Some Results On Limit Theory For Linear Processes Generated By Mixing Dependent Sequences

Theory of Probability is a subject that uses mathematical tools to study the law of random phenomena.It has very important applications in economics,finance,medicine,engineering and other fields.In the 1930s,Kolmogorov proposed the axiomatic system of probability theory by means of measure theory.Subsequently,probability theory has been rapidly developed.Its main branch,probability limit theory,has become one of the hot research directions of modern probability theory.Limit theory for sequences of independent random variables has been developed successfully.Since independence is a strong property,many sequences do not satisfy independence.Therefore,it is necessary to study sequences of random variables which are weaker than an independent one.Since then,experts and scholars have given a variety of concepts of mixing and dependent random variable sequences,conducted extensive and in-depth research on the limit theory of mixing dependent random variable sequences,and obtained some classical results.Because linear process is one of the most representative models in time series analysis,it is widely used in economics,engineering,physics and other fields.Therefore,the study of the limit theory of linear processes has attracted the attention of many probabilistic scholars,which is of great significance.In this paper,we consider some results on limit theory for linear processes generated by mixing dependent sequences.In Chapter 2,we mainly study self-normalized limit theorems for linear processes generated by ?-mixing random variables.Firstly,it is assumed that the tail term of the series formed by the sequence of coefficients of a linear process satisfies the appropriate conditions.Secondly,the methods of truncation of random variables,decomposition of indicator function,exchange of summation order,sub-sequence and so on are used.At last,according to the probability inequality of the sequences of ?-mixing random variables,the self-normalized central limit theorems for linear processes generated by?-mixing random variables are obtained.On this basis,we prove the self-normalized almost sure central limit theorems for linear processes generated by ?-mixing random variables.The results extend the application range of the self-normalized almost sure central limit theorem.In Chapter 3,we get limit distribution for products of sums of partial sums of linear processes generated by NSD sequence.First of all,the central limit theorem of weighted sum of linear process generated by NSD sequence is obtained under weaker weighted sequence conditions by using linear process decomposition and the central limit theorem of weighted sum of NSD random variable sequence.Second,the polynomial expansion of logarithmic function is used to transform the product of the sum of random variables into the form of summation.Then,by using probability inequality and the central limit theorem of weighted sums of linear processes generated by NSD sequences,the limit distributions of the products of partial sums and the products of sums of partial sums of linear processes generated by NSD sequences are studied respectively,which improve the limit theory of the products of partial sums of random variables.In Chapter 4,we establish the complete moment convergence of a linear process with random coefficients generated by a class of random variables satisfying the Rosenthal-type maximal inequality.At the moment of studying complete moment convergence,many experts and scholars at home and abroad usually assume that the random variable sequence is a specific sequence.In this section,we mainly apply some moment inequalities,and study the complete moment convergence of a linear process with random coefficients generated by a class of random variables satisfying the Rosenthal-type maximal inequality by means of inequality expansion and contraction.At the same time,the complete moment convergence of linear process with constant coefficients is extended to the complete moment convergence of linear processes with random coefficients,which enriches the theoretical results of complete moment convergence.