Generalization of Almost Sure Convergence Properties of Pairwise NQD Random Sequences

Received: March 29, 2011 Accepted: April 19, 2011 doi:10.5539/jmr.v3n3p73 The research is financed by the Guangxi National Natural Science Foundation of China (No. 2010GXNSFA013121) Abstract Some sufficient conditions on the almost sure convergence of NQD pairwise random sequences are obtained by using the properties of some slowly varying functions.


Introduction
This definition was introduced by Lehmann(1966).Obviously, pairwise NQD random sequences was a widely sequence of random variables.A series of negatively correlative sequences of NA(1983), LNQD, ND random variables based on it.Currently, a number of writers have studied a series of useful results of the limit of pairwise NQD random sequences.Wancheng Gao (2005)studied the weak law of large number of to the case pairwise NQD of random sequences, and the teacher Yuebao Wang (1998) studied different distributions strong stability of pairwise NQD random sequences, and Qunying Wu (2005)for the three series theorem of pairwise NQD random sequences.
The main purpose of this paper is to study and extend almost sure convergence of NQD pairwise random sequences that there exist some slowly varying function (1976).

Definition
Two random variables X and Y are said to be negative quadrant dependent (NQD, in short) if for any x, y ∈ R, P(X < x, Y < y) ≤ P(X < x)P(Y < y). (1) Where i j , X i and X j are said to be NQD,A sequence {X n ; n ≥ 1}of random variables is said to be pairwise NQD.

Lemma 1
Let random variables X and Y be NQD, then (1) EXY ≤ EXEY; (2) P(X < x, Y < y) ≤ P(X < x)P(Y < y); (3) I f f and g are both nondecreasing (or both noncreasing) f unctions, then f (X) and g(Y) are NQD. (2) Assume that f (a, k)is the function of joint distribution X a+1 , X a+2 , ...X a+k , (a ≥ 0, k > 1)that satisfies: If there exists the slowly varying function l(x),such that, where Proof of Lemma 2. We have Mathematic Induction , if n = 1, form (4), we get (5).Assume that (5) exists n < N and a > 0 , then two conditions of 1 Therefore, Applying on both sides of expectations by inequation of Cauchy-Schwarz,we get Then we have, And, Therefore,when N is even number ,we get 1 < n ≤ N 2 and N 2 ≤ n < N. Then the conclusion is also satisfied.Finally, we have (5).

Lemma 4
(1) I f (2) I f P(A k A m ) ≤ P(A k )P(A m ), k m, and 3. Main results and the proofs

Theorem
Suppose {X n ; n ≥ 1}is pairwise NQD random sequences, and satisfyied There exists that l(x)(x → ∞)is the slowly varying function of monotonically nondecreasing , then Proof of Theorem.Assume EX n = 0,if positive integer m > n → ∞, by(lemma 3) ,we have Hence, {S n , n ≥ 1} is a sequence of Cauchy satisfying L 2 , because of completeness of L 2 ,∃r.v.S satisfied that Applying ( 20) and the properties of slowly varying function, we obtain EX 2 i l 2 (i) Since(lemma 4),if k → ∞,we get max Finally, we have S n a.s.
This completes the proof of Theorem.