Sufﬁcient and Necessary Conditions of Complete Convergence for Weighted Sums of PNQD Random Variables

The complete convergence for pairwise negative quadrant dependent (cid:2) PNQD (cid:3) random variables is studied. So far there has not been the general moment inequality for PNQD sequence, and therefore the study of the limit theory for PNQD sequence is very di ﬃ cult and challenging. We establish a collection that contains relationship to overcome the di ﬃ culties that there is no general moment inequality. Su ﬃ cient and necessary conditions of complete convergence for weighted sums of PNQD random variables are obtained. Our results generalize and improve those on complete convergence theorems previously obtained by Baum and Katz (cid:2) 1965 (cid:3) and Wu (cid:2) 2002 (cid:3) .


Introduction and Lemmas
Random variables X and Y are said to be negative quadrant dependent NQD if 1.1 for all x, y ∈ R. A sequence of random variables {X n ; n ≥ 1} is said to be pairwise negative quadrant dependent PNQD if every pair of random variables in the sequence is NQD.This definition was introduced by Lehmann 1 .Obviously, PNQD sequence includes many negatively associated sequences, and pairwise independent random sequence is the most special case.In many mathematics and mechanic models, a PNQD assumption among the random variables in the models is more reasonable than an independence assumption.PNQD series have received more and more attention recently because of their wide applications Lemma 1.1 see 1 .Let X and Y be NQD random variables.Then iii if f and g are Borel functions, both of which being monotone increasing (or both are monotone decreasing), then f X and g Y are NQD.
Lemma 1.4.Let {X n ; n ≥ 1} be a sequence of PNQD random variables.Then for any x ≥ 0, there exists a positive constant c such that for all n ≥ 1, Proof.We can prove the Lemma by Lemma A.6 of Zhang and Wen 11 .

Main Results and the Proof
In the following, the symbol c stands for a generic positive constant which may differ from one place to another.Let a n b n a n b n denote that there exists a constant c > 0 such that a n ≤ cb n a n ≥ cb n for all sufficiently large n, and let X i ≺ X X i X denote that there exists a constant c > 0 such that Theorem 2.1.Let {X n ; n ≥ 1} be a sequence of PNQD random variables with X i ≺ X.Let {a nk ; k ≤ n, n ≥ 1} be a sequence of real numbers such that 2.1 Let for αp > 1, 0 < p < 2, α > 0, and where

2.4
Hence, Theorem 4 in Wu 6 is a particular case of our Theorem 2.1.
Remark 2.4.When {X n ; n ≥ 1} is i.i.d. and a ni n −α , for all i ≤ n, n ≥ 1, then Theorems 2.1 and 2.2 become Baum and Katz 10 complete convergence theorem.Hence, our Theorems 2.1 and 2.2 improve and extend the well-known Baum and Katz theorem.

2.6
Firstly, we prove that max 1≤k≤n

2.7
For any ω ∈ D n , we have and for any 1

2.9
Hence where the symbol A denotes the number of elements in the set A.
When a b 0, then |a ni X i ω | ≤ n α q−1 for any 1 ≤ i ≤ n; thus, Y ni ω X i ω , and therefore by 2.8 , When a 1, b 0 or a 0, b 1 , then there exists only an i 0 :

2.12
When a b 1, then there exist

2.14
Hence, 2.7 holds, that is: Therefore, in order to prove 2.3 , we only need to prove that
By Lemma 1.1 ii , X i ≺ X, and the definition of q, αp 1
In order to prove 2.18 , firstly, we prove that max 1≤k≤n

2.33
This completes the proof of Theorem 2.2.