The Limit Theorems Of Some Random Sequences And The Inequalities Of Conditional Demimartingales

Probability theory is a science of quantitatively studying regularity of ran-dom phenomena, which is theoretically rigorous, used widely, and developed rapidly. Probability limit theory is one of the important branches and an very essentially theoretical basis of other branches of probability and mathematical statistics. The strong laws of large numbers and the weak laws of large numbers are two important subjects studied of probability limit theory. This thesis fo-cuses mainly on limit theorems of some sequences of random variables, such as the strong laws of large numbers and the weak laws of large numbers.In Chapter2, we investigate the complete convergence and complete mo-ment convergence of maximal partial sum for martingale differences, thus, we obtain the convergence rates in Marcinkiewicz-Zygmund-type strong law of large numbers for martingale differences. The results include Baum-Katz-type Theo-rem and Hsu-Robbins-type Theorem as special cases, generalize the results for the partial sum of Stocia (2007,2011) to the case of maximal partial sum and expand the scope of the parameters. In addition, we also discuss the complete moment convergence for randomly weighted sums of martingale differences, which generalize the non-random weights to the case of random weights. Meanwhile, Marcinkiewicz-Zygmund-type strong law of large numbers of randomly weighted sums for martingale differences is obtained.In Chapter3, we study the complete convergence of weighted sums for array of rowwise AANA random variables, which complement and improve the corre-sponding ones of Baek et al.(2008). In addition, under the more extensive scope of parameters and the weaker moment conditions, the complete convergence and the complete moment convergence fof weighted sums for array of rowwise AANA random variables are obtained. As an application, Baum-Katz-type Theorem of array of rowwise AANA random variables and Marcinkiewicz-Zygmund-type strong law of large numbers of sequences of AANA random variables are pre-sented under the more extensive scope of parameters and the weaker moment conditions.In Chapter4, the concept of a kind of uniform integrability is given and the moment convergence and the weak laws of large number for martingale dif-ferences, pairwise m-dependent sequences and NOD sequences are investigated under the conditions of this uniform integrability. The results extend and im-prove the corresponding ones of Sung et al.(2008). As an application, we obtain the moment consistency of estimators of regression functions under the nonpara-metric regression model under errors satisfying the conditions of this uniform integrability.The concept of the conditional uniform integrability for array of random variables in Chapter5. We discuss the conditional moment convergence of some arrays of conditionally dependent random variables, which generalize and improve the corresponding ones of Ordonez Cabrera&Volodin (2005) and Chandra&Goswami (2006) and generalize the corresponding ones of Ordonez Cabrera et al.(2012).In the last chapter of this thesis we investigate some inequalities of con-ditional demimartingales and the function of conditional demimartingales, such as maximal inequalities, minimal inequalities and Doob-type inequality. Using a formula of conditional expectation and a maximal inequality of conditional demi-martingales, the maximal Φ-inequalities of nonnegative conditional demimartin-gales and some maximal inequalities of concave Young functions for conditional demimartingales are obtained. As an application, the strong law of large numbers of sequences of conditional PA random variables is presented.