Some Limit Theorems For Certain Mixing Random Sequences

Theory of probability describes the regularity of things influenced by massive stochas-tic factors. Therefore, researching into the limite of partial sums of random variables isextremely important for us to make clear the essence of the stochastic phenomenon.Early in the 1930s-1940s, the limite theory of probability saw a full development,which was concerning with the sequence of independent random variables. The basic con-clusion of the theory was given in the monograph by Gnedenko and Kolmogorov in 1954,Limit Distributions for Sums of Independent Random Variables.Never-theless, in many practical cases, on one hand, samples or some functions of independentsamples are not independent. On the other hand, fundamental research and branchesof some other science sometimes call for the dependence of random variables, such as inMarkov chain, theory of random fields and analysis of time series,etc. As a result, in themiddle 1950s, after the classical limite theory of independent random variables were fullydeveloped, the concept of dependent random variables came into discussion in probabilitytheory and certain branches of statistics. This concept intrigued many probability statisti-cians who began to research in it and made a lot of researching achievements. Many of theresults before 1997 were collected in the monograph by Lu Chuanrong and Lin Zhengyan,Limit Theory for Sums of Dependent Mixing Sequences.The core research topic of probability theory is the strong law of large numbers(SLLN),discussion of whose rate of convergence is of great importance. There are two basic ap- proaches to proving SLLN. The first is to prove the desired result for a subsequence andthen reduce the problem for the whole sequence to that for the subsequence. In so doing,amaximal inequality for cumulative sums is usually needed for the second step and theother one is to use directly a maximal inequality for normed sums. Inequalities of thiskind are said to be of Hájek-Rényi-type(HR). Due to the academic difficulty of obtainingthe HR inequality, the first method is used more often. However, once the HR inequalityis obtained, SLLN becomes an obvious problem. In 2000, Fazekas and Klesov built up amodel of HR inequality in literature[2], which was used to reach some conclusions relatingto dependent variables. So it is more easy to use the second proof mentioned above.This essay is composed of three chapters. Chapter 1 introduces some concepts, defini-tions, inequalities helpful for reaching the conclusion, lemmas as well as some correspond-ing deductions. In Chapter 2, by using the lemmas and their corresponding deductionsgiven in Chapter 1, the author reaches several conclusions, for example, SLLN, estimatesof the rate of convergence in SLLN, and that partial sums of PA sequences, which average0 in certain conditions, is almost sure convergent,etc. In Chapter 3, the author comparessome conclusions about PA sequences, and takes a further step to discuss SLLN, estimatesof the rate of convergence in SLLN in the case of strongly positive dependent (SPD) se-quences,(?)-mixing sequences.