Some Limit Theorems For AANA Sequence And The Bahadur Representation For NOD Sample

The limit theory of dependent sequences is one of the central issues for studying probabilty, which has wide applications in reliability theory, complex systems, multivari-ate statistical analysis and other fields. So it has received more and more attention by many scholars. In this thesis, with the help of some techniques such as the continuity of probability, Borel-Cantelli lemma, Bernstein type inequality, maximal inequality, the truncated method for random variables, etc., we investigate some limit theorems for three kinds of random variables such as NA, NOD and AANA, and obtain some new results, for example, the strong limit theorems for weighted sums of NA random variables, the Bahadur representation for sample quantile under NOD random variables, convergence properties for NOD random variables, the results of Marcinkiewicz type strong law of large numbers for the partial sums of AANA random variables, and so on. Our results generalize and improve the corresponding results of the cited references.The specific results are as follows:firstly, by discussing some basic properties of the NA random variables, we have the equivalence of almost sure convergence and complete convergence. As an application of these properties, some strong convergence results for the weighted sums are obtained, which generalize the corresponding results for independent sequences of references [31,33] without adding extra conditions. On the other hand, by using the truncated method, we obtain strong limit theorems for weighted sums of NA random variables. The results listed above are new results for NA sequence.Secondly, with the help of techniques of Bernstein type inequality and some properties for the inverse function of the distribution function, we focuse on the study of the Bahadur representation for sample quantile under NOD sequence.For a fixed p∈(0,1), letξp=F-1(p), ξp,n=Fn-1(p).Let{Xn,n≥1} be a sequence of NOD random variables. Assume that the com-mon marginal distribution function F(x) is continuously differentiable with the derivative function f(x) in a neighborhood Np ofsuch that0<d=sup{f(x):x∈Np}<∞. Put τn=(?)(logn)3/2/n1/2(loglogn)1/2, Dn=《ξp-τn,ξp+τn], then with probability1(wp1)Let {Xn,n>1} be a sequence of NOD random variables. Assume that the com-mon marginal distribution function F(x) is continuously differentiable with the derivative function f(x) in a neighborhood Np of ξp such that0<d=sup{/(x): x∈Np}<∞. Futhermore, suppose that F'(ξp)=f{ξp)>0and f'(x) is bounded in some neighborhood of ξP. Put τn=(?)(log logn)1/2, Dn=[ξp-τn,ξp+τn}. ThenWe have a better bound than the one obtained by Ling'2'3'. Since NOD is weaker than NA, our results generalize and improve the corresponding results obtained by Ling'23).In addition, by applying the maximal type inequality of NOD random variables and the truncated method of random varialbles, the complete convergence theorems for arrays of rowwise NOD random variables are studied. Our results not only generalize ones of Hu and Taylor'08] for independent random variables, but also extend the corresponding ones of Zhu, Wang and Wu, Shen and so on. Last, the strong convergence result for the weighted sums for NOD sequence; is obtained.Finally, we study another sequence of AANA random variables, which is also weaker than NA sequence, and give out Hajek-Renyi type inequalities for the partial sums for AANA random variables. Our results weaken the conditions of the theorems in Ko ct al. and give the accurate coefficient of Cp. So our results generalize and improve the results of Theorem2.4, Theorem2.5and Theorem2.6in Ko et al., respectively. Mean-while, the large deviation and Marcinkiewicz type strong law of large numbers for AANA sequence are shown, which generalize the corresponding results of independent and iden-tically distributed random vaviables and NA random variables.