Convergence Properties Of Pairwise NQD And ρ~--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. During the past forty years, complete convergence and strong convergence have become the most important and popular orientations of the current study of Probability Limit Theory. Starting with the above mentioned points, we obtain some limit theorems of two types of important random variables, and draw some precise results in this thesis.As is known to all, everything has correlations between one another if the world. If we can properly describe these correlations by mathematics, we can analyze subjects accurately by the precise tool——mathematics. Hence one can see that, the study on dependent random variables has momentous significance. In fact, the study on the limit properties of dependent random variables may be dated to 1920's and 1930's, at that time ,scholars such as Bernstein(1927), Hopf(1937), Robbins(1948) had carried on studies on this topic. Till now, new kinds of dependent random variables and their corresponding conclusions have emerged in a endless stream. This article is deemed to take two common kinds of dependent random variables. It is divided into three chapters as follows:In Chapter one, some limit properties of Pairwise NQD sequences have been discussed. The week law of large numbers, L pconvergence and complete convergence of the maximum of sums of pairwise NQD random matrix sequences are discussed. Under the condition that the {X_nk;1≤k≤k_n↑∞, n≥1} is Cesàro uniformly integrable, the authors are able to give the week law of large numbers, L pconvergence and complete convergence of the maximum of sums of pairwise NQD random matrix sequences, which generalize the corresponding limit results for independent random matrix sequences to pairwise NQD random matrix sequences.In Chapter two and Chapter three, some limit properties ofρ^- -mixing random sequences have been discussed. In Chapter two, the complete convergence and Marcinkiewicz strong laws forρ^- -mixing random sequences are discussed. As a result, Baum and Katz complete convergence theorem and Marcinkiewicz strong laws are extended to the case ofρ^- -mixing random sequences. In Chapter three, we establish some sufficient conditions of the complete convergence and strong convergence for weighted sums ofρ^- -mixing random sequences.The results obtained extend the theorem of Thrum and Stout.