- Open Access
Some convergence properties for weighted sums of pairwise NQD sequences
© Xu and Tang; licensee Springer 2012
- Received: 6 June 2012
- Accepted: 25 September 2012
- Published: 1 November 2012
Some properties for pairwise NQD sequences are discussed. Some strong convergence results for weighted sums of pairwise NQD sequences are obtained, which generalize the corresponding ones of Wang et al. (Bull. Korean Math. Soc. 48(5):923-938, 2011) for negatively orthant dependent sequences.
- strong convergence
- almost sure convergence
- pairwise NQD random variables
A sequence of random variables is said to be pairwise NQD if and are NQD for all and .
The concept of pairwise NQD was introduced by Lehmann  and many applications have been found. See, for example, Matula , Wang et al. , Wu , Li and Wang , Gan , Huang et al. , Chen , and so forth. Obviously, the sequence of pairwise NQD random variables is a family of very wide scope, which contains a pairwise independent random variable sequence. Many known types of negative dependence such as negative upper (lower) orthant dependence and negative association (see Joag-Dev and Proschan ) have been developed on the basis of this notion. Among them the negatively associated class is the most important and special case of a pairwise NQD sequence. So, it is very significant to study probabilistic properties of this wider pairwise NQD class which contains negatively orthant dependent (NOD) random variables as special cases. The main purpose of this paper is to study strong convergence results for weighted sums of pairwise NQD random variables, which generalize the previous known results for negatively associated random variables and negatively orthant dependent random variables, such as those of Wang et al. [10, 11].
Throughout the paper, denote , . C denotes a positive constant which may be different in various places. Let denote that there exists a constant such that for sufficiently large n. The main results of this paper depend on the following lemmas:
Lemma 1.1 (Lehmann )
, for any ;
If f and g are both nondecreasing (or nonincreasing) functions, then and are NQD.
Lemma 1.2 (Matula )
If , then ;
If , and , then .
Lemma 1.3 (Wu )
Let be a sequence of pairwise NQD random variables. If , then converges almost surely.
In this section, we will provide some properties for pairwise NQD random variables.
Property 2.1 Let be a sequence of pairwise NQD random variables, for any , .
Proof ‘⇐’ If , then a.s. follows immediately from the Borel-Cantelli lemma.
it follows that , . By Lemma 1.1(iii), we can see that and are both pairwise NQD. By Lemma 1.1(ii) and Lemma 1.2(ii), we have , . Therefore, . □
Property 2.2 Let be a sequence of pairwise NQD random variables and be a sequence of positive numbers. Denote , and for , then implies .
Proof It is easily seen that , and for each . Thus, implies that , or . By Lemma 1.2(ii), we have or , which implies that . □
Property 2.3 Under the conditions of Property 2.2, .
In this section, we will provide some sufficient conditions to prove the strong convergence for weighted sums of pairwise NQD random variables.
in order to show a.s., we only need to show that the first three terms above are or a.s.
By Lemma 1.3 and Kronecker’s lemma, we have a.s.
Hence, the desired result (3.3) follows from the statements above immediately. □
which implies (3.11) by Toeplitz’s lemma. The proof is completed. □
The authors are most grateful to the editor Andrei I Volodin and the anonymous referee for careful reading of the manuscript and valuable suggestions which helped significantly improve an earlier version of this paper. This work was supported by the Natural Science Project of Department of Education of Anhui Province (KJ2011z056).
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