Convergence Properties Of Mixing Random Sequences

Theory of Probability is a science of quantitatively studying regularity of random phenomena, which is extensively applied in natural science, technological science, social science and managerial science etc. Hence, it has been developing rapidly since l930's and many new branches have emerged from time to time. Probability Limit Theory is one of the branches and also an important theoretical basis of science of Probability and Statistics.However the limit of probability theory of independent random variable is the one classical theory of the limit of probability theories. It has been perfectly developed during 1930s and 1940s and has been summed up in the monograph written by Gnedenko and Kolomlgorov. After that, many statisticians pose and discuss the convergence properties of all types of mixing sequence such as weak convergence,strong convergence and complete convergence,partly because of the demand of statistical problems, partly because of the theoretical research and other branches of a dependency may require.The complete convergence,which was introduced by China's famous mathematical statistician , Xu Baolu and American statistician,Robbins,is a very important convergence nature. In addition, strong convergence and strong consistency and strong laws of the sequence of random variables are also another important nature.This paper mainly discussed the natures of the complete convergence and strong convergence on the three kinds of mixing random variables.In chapter 1, an exponential inequality is established for identically distributed negatively dependent random variables. By this exponential inequality, We obtain the convergence rate for the strong law of large numbers. In this chapter, we have extended Soo Hak Sung's results about negatively associated random variables.In chapter 2, it is discussed that the natures of the complete convergence and strong convergence on the pairwise NQD random sequence. The concept of the pairwise NQD random sequence was introduced by Lehmann who is the famous statistician in 1966. It is necessary to study the pairwise NQD random sequence because it is wider than NA and ND sequence that we are known. In this chapter, we obtain some results about complete convergence,almost sure convergence,strong law of large numbers on the pairwise NQD random sequence by using the nature of the regularly varying function,slowly varying function and so on.In chapter 3, we get a theorem about the complete convergence for weighted sums of arrays of rowwise(?)mixing random variables by using the moment's inequality of(?)mixing random sequences, and obtain a series of corollary which is the special case in the theorem that we have proofed.