Research Article | Open Access
Sufficient and Necessary Conditions of Complete Convergence for Weighted Sums of PNQD Random Variables
The complete convergence for pairwise negative quadrant dependent (PNQD) random variables is studied. So far there has not been the general moment inequality for PNQD sequence, and therefore the study of the limit theory for PNQD sequence is very difficult and challenging. We establish a collection that contains relationship to overcome the difficulties that there is no general moment inequality. Sufficient and necessary conditions of complete convergence for weighted sums of PNQD random variables are obtained. Our results generalize and improve those on complete convergence theorems previously obtained by Baum and Katz (1965) and Wu (2002).
1. Introduction and Lemmas
Random variables and are said to be negative quadrant dependent (NQD) if for all . A sequence of random variables is said to be pairwise negative quadrant dependent (PNQD) if every pair of random variables in the sequence is NQD. This definition was introduced by Lehmann . Obviously, PNQD sequence includes many negatively associated sequences, and pairwise independent random sequence is the most special case.
In many mathematics and mechanic models, a PNQD assumption among the random variables in the models is more reasonable than an independence assumption. PNQD series have received more and more attention recently because of their wide applications in mathematics and mechanic models, percolation theory, and reliability theory. Many statisticians have investigated PNQD series with interest and have established a series of useful results. For example, Matula , Li and Yang , and Wu and Jiang  obtained the strong law of large numbers, Wang et al.  obtained the Marcinkiewicz's weak law of large numbers, Wu  obtained the strong convergence properties of Jamison weighted sums, the three-series theorem, and complete convergence theorem, and Li and Wang  obtained the central limit theorem. It is interesting for us to extend the limit theorems to the case of PNQD series. However, so far there has not been the general moment inequality for PNQD sequence, and therefore the study of the limit theory for PNQD sequence is very difficult and challenging. In the above-mentioned conclusions, only the Kolmogorov-type strong law of large numbers obtained by Matula [2, Theorem 1] and Baum and Katz-type complete convergence theorem obtained by Wu [6, Theorem 4] achieve the corresponding conclusions of independent cases, and the rest did not achieve the optimal results of independent cases.
Complete convergence is one of the most important problems in probability theory. Recent results of the complete convergence can be found in Wu , Chen and Wang , and Li et al. . In this paper, we establish a collection that contains relationship to overcome the difficulties that there is no the general moment inequality and obtain the complete convergence theorem for weighted sums of PNQD sequence, which extend and improve the corresponding results of Baum and Katz  and Wu .
Lemma 1.1 (see ). Let and be NQD random variables. Then (i), (ii), for all , (iii)if and are Borel functions, both of which being monotone increasing (or both are monotone decreasing), then and are NQD.
Lemma 1.2 (see [6, Lemma 2]). Let be a sequence of PNQD random variables with . Then
Lemma 1.3 (see [2, Lemma 1]).
(i) If , then .
(ii) if , and , then .
Lemma 1.4. Let be a sequence of PNQD random variables. Then for any , there exists a positive constant such that for all ,
Proof. We can prove the Lemma by Lemma of Zhang and Wen .
2. Main Results and the Proof
In the following, the symbol stands for a generic positive constant which may differ from one place to another. Let () denote that there exists a constant such that () for all sufficiently large , and let () denote that there exists a constant such that () for all and .
Theorem 2.1. Let be a sequence of PNQD random variables with . Let be a sequence of real numbers such that
Let for , and , for . If
then where .
Remark 2.4. When is i.i.d. and , for all , then Theorems 2.1 and 2.2 become Baum and Katz  complete convergence theorem. Hence, our Theorems 2.1 and 2.2 improve and extend the well-known Baum and Katz theorem.
Proof of Theorem 2.1. Without loss of generality, assume that for . Let such that . For all , let
Firstly, we prove that
For any , we have and for any , Hence where the symbol denotes the number of elements in the set .
When , then for any ; thus, , and therefore by (2.8),
When (or ), then there exists only an : such that (or ), the remaining ; thus, . If , then . If , then by (2.8),
When , then there exist such that , the remaining ; thus, . Without loss of generality, assume that . If , then ; if , then by (2.8), If , then by (2.8), Hence, (2.7) holds, that is: Therefore, in order to prove (2.3), we only need to prove that By (2.1), (2.2), , and , That is, (2.16) holds.
By Lemma 1.1(ii), , and the definition of , That is, (2.17) holds.
In order to prove (2.18), firstly, we prove that
When , then ; from and the definition of , we have
When , and , then from (2.2), thus,
When , and , by , and , we get Hence, (2.21) holds; that is, for any , we have for all sufficiently large . Thus,
Let . Obviously, is monotonic on . By Lemma 1.1(iii), is also a sequence of PNQD random variables with , by Lemma 1.2 and : This completes the proof of Theorem 2.1.
Proof of Theorem 2.2. Noting that and , from (2.3), Thus, by , we get This implies that Hence, for all sufficiently large , By Lemma 1.4, which together with (2.27), By , we obtain This completes the proof of Theorem 2.2.
The author is very grateful to the referees and the editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper. Supported by the National Natural Science Foundation of China (11061012), and project supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ( 47), and the support program of Key Laboratory of Spatial Information and Geomatics (1103108-08).
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Copyright © 2012 Qunying Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.