On the complete convergence for weighted sums of a class of random variables
- Jinghuan Zhao1Email author,
- Yan Sun2 and
- Yu Miao1
https://doi.org/10.1186/s13660-016-1165-2
© Zhao et al. 2016
Received: 6 April 2016
Accepted: 7 September 2016
Published: 13 September 2016
Abstract
In this article, some new results as regards complete convergence for weighted sums \(\sum_{i=1}^{n}a_{ni}X_{i}\) of random variables satisfying the Rosenthal type inequality are established under some mild conditions. These results extend the corresponding theorems of Deng et al. (Filomat 28(3):509-522, 2014) and Gan and Chen (Acta Math. Sci. 28(2):269-281, 2008).
Keywords
MSC
1 Introduction
In this paper we are interested in the complete convergence of a sequence of random variables which satisfies the Rosenthal type inequality. First let us recall some definitions and well-known results.
1.1 Complete convergence
The result of Hsu-Robbins-Erdös’ strong law is a fundamental theorem in probability theory and has been intensively investigated in several directions by many authors in the past decades. One of the most important results is Baum and Katz’ [5] strong law of large numbers.
It is well known that the analysis of weighted sums plays an important role in the statistics, such as jackknife estimate, nonparametric regression function estimate and so on. Many authors considered the complete convergence of the weight sums of random variables. Thrum [6] studied the almost sure convergence of weighted sums of i.i.d. random variables; Li et al. [7] obtained complete convergence of weighted sums without the identically distributed assumption. Liang and Su [8] extended the results of Thrum [6], and Li et al. [7] showed the complete convergence of weighted sums of negatively associated sequence. The reader can refer to further literature on complete convergence of weighted sums, such as Xue et al. [9] for the NSD sequence, Gan and Chen [10] for the NOD sequence and so on.
1.2 Rosenthal type inequality
The concept of stochastic domination is presented as follows.
Definition 1.1
In the present paper, we shall study the complete convergence of weighted sums of random sequence under the assumption that the random variables satisfy the Rosenthal type inequality. Our main results are stated in Section 2 and the proofs are given in Section 3. Throughout this paper, let C denote a positive constant, which may take different values whenever it appears in different expressions. \(a_{n}=O(b_{n})\) means \(|a_{n}/b_{n}|\leq C\) and \(I(\cdot)\) stands for the indicator function.
2 Main results
Theorem 2.1
Remark 2.1
Remark 2.2
Remark 2.3
Remark 2.4
Theorem 2.2
Theorem 2.3
Corollary 2.1
3 Proofs of main results
In order to prove the main theorems, we need the following lemma which includes the basic properties for stochastic domination. One can refer to Shen [20], Wang et al. [21], Wu [22], or Shen and Wu [23] for the proof.
Lemma 3.1
Proof of Theorem 2.1
Proof of Theorem 2.2
Proof of Theorem 2.3
Proof of Corollary 2.1
4 Conclusions
The present work is meant to establish some new results as regards complete convergence for weighted sums of random variables which satisfy the Rosenthal type inequality. These results extend some known results.
Declarations
Acknowledgements
The work is supported by Science and Technology Project of Beijing Municipal Education Commission (KM201611232019), NSFC (71501016) and IRTSTHN (14IRTSTHN023), NSFC (11471104).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Deng, X, Ge, MM, Wang, XJ, Liu, YF, Zhou, Y: Complete convergence for weighted sums of a class of random variables. Filomat 28(3), 509-522 (2014) MathSciNetView ArticleGoogle Scholar
- Gan, SX, Chen, PY: Some limit theorems for sequences of pairwise NQD random variables. Acta Math. Sci. 28(2), 269-281 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Hsu, PL, Robbins, H: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33(2), 25-31 (1947) MathSciNetView ArticleMATHGoogle Scholar
- Erdös, P: On a theorem of Hsu and Robbins. Ann. Math. Stat. 20, 286-291 (1949) MathSciNetView ArticleMATHGoogle Scholar
- Baum, LE, Katz, M: Convergence rates in the law of large numbers. Trans. Am. Math. Soc. 120, 108-123 (1965) MathSciNetView ArticleMATHGoogle Scholar
- Thrum, R: A remark on almost sure convergence of weighted sums. Probab. Theory Relat. Fields 75(3), 425-430 (1987) MathSciNetView ArticleMATHGoogle Scholar
- Li, DL, Rao, MB, Jiang, TF, Wang, XC: Complete convergence and almost sure convergence of weighted sums of random variables. J. Theor. Probab. 8(1), 49-76 (1995) MathSciNetView ArticleMATHGoogle Scholar
- Liang, HY, Su, C: Complete convergence for weighted sums of NA sequences. Stat. Probab. Lett. 45(1), 85-95 (1999) MathSciNetView ArticleMATHGoogle Scholar
- Xue, Z, Zhang, LL, Lei, YJ, Chen, ZG: Complete moment convergence for weighted sums of negatively superadditive dependent random variables. J. Inequal. Appl. 2015, 117 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Gan, SX, Chen, PY: Some limit theorems for weighted sums of arrays of NOD random variables. Acta Math. Sci. Ser. B Engl. Ed. 32(6), 2388-2400 (2012) MathSciNetMATHGoogle Scholar
- Shao, QM: A comparison theorem on inequalities between negatively associated and independent random variables. J. Theor. Probab. 13(2), 343-356 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Utev, S, Peligrad, M: Maximal inequalities and an invariance principle for a class of weakly dependent random variables. J. Theor. Probab. 16(1), 101-115 (2003) MathSciNetView ArticleMATHGoogle Scholar
- Shen, AT: Probability inequalities for END sequence and their applications. J. Inequal. Appl. 2011, 98 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Stout, WF: Almost Sure Convergence. Academic Press, New York (1974) MATHGoogle Scholar
- Hu, TZ: Negatively superadditive dependence of random variables with applications. Chinese J. Appl. Probab. Statist. 16(2), 133-144 (2000) MathSciNetMATHGoogle Scholar
- Wang, XJ, Deng, X, Zheng, LL, Hu, SH: Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications. Statistics 48(4), 834-850 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Asadian, N, Fakoor, V, Bozorgnia, A: Rosenthal’s type inequalities for negatively orthant dependent random variables. JIRSS 5(1-2), 69-75 (2006) Google Scholar
- Wu, QY: Complete convergence for weighted sums of sequences of negatively dependent random variables. J. Probab. Stat. 2011, Article ID 202015 (2011) MathSciNetMATHGoogle Scholar
- Yuan, D, An, J: Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications. Sci. China Ser. A, Math. 52(9), 1887-1904 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Shen, AT: On the strong convergence rate for weighted sums of arrays of rowwise negatively orthant dependent random variables. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 107(2), 257-271 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Wang, XJ, Xu, C, Hu, TC, Volodin, A, Hu, SH: On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. Test 23(3), 607-629 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Wu, QY: Probability Limit Theory for Mixing Sequences. Science Press of China, Beijing (2006) Google Scholar
- Shen, AT, Wu, RC: Strong and weak convergence for asymptotically almost negatively associated random variables. Discrete Dyn. Nat. Soc. 2013, Article ID 235012 (2013) MathSciNetMATHGoogle Scholar