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Some convergence properties for weighted sums of pairwise NQD sequences
Journal of Inequalities and Applications volume 2012, Article number: 255 (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.
Definition 1.1 Two random variables X and Y are said to be negatively quadrant dependent (NQD) if for any ,
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 )
Let X and Y be NQD, then
, for any ;
If f and g are both nondecreasing (or nonincreasing) functions, then and are NQD.
Lemma 1.2 (Matula )
Let be a sequence of events defined on a fixed probability space .
If , then ;
If , and , then .
Lemma 1.3 (Wu )
Let be a sequence of pairwise NQD random variables. If , then converges almost surely.
2 Properties for pairwise NQD random variables
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.
‘⇒’ Let a.s., we can see that a.s., a.s. Denote
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, .
3 Strong convergence properties for weighted sums of pairwise NQD random variables
In this section, we will provide some sufficient conditions to prove the strong convergence for weighted sums of pairwise NQD random variables.
Theorem 3.1 Let and be a sequence of mean zero pairwise NQD random variables with identical distribution
where is a slowly varying function at infinity and as for all . Let and be sequences of positive constants satisfying . Denote and for . Assume that
Proof By the Borel-Cantelli lemma and Kronecker’s lemma, it is easily seen that (3.2) implies that
then is still pairwise NQD from Lemma 1.1(iii). Since
in order to show a.s., we only need to show that the first three terms above are or a.s.
By inequality, Theorem 1b in Feller [, p.281] and (3.2), we can get
By Lemma 1.3 and Kronecker’s lemma, we have a.s.
By (3.2) again,
which implies that
By Kronecker’s lemma, it follows that
By Theorem 1a in Feller [, p.281] and (3.2) again, we have
which implies that
By Kronecker’s lemma, it follows that
Hence, the desired result (3.3) follows from the statements above immediately. □
Theorem 3.2 Let , be a sequence of pairwise NQD random variables with identical distribution and be sequences of positive numbers with . Denote and for . Assume that
Proof Let and
By (3.7), it is easily seen that as (otherwise, there exist infinite subscripts i and some such that , hence, , which is contrary to from (3.7)). It follows by (3.6) and (3.7) that
By the inequality above and Borel-Cantelli lemma, we can see that . Therefore, in order to prove (3.8), we only need to prove
By (3.6) and (3.7) again,
Therefore, by the inequality above, Lemma 1.3 and Kronecker’s lemma, we have
In order to prove (3.9), it suffices to prove that
It is easily seen that for implies that , thus
By the Lebesgue dominated convergence theorem and , we have
which implies (3.11) by Toeplitz’s lemma. The proof is completed. □
Remark 3.1 In Theorem 3.2, the condition can be relaxed to when . It suffices to prove (3.11). In fact, it follows by (3.6) and (3.7) that
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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).
The authors declare that they have no competing interests.
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Cite this article
Xu, H., Tang, L. Some convergence properties for weighted sums of pairwise NQD sequences. J Inequal Appl 2012, 255 (2012). https://doi.org/10.1186/1029-242X-2012-255
- strong convergence
- almost sure convergence
- pairwise NQD random variables