Strong Limit Theorems For Mixing Sequences

Probability theory is branch of mathematics dealing with chance phenomena and has clearly discernible links with the real world.It is the frame work foundations of many applying subject,such as Information theory,Mathematics risk theory and Insurance theory for Actuaries etc. The strong limit theorems for partial sums of random variables is one of the central question for studying probability.Martingales and stopping times are the basis of Finance theory,Ruin theory,Risk theory and Insurance theory.It is important meaningful to study the strong limit theorems for the sequences of random variables by using martingale and stopping times.The main purpose of this thesis is to study strong convergence and deviation theorems for random variables.This thesis includes four chapters.In chapter 1 and chapter 2,we give an introduction of the basic notions,main results and approaches used in this paper.In chapter 3,we mainly research the strong convergence for(?)-mixing random variable sequences.some results on the convergence of(?)-mixing random sequences have been presented.We study the almost sure convergence for(?)-mixing random sequences.As a result,the authors generalizes partial result of Wu.We obtain the mainstream and some corresponding conclusions by use of truncation methods and the three series theorem of(?)-mixing. In chapter 4,we introduce the notion of log-likelyhood ratio of stochastic sequences,as a measure of dissimilarity between their joint distribution and the product of their marginals.We obtain some deviation theorems for the Jamison type weighted sums by using of the generating function method.