Certain Limit Properties Of Two Types Of Dependent Random Variables

This thesis is mainly about the limit properties of the two dependent variables: NQD and NOD, and it is composed by two chapters.Chapter I is about the strong convergence properties of pairwise NQD sequences. The definition of pairwise NQD was first given by Lehmann(1966). The class of NQD random variables is a very broad class, including NA sequences which is widely studied by many scholars, moreover, it popularized the results of pairwise independent random variables.In this paper, we discuss the strong convergence of pairwise NQD sequences, which generalized the result of Martikainen (1995), and prove that the conclusion still holds 为 NQD random variables under the same conditions. The following is the main result. Theorem 1 Let { X_n,n ≥ 1} be a sequence of pairwise NQD identically distributed randomvariables, 1 <γ<2,Ï„ >0 and Ï„ >4γ-6 , if E | X₁ |γ (log+ | X₁ |)Ï„ < ∞,then(S_n-ES_n) /n â†' 0,a.s..Chapter II, we discuss the weak convergence and strong convergence of NOD sequences. The definition of NOD was due to Joag-Dev and Proschan (1983). The class of NOD random variables is a broad class including independent random variables and NA sequences. But NOD is a little stronger than NQD. In this chapter, I get some theorems of NOD, which extends the related results for independent random variables and NA sequences. The following law of the large numbers is one of our main results. Theorem 2 Let 1 ≤ p < 2 , and H(t) be a positive and increasing function, which takes values ininterval (0, ∞) , moreover, as t â†' ∞, H(t) â†' ∞. Suppose ↘ 0, and {X_i} is a sequence ofNOD being stochastically domained by X. If E | X |^p< ∞, Then,Also, we have the following complete convergence. Theorem 3 Let 1 ≤ p < 2,αp > 1, and {X_i} be a sequence of NOD random variables beingstochastically domained by X. Suppose E | X |^p< ∞. Then...