Almost Sure Convergence of Pairwise NQD Random Sequence

In this paper, considering the sequences of pairwise NQD that is applied broadly through introducing the slowly varying function, we extend a series of conclusions.


Introduction and Lemmas
Definition 1 ( Lehmann E L. 1966 ) : Let random variables X and Y are said to be NQD (Negatively Quadrant Where  i j  , X and Y are said to be NQD (Negatively Quadrant Dependent), A sequence of random variables  ; 1 n X n  is said to be pairwise NQD.
This definition was introduced by Lehmann.Obviously, Pairwise NQD random sequences contains a kind of negatively correlative sequences of NA, LNQD, ND random variables.For NA random sequences, a number of writers have obtained as same as the convergence property in many independent conditions, the properties of limit behavior of LNQD, ND sequences seldom appear in literature.Matula (Matula P. 1992)  In the following, Almost Sure Convergence of NQD pair-wise random Sequences are extened on the condation that slowly varying function.
For positive functions, as we know, if there exist a Positive Function such as   0 (3) In this paper, c is usually said to be different real constants and ( ) l x is the slowly varying function.Lemma 1 (Lehmann E L. 1966) Assume that random variables X and Y are NQD , x y R  (3)If the functions of r and s with non descending(non incremental),then ( ) r X and ( ) s Y are NQD.
Lemma 2: Assume that ( , ) g a k is the function of Joint Distribution ( 0, 1) If there exists the slowly varying function ( ) l x Such that ( ) ( ) ( ) l t l n l tn   , for 0 t   , we can easily get the following results such that Proof.Because of randomicity of a, using a fixed.If t is even, either let t=2,we have Mathematic Induction , if n=1, we get Applying on both sides of expectations by inequation of Cauchy-Schwarz. (2) Consequence, the conclusion is satisfied by mathematical Induction, where t  even, 0, 1 a n   .

Main results and the proofs.
Theorem 1: Let ( , ) f a k and ( , ) g a k are the functions of Joint Distribution ( 0, 1)  that satisfied and ( , ) f a k is satisfied with supposed that(1)and(2).Assume: There exist that ( ) l x is the slowly varying function,such that 2 ( , ) ( , ) ( ) Proof.From(6), we have Applying the properties of Lemma2 and (3)(4)(5)and the slowly varying function,we have It is easy to check that(1) (2) (3) (4) and (5) hold,using Theorem , we obtain the infer.
obtained strong law of large numbers for Kolmogorov as same as them under the independent, the distinguishing theorem of the Complete Convergence of Baum and Kata(WANG Y B, Chun Su, & Xuguo Liu.1998) was obtained by author Yuebao Wang who considered Wu (WU Q Y. 2005) obtained the weak law of large numbers and the criteria theorem of Complete Convergence of Baum and Kata (WU Q Y. 2002) on the condation that pairwise NQD.These properties achieved the results of independence condition, and the results of properties of Jamison(Jamsion B, O Rey S, & Pruitt W. 1965) Weighted Sums is obtained.Currently, a number of writers have studied a series of useful results of the limit of pairwise NQD random Sequence.Yuebao Wang (WANG Y B, YAN Ji-gao, & CHENG Feng-yang etal.2001) studied the strong stability of different distribution pairwise NQD, Wancheng Gao(WAN Cheng-gao.2005) studied f the weak law of large numbers of pairwise NQD and in condition of rank 2 Cesàro's uniformly integrability is the convergence properties of r L and Yanping Chen(CHEN Ping-yan.2008) studied the convergence properties of r L satisfied pairwise NQD in the condition of uniformly integrability is (integrability as same as them under the condition of independent, and the teacher Yuebao Wang Wang(WANG Y B, YAN J G, & CAI X Z. 2001) studied different distributions strong stability of pair-wise NQD, and Qunying Wu(WU Q Y. 2002) studied the three series theorem of pair-wise NQD.
 is a sequence of Cauchy satisfying 2 L , because of completeness of 2 L , there exist a