A note on the complete convergence for weighted sums of negatively dependent random variables
© Sung; licensee Springer 2012
Received: 7 April 2012
Accepted: 27 June 2012
Published: 11 July 2012
The complete convergence theorems for weighted sums of arrays of rowwise negatively dependent random variables were obtained by Wu (Wu, Q: Complete convergence for weighted sums of sequences of negatively dependent random variables. J. Probab. Stat. 2011, Article ID 202015, 16 pages) and Wu (Wu, Q: A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables. J. Inequal. Appl. 2012, 50). In this paper, we complement the results of Wu.
By the Borel-Cantelli lemma, this implies that almost surely (a.s.). The converse is true if are independent random variables. Therefore, the complete convergence is a very important tool in establishing almost sure convergence. There are many complete convergence theorems for sums and weighted sums of independent random variables.
Volodin et al.  and Chen et al.  ( and , respectively) proved the following complete convergence for weighted sums of arrays of rowwise independent random elements in a real separable Banach space.
where C is a positive constant.
If , then (1.3) is immediate. Hence Theorem 1.1 is of interest only for .
for all real numbers . An infinite family of random variables is negatively dependent if every finite subfamily is negatively dependent.
Theorem 1.2 (Wu )
- (i)If and , then(1.4)
If and , then (1.4) holds.
Using the moment inequality of negatively dependent random variables, Wu  obtained a complete convergence result for weighted sums of identically distributed negatively dependent random variables.
Theorem 1.3 (Wu )
For , (1.7) implies (1.8).
The proof of the sufficiency is correct when . However, condition (1.5) does not hold, since the left-hand side of (1.5) goes to the limit as , but the right-hand side diverges. Hence, there are no arrays satisfying (1.5).
Throughout this paper, the symbol C denotes a positive constant which is not necessarily the same one in each appearance. It proves convenient to define , where lnx denotes the natural logarithm.
2 Preliminary lemmas
In this section, we present some lemmas which will be used to prove our main results.
The following two lemmas are well known and their proofs are standard.
The following Lemma 2.2(i)-(iii) can be found in Sung .
for any .
for any such that .
The Marcinkiewicz-Zygmund and Rosenthal type inequalities play an important role in establishing complete convergence. Asadian et al.  proved the Marcinkiewicz-Zygmund and Rosenthal inequalities for negatively dependent random variables.
Lemma 2.3 (Asadian et al. )
The last lemma is a complete convergence theorem for an array of rowwise negatively dependent mean zero random variables.
for all .
- (ii)There exists such that
Then for all .
3 Main results
In this section, we obtain two complete convergence results for weighted sums of arrays of rowwise negatively dependent random variables.
We will prove (3.3) and (3.4) with three cases.
Case 1 ().
The sixth inequality follows from Lemma 2.2.
Since and as , (3.5) holds.
Case 2 ().
As in Case 1, we have that .
Case 3 ().
Hence (3.3) holds by Lemma 2.4.
To prove (3.4), we take such that . The proof of the rest is similar to that of (3.3) and is omitted. □
Remark 3.1 When , Theorem 3.1 holds without the condition of negative dependence (see Theorem 2(i) in Sung ). Theorem 3.1 extends the result of Sung  for independent random variables to negatively dependent case.
But, Theorem 1.2(i) does not deal with the case of .
The following theorem shows that if the moment condition of Theorem 3.1 is replaced by a stronger condition , then condition (3.1) can be replaced by the weaker condition (3.7).
Proof As in the proof of Theorem 3.1, it suffices to prove (3.3) and (3.4). The proof of (3.3) is same as that of Theorem 3.1 except that q is replaced by p.
Hence (3.4) holds by Lemma 2.4. □
Remark 3.3 If , then . Hence Theorem 1.2(ii) follows from Theorem 3.2 by taking . But, Theorem 1.2(ii) does not deal with the case of .
As mentioned in the Introduction, (1.5) does not hold. Hence it is of interest to find a complete convergence result similar to Theorem 1.3 without condition (1.5). The following corollary does not assume condition (1.5).
Hence the result follows from Theorem 3.2. □
Remark 3.4 When , Corollary 3.1 holds without the condition of negative dependence. Although (3.8) is weaker than (1.8), (3.8) can be strengthened to (1.8) if the negative dependence is replaced by the stronger condition of negative association.
The author would like to thank the referees for helpful comments and suggestions. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0013131).
- Hsu PL, Robbins H: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 1947, 33: 25–31. 10.1073/pnas.33.2.25MathSciNetView ArticleMATHGoogle Scholar
- Volodin A, Giuliano Antonini R, Hu TC: A note on the rate of complete convergence for weighted sums of arrays of Banach space valued random elements. Lobachevskii J. Math. 2004, 15: 21–33.MathSciNetMATHGoogle Scholar
- Chen P, Sung SH, Volodin AI: Rate of complete convergence for arrays of Banach space valued random elements. Sib. Adv. Math. 2006, 16: 1–14.MathSciNetGoogle Scholar
- Wu Q: A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables. J. Inequal. Appl. 2012., 2012: Article ID 50Google Scholar
- Lehmann EL: Some concepts of dependence. Ann. Math. Stat. 1966, 37: 1137–1153. 10.1214/aoms/1177699260View ArticleMathSciNetMATHGoogle Scholar
- Wu Q: Complete convergence for weighted sums of sequences of negatively dependent random variables. J. Probab. Stat. 2011., 2011: Article ID 202015Google Scholar
- Liang HY, Su C: Complete convergence for weighted sums of NA sequences. Stat. Probab. Lett. 1999, 45: 85–95. 10.1016/S0167-7152(99)00046-2MathSciNetView ArticleMATHGoogle Scholar
- Sung SH: Complete convergence for weighted sums of random variables. Stat. Probab. Lett. 2007, 77: 303–311. 10.1016/j.spl.2006.07.010View ArticleMathSciNetMATHGoogle Scholar
- Asadian N, Fakoor V, Bozorgnia A: Rosenthal’s type inequalities for negatively orthant dependent random variables. J. Iran. Stat. Soc. 2006, 5: 69–75.Google Scholar
- Dehua Q, Chang KC, Giuliano Antonini R, Volodin A: On the strong rates of convergence for arrays of rowwise negatively dependent random variables. Stoch. Anal. Appl. 2011, 29: 375–385. 10.1080/07362994.2011.548683MathSciNetView ArticleMATHGoogle Scholar
- Sung SH: A note on the complete convergence for arrays of dependent random variables. J. Inequal. Appl. 2011., 2011: Article ID 76Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.