds of dependent sequences (such as NOD sequences,ψ-mixing sequences, ρ*-mixing sequences and pairwise NQD sequences) areinvestigated. Some probability inequalities and moment inequalities for dependentsequences are presented and used. Meanwhile, by using the truncated method, completeconvergence theorems and strong law of large numbers for weighted sums of dependentsequences are obtained. Strong convergence properties of moving average processesgenerated by ρ*-mixing sequences and pairwise NQD sequences respectively are alsofurther studied, and some new results are obtained. Based on the properties and someinequalities of dependent sequences, and the limit theory research technique, thedissertation mainly completes the following research work.1. Research background of this dissertation is introduced briefly, and some limitresults for four kinds of dependent sequences are reviewed. Some important inequalitiesand lemmas which are closely related to the focuses in this dissertation are introduced.And the main content of this dissertation is summarized.2. By using Rosenthal type moment inequality and some exponential inequalitiesfor NOD sequences, Baum-Katz type complete convergence theorem and strong law oflarge numbers for weighted sums of NOD sequences are obtained. Also, under a weakermoment condition, some results of complete convergence for weighted sums of NODsequences are studied and obtained, which generalize and improve the applicableconditions and scope of the corresponding results of Baum and Katz for completeconvergence of independent and identically distributed random variables; Liang and Su;Li et al.; Cai for complete convergence for weighted sums of NAsequences; Wang et al.;Ko and Kim; Sani et al. for convergence properties of NOD sequences, to some certainextent.3. Strong limit convergence properties for ψ-mixing sequences are mainlystudied in this dissertation, Three series theorem, Khintchine-Kolmogorov typeconvergence theorem, almost sure convergence theorem of weighted sums, and almostsure convergence theorem for product sums of ψ-mixing sequences are obtained. Complete convergence for weighted sums of ψ-mixing sequences is also investigatedin this dissertation and some new research results are obtained. Because of very fewresults for weighed product sums of ψ-mixing sequences in known references, theresults for weighed product sums of non-identically distributed ψ-mixing sequencesobtained in this dissertation are new results.4. This dissertation focuses on strong convergence properties for weighted sums ofρ*-mixing sequences and complete moment convergence for moving averageprocesses generated by ρ*-mixing sequences. Inspired by Bai and Cheng for strongconvergence of independent and identically distributed random variable sequences andCai; Qiu for convergence properties for weighted sums of ρ*-mixing sequencesrespectively, and by using the moment inequality for ρ*-mixing sequences and thetruncated method, almost sure convergence theorem and complete convergence theoremfor the maximum weighted sums of non-identically distributed ρ*-mixing sequencesare studied, the results obtained extend and improve the applicable scope of thecorresponding results of the above references for independent and identically distributedrandom variables and ρ*-mixing sequences. Moreover, this dissertation discussescomplete moment convergence for moving average processes generated by ρ*-mixingsequences. By introducing a slowly varying function, complete moment convergencetheorems for the maximal partial sums of moving average processes under two cases ofr=1and r>1are obtained, respectively. The results obtained improve thecorresponding results of Ko and Kim for the general partial sums of moving averageprocesses generated by ρ*-mixing sequences and generalize and extend thecorresponding results of Li and Zhang for moving average processes generated by NAsequences to the case of moving average processes generated by ρ*-mixingsequences.5. Some convergence properties for pairwise NQD sequences are studied. By usingthe Kolmogorov type inequality for pairwise NQD sequences and the truncated method,some results of complete convergence properties for non-identically distributed pairwiseNQD sequences are investigated and obtained in this dissertation, which supplementand improve the corresponding complete convergence theorems of Wu; Gan and Chenfor pairwise NQD sequences. By constructing and assuming some moment conditionsexistence, strong law of large numbers for pairwise NQD sequences are studied, andsome new results are obtained, which generalize the applicable scope of thecorresponding results of Liu; Wan for pairwise NQD sequences, respectively.Furthermore, influenced and inspired by Chen et al.; Zhang and Wang for results of pairwise NQD sequences, complete convergence properties for the maximal partialsums of moving average processes generated by non-identically distributed pairwiseNQD sequences are studied in this dissertation. Because of pairwise NQD sequencesbeing a much more widely kind of random variable sequences, very few studies of limitproperties for moving average processes generated by pairwise NQD sequences inknown references, the results obtained in this dissertation are new results for pairwiseNQD sequences.6. This dissertation concludes the studies and points out future research directions.