- Open Access
Strong convergence results for arrays of rowwise pairwise NQD random variables
© Tang; licensee Springer. 2013
- Received: 5 September 2012
- Accepted: 22 February 2013
- Published: 14 March 2013
Let be an array of rowwise pairwise NQD random variables. Some sufficient conditions of complete convergence for weighted sums of arrays of rowwise pairwise NQD random variables are presented without assumption of identical distribution. Our results partially extend the corresponding ones for independent random variables and negatively associated random variables.
- arrays of rowwise pairwise NQD random variables
- weighted sums
- complete convergence
Throughout the paper, let be the indicator function of the set A. C denotes a positive constant which may be different in various places and stands for . Denote , where lnx is the natural logarithm.
The concept of complete convergence was introduced by Hsu and Robbins  as follows. A sequence of random variables is said to converge completely to a constant C if for all . In view of the Borel-Cantelli lemma, this implies that almost surely (a.s.). The converse is true if the are independent. Hsu and Robbins  proved that the sequence of arithmetic means of independent and identically distributed (i.i.d.) random variables converges completely to the expected value if the variance of the summands is finite. Since then many authors, such as Spitzer , Baum and Katz , Gut  and so forth, have studied the complete convergence for partial sums and weighted sums of random variables. The main purpose of the present investigation is to provide the complete convergence results for weighted sums of arrays of rowwise pairwise negatively quadrant dependent random variables.
Firstly, let us recall the definition of pairwise negatively quadrant dependent random variables.
A sequence of random variables is said to be pairwise NQD if and are NQD for all and .
An array of random variables is called rowwise pairwise NQD random variables if for every , are pairwise NQD random variables.
The concept of pairwise NQD was introduced by Lehmann . Obviously, a sequence of pairwise NQD random variables is a family of very wide scope, which contains a pairwise independent random variable sequence and a negatively orthant dependent (NOD) random variable sequence as special cases. For more details about NOD random variables, one can refer to Joag-Dev and Proschan , Wang et al. [7, 8] and so forth. 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. Since Lehmann’s article appeared, a number of limit theorems for pairwise NQD random variables have been established. For example, Matula  obtained the Kolmogorov strong law of large numbers for a pairwise NQD random variable sequence with identical distribution, Wang et al.  and Wu  investigated some limit properties for such a sequence, Li and Wang  obtained the central limit theorem, Gan and Chen  studied further some limit properties for the pairwise NQD sequence without limitation of an identically distributed condition and obtained Baum-Katz (Baum and Katz ) type complete convergence and the strong stability of Jamison (Jamison et al. ) type weighted sums for the pairwise NQD sequence which may have different distributions, Huang et al.  studied the complete convergence for sequences of pairwise NQD random variables and so forth.
Our goal in this paper is to further study the complete convergence for weighted sums of arrays of rowwise pairwise NQD random variables under some moment conditions. We will give some sufficient conditions for complete convergence for an array of rowwise pairwise NQD random variables without assumption of identical distribution. The results presented in this paper are obtained by using the truncated method and the generalized Kolmogorov type inequality of pairwise NQD random variables.
for all , and .
The following lemmas are useful for the proof of the main results.
Lemma 1.1 (cf. Lehmann )
, for any ;
If f and g are both nondecreasing (or nonincreasing) functions, then and are NQD.
Lemma 1.2 (cf. Wu )
where and are positive constants.
Proof The proof can be found in Wu . So, we omit the details. □
The following two lemmas are from Sung .
which implies (2.5).
Therefore, follows from the statements above. This completes the proof of the theorem. □
Remark 2.1 The key to the proof of Theorem 2.1 is the Kolmogorov type inequality for pairwise NQD random variables. For many sequences of random variables, such as independent sequence, negatively associated sequence (see Shao ), negatively dependent sequence (see Asadian et al. ), -mixing sequence (see Utev and Peligrad ), φ-mixing sequence (see Wang et al. ) and so forth, the Kolmogorov type inequality also holds. So, Theorem 2.1 also holds for these sequences.
holds for any under the conditions of Theorem 2.1. The authors suggest that a solution can be obtained if a better moment inequality than the one presented above in Lemma 1.2 could be established.
The authors are most grateful to the editor Andrei Volodin and the anonymous referee for careful reading of the manuscript and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the Project of the Feature Specialty of China (TS11496) and the Scientific Research Projects of Fuyang Teachers College (2009FSKJ09).
- Hsu PL, Robbins H: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 1947, 33(2):25–31. 10.1073/pnas.33.2.25MathSciNetView ArticleGoogle Scholar
- Spitzer FL: A combinatorial lemma and its application to probability theory. Trans. Am. Math. Soc. 1956, 82(2):323–339. 10.1090/S0002-9947-1956-0079851-XMathSciNetView ArticleGoogle Scholar
- Baum LE, Katz M: Convergence rates in the law of large numbers. Trans. Am. Math. Soc. 1965, 120: 108–123. 10.1090/S0002-9947-1965-0198524-1MathSciNetView ArticleGoogle Scholar
- Gut A: Complete convergence for arrays. Period. Math. Hung. 1992, 25(1):51–75. 10.1007/BF02454383MathSciNetView ArticleGoogle Scholar
- Lehmann EL: Some concepts of dependence. Ann. Math. Stat. 1966, 37: 1137–1153. 10.1214/aoms/1177699260View ArticleGoogle Scholar
- Joag-Dev K, Proschan F: Negative association of random variables with applications. Ann. Stat. 1983, 11: 286–295. 10.1214/aos/1176346079MathSciNetView ArticleGoogle Scholar
- Wang XJ, Hu SH, Yang WZ, Ling NX: Exponential inequalities and inverse moment for NOD sequence. Stat. Probab. Lett. 2010, 80(5–6):452–461. 10.1016/j.spl.2009.11.023MathSciNetView ArticleGoogle Scholar
- Wang XJ, Hu SH, Shen AT, Yang WZ: An exponential inequality for a NOD sequence and a strong law of large numbers. Appl. Math. Lett. 2011, 24: 219–223. 10.1016/j.aml.2010.09.007MathSciNetView ArticleGoogle Scholar
- Matula P: A note on the almost sure convergence of sums of negatively dependent random variables. Stat. Probab. Lett. 1992, 15: 209–213. 10.1016/0167-7152(92)90191-7MathSciNetView ArticleGoogle Scholar
- Wang YB, Su C, Liu XG: On some limit properties for pairwise NQD sequences. Acta Math. Appl. Sin. 1998, 21: 404–414.MathSciNetGoogle Scholar
- Wu QY: Convergence properties of pairwise NQD random sequences. Acta Math. Sin. 2002, 45: 617–624.Google Scholar
- Li YX, Wang JF: An application of Stein’s method to limit theorems for pairwise negative quadrant dependent random variables. Metrika 2008, 67(1):1–10.MathSciNetView ArticleGoogle Scholar
- Gan SX, Chen PY: Some limit theorems for sequences of pairwise NQD random variables. Acta Math. Sci. 2008, 28(2):269–281.MathSciNetView ArticleGoogle Scholar
- Jamison B, Orey S, Pruitt W: Convergence of weighted averages of independent random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb. 1965, 4: 40–44. 10.1007/BF00535481MathSciNetView ArticleGoogle Scholar
- Huang HW, Wang DC, Wu QY, Zhang QX: A note on the complete convergence for sequences of pairwise NQD random variables. Arch. Inequal. Appl. 2011., 2011: Article ID 92. doi:10.1186/1029–242X-2011–92Google Scholar
- Wu QY: Probability Limit Theory for Mixing Sequences. Science Press of China, Beijing; 2006.Google Scholar
- Sung SH:On the strong convergence for weighted sums of -mixing random variables. Stat. Pap. 2012. doi:10.1007/s00362–012–0461–2Google Scholar
- Shao QM: A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theor. Probab. 2000, 13(2):343–356. 10.1023/A:1007849609234View ArticleGoogle Scholar
- Asadian N, Fakoor V, Bozorgnia A: Rosenthal’s type inequalities for negatively orthant dependent random variables. JIRSS 2006, 5(1–2):69–75.Google Scholar
- Utev S, Peligrad M: Maximal inequalities and an invariance principle for a class of weakly dependent random variables. J. Theor. Probab. 2003, 16(1):101–115. 10.1023/A:1022278404634MathSciNetView ArticleGoogle Scholar
- Wang XJ, Hu SH, Yang WZ, Shen Y: On complete convergence for weighted sums of φ -mixing random variables. Arch. Inequal. Appl. 2010., 2010: Article ID 372390Google 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.