Some Convergence Properties Of The Sequences Of NA And (ρ|~) Mixing Random Variables

Probability theory is the subject of random phenomena from a number of regulardiscipline. It has a wide range of applications in the natural sciences, technical sciences,social sciences, management science. The probability limit theory is one of the mainbranch, it is also an important foundation of other branches of probability theory andmathematical statistics. Over the years, people are committed to working on limit theory ofindependent sequence of identically distributed random variables, and have achieved greatresults. In the mid-1950s, following the classical limit theory for independent randomvariables, a lot of probability statisticians have been proposed to discuss the convergenceproperties of a variety of mixed sequence. The dependent variable limit theory proposed,on the one hand, the needs of the statistical issues, on the other hand from the need oftheoretical studies and other branches of the dependency requirements. Thus for the needsof theoretical and practical applications, many probability statisticians have been proposedto discuss the convergence properties of a variety of dependent sequences. Such as theweak convergence of dependent sequences, strong convergence, complete convergence andso on. NA sequence and(ρ|~) mixed sequence is the two more important sequences of thedependent sequence, in this thesis it discussed the convergence properties of the NA and(ρ|~) mixing sequence random variables,and promoted the strong law of large numbers in thecase of NA, then obtained the completely convergence and the average convergence of(ρ|~) mixed sequence.In the first chapter, study the strong law of large numbers in the case of the NA. Thestrong law of large numbers is a very classical result in the theory of probability limit, sothe strong law of large numbers attracted the interest of domestic and foreign scholars,andmade a lot of independent and sequence-dependent classical results. NA random variablessequence is an important case of non-independent random variable sequence, the conceptof NA is put forward by Joag-Dev and Proschan in1983, it has a wide range ofapplications in reliability theory, the permeability theory and multivariate statisticalanalysis, thus has aroused widespread interest. Strong law of large numbers of theindependent case, strong law of large numbers to the NA case is necessary. This chapterfirst describes the NA, Sums for NQD, the concept of ND and commonly used the momentinequality of NA and then promotes the strong law of large numbers in the NA case, the results improve Hu and Taylor’s conclusion.The second chapter, study the maximum complete convergence and averageconvergence of(ρ|~)-mixed sequences. The concept of(ρ|~)-mixed sequences is introduced in1990by Bradley, which is an extremely wide range of dependent mixed-sequence. It isvaluable to research. such as: Wuqunying,study the convergence properties of(ρ|~)-mixedsequences, give the basic inequality of(ρ|~)-mixed sequence and obtain the completeconvergence theorem of the same distribution of mixed sequence of Baum and Katz, thestrong law of large numbers of Marcinkiewicz, three theorem convergence properties.Bradley studied its weak limit theorem; Bryc、Smolenski and Yang Shanchao, discussedthe strong convergence of the(ρ|~)-mixed sequence. This chapter, use the moment inequalityof(ρ|~)-mixed sequence, study the limit properties of(ρ|~)-mixed sequence and maximumaverage convergenceand complete convergence under the circumstances cesaro integrable,and Scesaro integrable and then obtain the average convergence and completeconvergence.Chapter III, study the weighted sums of(ρ|~)-mixed sequences by using the momentinequality of(ρ|~)-mixed sequences, increase the weight ofan k, obtain the averageconvergence of(ρ|~) mixing sequences, and improve the relevant conclusions of the mixedsequence.