Abstract and Applied Analysis

Volume 2014 (2014), Article ID 949608, 7 pages

http://dx.doi.org/10.1155/2014/949608

Research Article

On the Strong Convergence and Complete Convergence for Pairwise NQD Random Variables

Aiting Shen,¹ Ying Zhang,¹ and Andrei Volodin²

¹School of Mathematical Science, Anhui University, Hefei 230601, China

²Department of Mathematics and Statistics, University of Regina, Regina, SK, Canada, S4S 0A2

Received 15 December 2013; Accepted 7 April 2014; Published 8 May 2014

Academic Editor: Angelo Favini

Copyright © 2014 Aiting Shen et al. 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.

Abstract.

Let be a sequence of positive constants with and let be a sequence of pairwise negatively quadrant dependent random variables. The complete convergence for pairwise negatively quadrant dependent random variables is studied under mild condition. In addition, the strong laws of large numbers for identically distributed pairwise negatively quadrant dependent random variables are established, which are equivalent to the mild condition . Our results obtained in the paper generalize the corresponding ones for pairwise independent and identically distributed random variables.

1. Introduction

Throughout the paper, let be a sequence of positive constants with , and let be a sequence of pairwise i.i.d. random variables. Denote for each . Now, we consider the following assumptions:

(i);(ii) a.s.;(iii) a.s.

Recently, Sung [1] proved that the three assumptions above are equivalent for pairwise i.i.d. random variables. In addition, he presented some results on complete convergence for pairwise i.i.d. random variables. For more details about the strong law of large numbers and complete convergence for independent random variables or dependent random variables, one can refer to Etemadi [2], Wang et al. [3], Chen et al. [4], Tang [5], and so forth.

We point out that the keys to the proofs of the main results of Sung [1] are the Khintchine-Kolmogorov-type convergence theorem and the second Borel-Cantelli lemma for pairwise independent events (e.g., see Theorem in [6] or Theorem in [7]), while these are not proved for pairwise negatively quadrant dependent random variables (pairwise NQD, in short; see Definition 1). If we want to generalize the main results of Sung [1] to the case of pairwise NQD random variables, we should propose new methods or prove the Khintchine-Kolmogorov-type convergence theorem and the second Borel-Cantelli lemma for pairwise NQD random variables. The answer is positive.

Firstly, let us recall the concept of pairwise negatively quadrant dependent random variables as follows.

Definition 1. The pair of random variables and is said to be negatively quadrant dependent (NQD, in short), if, for all ,

A sequence of random variables is said to be pairwise NQD, if is NQD for every , .

An array of random variables is called rowwise pairwise NQD random variables if for every , is a sequence of pairwise NQD random variables.

The concept of pairwise NQD random variables was introduced by Lehmann [8], which includes pairwise independent random sequence and some negatively dependent sequences, such as negatively associated sequences (see [9–13]), negatively orthant dependent sequences (see [9, 14–18]), and linearly negative quadrant dependent sequences (see [19–21]). Hence, studying the probability limiting behavior of pairwise NQD random variables and its applications in probability theory and mathematical statistics are of great interest. Many authors have dedicated themselves to the study of it. Matula [10] gained the Kolmogorov-type strong law of large numbers for the identically distributed pairwise NQD sequences; Wu [22] gave the generalized three-series theorem for pairwise NQD sequences and proved the Marcinkiewicz strong law of large numbers; Chen [23] discussed Kolmogorov-Chung strong law of large numbers for the nonidentically distributed pairwise NQD sequences under very mild conditions; Wan [24] and Huang et al. [25] obtained the complete convergence for pairwise NQD random sequences; Wang et al. [26], Li and Yang [27], Gan and Chen [28], Shi [29], Xu and Tang [30], and Tang [31] studied the strong convergence properties for pairwise NQD random variables; Sung [21] established the convergence for weighted sums of arrays of rowwise pairwise NQD random variables under weaker uniformly integrable conditions; and so on. The main purpose of the paper is to establish the second Borel-Cantelli lemma for pairwise NQD random variables and generalize the main results of Sung [1] to the case of pairwise NQD random variables without adding any extra conditions.

Our main results are as follows. The first two results are the complete convergence for pairwise NQD random variables.

Theorem 2. Let be a sequence of positive constants with . Let be a sequence of pairwise NQD random variables with identical distribution. If , then

Theorem 3. Let be a sequence of positive constants with . Let be a sequence of pairwise NQD random variables with identical distribution. If , then

The following two theorems are the results on strong convergence for pairwise NQD random variables.

Theorem 4. Let be a sequence of positive constants with . Let be a sequence of pairwise NQD random variables with identical distribution. Then, the following statements are equivalent:

(i),(ii) a.s.

Theorem 5. Let be a sequence of positive constants with . Let be a sequence of pairwise NQD random variables with identical distribution. Then, the following statements are equivalent:

(i),(ii) a.s.,(iii) a.s.

With Theorem 5 and the second Borel-Cantelli lemma for pairwise NQD random variables (see Corollary 16) in hand, we can get the following result for pairwise NQD random variables.

Corollary 6. Let be a sequence of positive constants with . Let be a sequence of pairwise NQD random variables with identical distribution and . Then,

Remark 7. Theorems 2 and 3 deal with the complete convergence for pairwise NQD random variables. Theorems 4 and 5 deal with the strong laws of large numbers for pairwise NQD random variables, which are equivalent to the mild condition . Pairwise NQD is a very wide dependence structure, which includes independent sequence as a special case. Hence, Theorems 2–5 generalize the corresponding ones for pairwise i.i.d. random variables to the case of pairwise NQD random variables.

Remark 8. Under the conditions of Theorem 3 and , we can get the Marcinkiewicz-Zygmund-type strong law of large numbers for pairwise NQD random variables as follows:

Remark 9. For a sequence of pairwise i.i.d. random variables with , Etemadi [2] proved that a.s. Note that is equivalent to and implies . Hence, Etemadi’s strong law of large numbers follows from Theorem 4 with .

Remark 10. Note that a.s. is equivalent to for any . Hence, Corollary 6 improves the corresponding result of Kruglov [32].

Throughout the paper, let be the indicator function of the set . denotes a positive constant not depending on , which may be different in various places. Denote , , and .

2. Preliminaries

In this section, we will present some important lemmas which will be used to prove the main results of the paper.

The first three lemmas come from Sung [1].

Lemma 11 (cf.[1]). Let be a sequence of positive constants with . Then the following properties hold.

(i) is a strictly increasing sequence with an .(ii) if and only if .(iii) if and only if for any .

Lemma 12 (cf. [1]). If is a sequence of positive constants with and is a random variable, then

Lemma 13 (cf. [1]). Let be a sequence of positive constants with and is a random variable. If , then .

The next one is the basic property for pairwise NQD random variables, which was given by Lehmann [8] as follows.

Lemma 14 (cf. [8]). Let and be NQD; then

(i);(ii), for any ;(iii)if and are both nondecreasing (or nonincreasing) functions, then and are NQD.

The following one is the generalized Borel-Cantelli lemma, which was obtained by Matula [10].

Lemma 15 (cf. [10]). Let be a sequence of events.

(i)If , then .(ii)If for and , then .

With the generalized Borel-Cantelli lemma accounted for, we can establish the second Borel-Cantelli lemma for pairwise NQD random variables as follows.

Corollary 16 (second Borel-Cantelli lemma for pairwise NQD random variables). Let be a sequence of positive constants with . Let be a sequence of pairwise NQD random variables. Then

Proof. “”. By Lemma 11, is equivalent to for all , which yields that a.s. by Borel-Cantelli lemma.

. Let a.s., which implies that a.s. and a.s.

For any , denote

Hence,

By Lemma 14(iii), we can see that and are both sequences of pairwise NQD random variables. It follows by Lemma 14 that, for any ,

By Lemma 15(ii) and (9)-(10), we can see that for . Hence,

which is equivalent to by Lemma 11. This completes the proof of the corollary.

The last one is the Kolmogorov-type strong law of large numbers for pairwise NQD random variables obtained by Chen [23], which plays an important role in proving the main results of the paper.

Lemma 17 (cf. [23]). Let be a sequence of pairwise NQD random variables with for each . Let be a sequence of real numbers satisfying . Suppose that

(i);(ii).

Then a.s.

3. Proofs of Theorems 2–5

Proof of Theorem 2. Note that the condition implies that

For fixed , denote for that

It is easily checked that

To prove the desired result (2), it suffices to show for . Note that ; we only need to prove and .

Note that are pairwise NQD random variables by Lemma 14(iii); we have by Markov's inequality, Lemma 14(i), and the assumption that

Combining with (12) and (15), we have

Finally, we will prove . It is easily seen that

In the following we prove . Note that and as ; we have , which implies that . Hence, . This completes the proof of the theorem.

Proof of Theorem 3. We use the same notations as those in Theorem 2. It is easy to see that

To prove the desired result (3), it remains to show for .

By Markov's inequality and the assumption , we have

By the assumptions of Theorem 3 and Lemma 13, we have , which implies that .

In the following, we will prove . It is easily checked that

Similar to the proof of in Theorem 2, we can get that .

Note that, for fixed , and are both pairwise NQD random variables. Hence, similar to the proof of in Theorem 2, we have

Similarly, we have . Therefore, follows by the statements above. This completes the proof of the theorem.

Proof of Theorem 4. Firstly, we will prove that (i) (ii).

For fixed , denote

Similar to the proof of in Theorem 2, we have

It follows by Lemma 12 that

Since for each , we have by (23) and (24) and Lemma 17 that

Note that

and the assumption implies that a.s.; we can get that

which together with Kronecker's lemma yield that

By (26) and (28), we have

It follows by the assumption again that

Therefore, the desired result (ii) follows by (29) and (30) immediately.

Next, we will prove that (ii) (i). Assume that

Then, we have

Note that

as , which implies that

It follows by (32) and (34) that a.s., which is equivalent to (i) by Corollary 16. The proof is completed.

Proof of Theorem 5. Firstly, we will prove that (ii) (i). It follows by (ii) that

which together with Corollary 16 imply that (i) holds.

On the other hand, assume that ; it follows by Lemma 13 that

The desired result (ii) follows by Theorem 4 and (36) immediately.

We have proved that (i) (ii); next we prove (i) (iii). It follows by Lemma 11(iii) that, for any ,

On the other hand, we have proved that (i) (ii); hence,

Therefore, (i) (iii) follows by the statements above immediately. This completes the proof of the theorem.

Proof of Corollary 6. The techniques used here are the second Borel-Cantelli lemma for pairwise NQD random variables (see Corollary 16) and Theorem 5. The proof is similar to that of Corollary 2.1 of Sung [1], so the details of the proof are omitted.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are most grateful to the Editor Angelo Favini and anonymous referee for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (11201001), the Natural Science Foundation of Anhui Province (1308085QA03), and the Students Innovative Training Project of Anhui University (201410357118).

References

S. H. Sung, “On the strong law of large numbers for pairwise i.i.d. random variables with general moment conditions,” Statistics & Probability Letters, vol. 83, no. 9, pp. 1963–1968, 2013.

N. Etemadi, “An elementary proof of the strong law of large numbers,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 55, no. 1, pp. 119–122, 1981.

X.J. Wang, S.H. Hu, Y. Shen, and N.X. Ling, “Strong law of large numbers and growth rate for a class of random variable sequences,” Statistics & Probability Letters, vol. 78, no. 18, pp. 3330–3337, 2008.

P.-Y. Chen, P. Bai, and S. H. Sung, “On complete convergence and the strong law of large numbers for pairwise independent random variables,” Acta Mathematica Hungarica, vol. 142, no. 2, pp. 502–518, 2014.

X. Tang, “Some strong laws of large numbers for weighted sums of asymptotically almost negatively associated random variables,” Journal of Inequalities and Applications, vol. 2013, article 4, 2013.

K. L. Chung, A Course in Probability Theory, Academic Press, New York, NY, USA, 2nd edition, 1974.

A. Gut, Probability: A Graduate Course, Springer, New York, NY, USA, 2005.

E. L. Lehmann, “Some concepts of dependence,” The Annals of Mathematical Statistics, vol. 37, pp. 1137–1153, 1966.

K. Joag-Dev and F. Proschan, “Negative association of random variables, with applications,” The Annals of Statistics, vol. 11, no. 1, pp. 286–295, 1983.

P. Matuła, “A note on the almost sure convergence of sums of negatively dependent random variables,” Statistics and Probability Letters, vol. 15, no. 3, pp. 209–213, 1992.

Q.-M. Shao, “A comparison theorem on moment inequalities between negatively associated and independent random variables,” Journal of Theoretical Probability, vol. 13, no. 2, pp. 343–356, 2000.

Q.Y. Wu and Y.Y. Jiang, “A law of the iterated logarithm of partial sums for NA random variables,” Journal of the Korean Statistical Society, vol. 39, no. 2, pp. 199–206, 2010.

X.J. Wang, X.Q. Li, S.H. Hu, and W.Z. Yang, “Strong limit theorems for weighted sums of negatively associated random variables,” Stochastic Analysis and Applications, vol. 29, no. 1, pp. 1–14, 2011.

Q.Y. Wu, “A Strong limit theorem for weighted sums of sequences of negatively dependent random variables,” Journal of Inequalities and Applications, vol. 2010, Article ID 383805, 8 pages, 2010.

Q.Y. Wu, “A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables,” Journal of Inequalities and Applications, vol. 2012, article 50, 2012.

X.J. Wang, S.H. Hu, W.Z. Yang, and N.X. Ling, “Exponential inequalities and inverse moment for NOD sequence,” Statistics & Probability Letters, vol. 80, no. 5-6, pp. 452–461, 2010.

X.J. Wang, S.H. Hu, and W.Z. Yang, “Complete convergence for arrays of rowwise negatively orthant dependent random variables,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales A: Matematicas, vol. 106, no. 2, pp. 235–245, 2012.

S. H. Sung, “On the exponential inequalities for negatively dependent random variables,” Journal of Mathematical Analysis and Applications, vol. 381, no. 2, pp. 538–545, 2011.

C. M. Newman, “Asymptotic independence and limit theorems for positively and negatively dependent random variables,” in Inequalities in Statistics and Probability, Y. L. Tong, Ed., vol. 5, pp. 127–140, Institute of Mathematical Statistics, Hayward, Calif, USA, 1984.

X.J. Wang, S.H. Hu, W.Z. Yang, and X.Q. Li, “Exponential inequalities and complete convergence for a LNQD sequence,” Journal of the Korean Statistical Society, vol. 39, no. 4, pp. 555–564, 2010.

S. H. Sung, “Convergence in r-mean of weighted sums of NQD random variables,” Applied Mathematics Letters, vol. 26, no. 1, pp. 18–24, 2013.

Q. Y. Wu, “Convergence properties of pairwise NQD random sequences,” Acta Mathematica Sinica, vol. 45, no. 3, pp. 617–624, 2002.

P. Y. Chen, “Strong law of large numbers for pairwise NQD random variables,” Acta Mathematica Scientia A, vol. 25, no. 3, pp. 386–392, 2005.

C. G. Wan, “Law of large numbers and complete convergence for pairwise NQD random sequences,” Acta Mathematicae Applicatae Sinica, vol. 28, no. 2, pp. 253–261, 2005.

H. Huang, D. Wang, Q. Wu, and Q. Zhang, “A note on the complete convergence for sequences of pairwise NQD random variables,” Journal of Inequalities and Applications, vol. 2011, article 92, 2011.

Y. B. Wang, C. Su, and X. G. Liu, “Some limit properties for pairwise NQD sequences,” Acta Mathematicae Applicatae Sinica, vol. 21, no. 3, pp. 404–414, 1998.

R. Li and W. Yang, “Strong convergence of pairwise NQD random sequences,” Journal of Mathematical Analysis and Applications, vol. 344, no. 2, pp. 741–747, 2008.

S.X. Gan and P.Y. Chen, “Some limit theorems for sequences of pairwise NQD random variables,” Acta Mathematica Scientia, vol. 28, no. 2, pp. 269–281, 2008.

J. H. Shi, “On the strong law of large numbers for pairwise NQD random variables with different distributions,” Acta Mathematicae Applicatae Sinica, vol. 34, no. 1, pp. 122–130, 2011.

H. Xu and L. Tang, “Some convergence properties for weighted sums of pairwise NQD sequences,” Journal of Inequalities and Applications, vol. 2012, article 255, 2012.

X. Tang, “Strong convergence results for arrays of rowwise pairwise NQD random variables,” Journal of Inequalities and Applications, vol. 2013, article 102, 2013.

V. M. Kruglov, “A strong law of large numbers for pairwise independent identically distributed random variables with infinite means,” Statistics & Probability Letters, vol. 78, no. 7, pp. 890–895, 2008.