A note on the strong limit theorem for weighted sums of sequences of negatively dependent random variables
© Huang and Wang; licensee Springer 2012
Received: 15 February 2012
Accepted: 10 August 2012
Published: 17 October 2012
In this paper, the strong limit theorem for weighted sums of sequences of negatively dependent random variables is further studied. As an application, the complete convergence theorem for sequences of negatively dependent random variables is obtained. Our results partly generalize and improve the corresponding results of Cai (Metrika 68:323-331, 2008) and Wang et al. (Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. a Mat., 2011, doi:10.1007/s13398-011-0048-0) to negatively dependent random variables under mild moment conditions.
for all . An infinite sequence of random variables is said to be ND if every finite subset is ND.
An array of random variables is called rowwise ND random variables if for every , is a sequence of ND random variables.
whenever and are increasing for every variable (or decreasing for every variable), such that the covariance exists. An infinite family of random variables is said to be NA if every finite subfamily is NA.
The concept of ND random variables was introduced by Ebrahimi and Ghosh, and the concept of NA random variables was introduced by Joag-Dev and Proschan . Obviously, independent random variables are ND. Joag-Dev and Proschan  pointed out that NA random variables are ND. They also presented an example in which possesses ND, but does not possess NA. So, we can see that ND is much weaker than NA. Because of the wide applications of ND random variables, the notions of ND random variables have received more and more attention recently. A large number of limit theorems for ND random variables have been established by many authors. We can refer to [2–12]etc. Hence, extending the limit properties of independent or NA random variables to the case of ND random variables is highly desirable and of considerable significance in theory and application.
As Bai and Cheng  remarked, many useful linear statistics based on a random sample are weighted sums of independent identically distributed (i.i.d.) random variables. Examples include least-squares estimators, nonparametric regression function estimators, jackknife estimates and so on. In this respect, studies of strong laws for these weighted sums have demonstrated significant progress in probability theory with applications in mathematical statistics; many authors studied the strong laws for weighted sums of random variables. In the case of independence, Bai and Cheng  proved the following strong laws of large numbers for weighted sums.
Theorem 1.1 (Bai and Cheng )
Theorem 1.2 (Bai and Cheng )
Wu generalized and improved the above Theorem 1.1 to ND random variables, removed the identically distributed condition as follows.
Theorem 1.3 (Wu )
Recently, Cai and Wang et al. have studied the complete convergence for NA random variables and arrays of rowwise ND random variables under the exponential moment conditions respectively. Their results generalize and improve the above Theorem 1.2 to NA and ND random variables. Inspired by Cai , Wang et al. , and other papers mentioned above, we investigate the limit behavior of ND random variables and obtain some complete convergence results. We use methods different from those of Cai and Wang et al. .
The main purpose of this paper is to further study the complete convergence for ND random variables and arrays of rowwise ND random variables under weaker moment conditions. As applications, the complete convergence for linear statistics of ND random variables and arrays of rowwise ND random variables are obtained without assumptions of being identically distributed. The results obtained not only generalize the above Theorem 1.2 to the case of ND and arrays of rowwise ND random variables, but also partly improve the corresponding results of Cai  and Wang et al. .
Throughout this paper, C will represent a positive constant whose value may change from one appearance to the next, and will mean .
for all and . Then is said to be stochastically dominated by X.
2 Main results and proofs
Now, we state and prove our main results of this paper.
In order to prove our results, we need the following lemmas.
Lemma 2.1 (Bozorgnia et al. )
Let random variables be ND, be all nondecreasing (or all nonincreasing) functions, then random variables are ND.
Lemma 2.2 (Asadian et al. )
The proof is similar to that of Lemma 2.3 of Sung . So, we omit it.
which implies that for .
If , it easily follows that for .
where for some and .
Next, we will prove in the following two cases.
Hence, the desired result (2.2) follows from (2.15)-(2.19) immediately. The proof of Theorem 2.1 is complete. □
Similar to the proof of Theorem 2.1, we can get the following results for arrays of rowwise ND random variables.
Theorem 2.2 Let be an array of rowwise ND random variables which is stochastically dominated by a random variable X. Let , , where the weights are a triangular array of real constants such that (2.1). Let . If for and , then (2.2) holds true.
It is still an open problem to obtain results of this type for ND random variables under the conditions of and . One suggests that a solution can be obtained if a better moment inequality than that presented above in Lemma 2.2 could be established.
The authors are very grateful to the anonymous referees and the editor Prof Andrei Volodin for their valuable comments and some helpful suggestions that improved the clarity and readability of the article. This work was supported by the Project Supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning (47), the National Natural Science Foundation of China (No:71271042, 11061012), the Plan of Jiangsu Specially-Appointed Professors and the Major Program of Key Research Center in Financial Risk Management of Jiangsu Universities of philosophy and social sciences (No:2012JDXM009).
- Ebrahimi N, Ghosh M: Multivariate negative dependence. Commun. Stat., Theory Methods 1981, 10(4):307–337. 10.1080/03610928108828041MathSciNetView ArticleMATHGoogle Scholar
- Joag-Dev K, Proschan F: Negative association of random variables with applications. Ann. Stat. 1983, 11(1):286–295. 10.1214/aos/1176346079MathSciNetView ArticleMATHGoogle Scholar
- Volodin A: On the Kolmogorov exponential inequality for negatively dependent random variables. Pak. J. Stat., Ser. A 2002, 18: 249–254.MathSciNetMATHGoogle Scholar
- Asadian N, Fakoor V, Bozorgnia A: Rosental’s type inequalities for negatively orthant dependent random variables. J. Iran. Stat. Soc. 2006, 5(1–2):66–75.Google Scholar
- Sung SH: A note on the complete convergence for arrays of dependent random variables. J. Inequal. Appl. 2011., 2011: Article ID 76. doi:10.1186/1029–242X-2011–76Google Scholar
- Kuczmaszewska A: On some conditions for complete convergence for arrays of rowwise negatively dependent random variables. Stoch. Anal. Appl. 2006, 24: 1083–1095. 10.1080/07362990600958754MathSciNetView ArticleMATHGoogle Scholar
- Amini M, Zarei H, Bozorgnia A: Some strong limit theorems of weighted sums for negatively dependent generalized Gaussian random variables. Stat. Probab. Lett. 2007, 77: 1106–1110. 10.1016/j.spl.2007.01.015MathSciNetView ArticleMATHGoogle Scholar
- Amini M, Azarnoosh HA, Bozorgnia A: The strong law of large numbers for negatively dependent generalized Gaussian random variables. Stoch. Anal. Appl. 2004, 22: 893–901.MathSciNetView ArticleMATHGoogle Scholar
- Amini M, Bozorgnia A: Complete convergence for negatively dependent random variables. J. Appl. Math. Stoch. Anal. 2003, 16: 121–126. 10.1155/S104895330300008XView ArticleMathSciNetMATHGoogle Scholar
- Wu QY: A strong limit theorem for weighted sums of sequences of negatively dependent random variables. J. Inequal. Appl. 2010., 2010: Article ID 383805. doi:10.1155/2010/383805Google Scholar
- Wu QY: Complete convergence for negatively dependent sequences of random variables. J. Inequal. Appl. 2010., 2010: Article ID 507293. doi:10.1155/2010/507293Google Scholar
- Wang XJ, Hu SH, Yang WZ: Complete convergence for arrays of rowwise negatively orthant dependent random variables. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. a Mat. 2011. doi:10.1007/s13398–011–0048–0Google Scholar
- Bai ZD, Cheng PE: Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett. 2000, 46: 105–112. 10.1016/S0167-7152(99)00093-0MathSciNetView ArticleMATHGoogle Scholar
- Cai GH: Strong laws for weighted sums of NA random variables. Metrika 2008, 68: 323–331. doi:10.1007/s00184–007–0160–5 10.1007/s00184-007-0160-5MathSciNetView ArticleMATHGoogle Scholar
- Bozorgnia A, Patterson RF, Taylor RL: Limit theorems for dependent random variables. World Congress Nonlinear Analysts’92 1996, 1639–1650.Google Scholar
- Sung SH: On the strong convergence for weighted sums of random variables. Stat. Pap. 2011, 52: 447–454. 10.1007/s00362-009-0241-9View ArticleMathSciNetMATHGoogle 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.