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.
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 .
2 Main results
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).
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