| The convergence property of dependent sequences is one of the research hot spots of modern probability limit theory.It is widely used in probability and statistics,finance and insurance,reliability theory,complex system,econometrics and other fields.In this disser-tation,we focus on the limit convergence properties for four kinds of dependent sequences,including NSD sequences,AANA sequences,ANA sequences and END sequences.By using moment inequalities and some probability inequalities of dependent sequences,we further study the complete convergence,complete moment convergence,strong law of large numbers and strong convergence of linear processes with random coefficients,and some new results are obtained.The dissertation mainly completes the following research work.Firstly,by using moment inequalities and probability inequalities of NSD sequences,the strong convergence properties for weighted sums of non-identically distributed NSD sequences are studied under a weaker moment condition,which extend and improve the corresponding ones of Cai[2]and Sung[1]for complete convergence of identically dis-tributed NA sequences.The methods of the proof are different from that of Sung[1].In addition,we investigate the complete convergence and complete moment convergence for arrays of rowwise NSD random variables by using the different methods from Wu[3]under some suitable conditions.The obtained results also generalize and improve the limit convergence properties of Gan and Chen[4]and Wu[3]for arrays of rowwise NA random variables.Secondly,the sufficient and necessary conditions for the complete moment conver-gence of weighted sums of non-identically distributed AANA sequences are investigated under certain conditions of constant arrays,which generalize and improve the correspond-ing ones of independent sequences and NA sequences in related references.Meanwhile,by using Rosenthal type moment inequality and some probability inequalities of AANA sequences,some results on complete moment convergence of the maximum weighted sums for arrays of rowwise AANA random variables are studied under some suitable con-ditions.Moreover,the case of 1+α+β<0 is also considered,which supplements and improves the corresponding one of Beak et al.[5]for complete convergence of weighted sums for arrays of NA sequences.Furthermore,we improve the results of Wang et al.[6]to the case of complete moment convergence.In addition,by proving an important lemma,the strong convergence properties of linear processes with random coefficients generated by AANA sequences and the Marcinkiewicz-Zygmund type strong law of large numbers are obtained,which generalize and improve the results of linear processes with constant coefficients in known references.Thirdly,by using the properties of slow varying function and Rosenthal type moment inequalities of ANA sequences,the complete convergence and complete moment conver-gence for maximum weighted sums of ANA sequences are established,which generalize and improve the Baum-Katz type strong law of large numbers.In addition,by proving an important lemma and using Marcinkiewicz--Zygmund type moment inequality of ANA sequences,the strong law of large numbers of linear processes with random coefficients generated by ANA sequences is studied.The convergence properties of linear processes with constant coefficients are extended to the case of random coefficients.Finally,by using moment inequalities and some probability inequalities of END se-quences,the complete convergence and complete moment convergence for weighted sums of END sequences and the Marcinkiewicz-Zygmund type strong law of large numbers are established,which generalize and improve the corresponding ones of independent se-quences and some dependent sequences in known references. |
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