Browse > Article
http://dx.doi.org/10.4134/BKMS.b160357

EQUIVALENT CONDITIONS OF COMPLETE MOMENT CONVERGENCE AND COMPLETE INTEGRAL CONVERGENCE FOR NOD SEQUENCES  

Deng, Xin (School of Mathematical Sciences Anhui University)
Wang, Xuejun (School of Mathematical Sciences Anhui University)
Publication Information
Bulletin of the Korean Mathematical Society / v.54, no.3, 2017 , pp. 917-933 More about this Journal
Abstract
In this paper, seven equivalent conditions of complete moment convergence and complete integral convergence for negatively orthant dependent (NOD, in short) sequences are shown under two cases: identical distribution and stochastic domination. The results obtained in the paper improve and generalize the corresponding ones of Liang et al. [10]). In addition, an extension of the Baum-Katz complete convergence theorem: six equivalent conditions of complete convergence is established.
Keywords
complete convergence; complete moment convergence; complete integral convergence; negatively orthant dependent sequence; stochastic domination;
Citations & Related Records
연도 인용수 순위
  • Reference
1 P. Y. Chen and D. C. Wang, Convergence rates for probabilities of moderate deviations for moving average processes, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 4, 611-622.   DOI
2 Y. S. Chow, On the rate of moment convergence of sample sums and extremes, Bull. Inst. Math. Acad. Sinica 16 (1988), no. 3, 177-201.
3 T. Z. Hu, Negatively superadditive dependence of random variables with applications, Chinese J. Appl. Probab. Statist. 16 (2000), no. 2, 133-144.
4 K. Joag-Dev and F. Proschan, Negative association of random variables with applications, Ann. Statist. 11 (1983), no. 1, 286-295.   DOI
5 D. Li and A. Spataru, Re nement of convergence rates for tail probabilities, J. Theoret. Probab. 18 (2005), no. 4, 933-947.   DOI
6 H. Y. Liang, D. L. Li, and A. Rosalsky, Complete moment and integral convergence for sums of negatively associated random variables, Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 3, 419-432.   DOI
7 D. H. Qiu, X. Liu, and P. Y. Chen, Complete moment convergence for maximal partial sums under NOD setup, J. Inequal. Appl. 2015 (2015), Article ID 58, 12 pages.   DOI
8 A. T. Shen, Some strong limit theorems for arrays of rowwise negatively orthant- dependent random variables, J. Inequal. Appl. 2011 (2011), Article ID 93, 10 pages.   DOI
9 A. T. Shen, Probability inequalities for END sequence and their applications, J. Inequal. Appl. 2011 (2011), Article ID 98, 12 pages.   DOI
10 A. T. Shen, On the strong convergence rate for weighted sums of arrays of rowwise negatively orthant dependent random variables, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM 107 (2013), no. 2, 257-271.
11 A. T. Shen, On asymptotic approximation of inverse moments for a class of nonnegative random variables, Statistics 48 (2014), no. 6, 1371-1379.   DOI
12 A. T. Shen, Y. Zhang, and A. Volodin, Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables, Metrika 78 (2015), no. 3, 295-311.   DOI
13 S. H. Sung, On the exponential inequalities for negatively dependent random variables, J. Math. Anal. Appl. 381 (2011), no. 2, 538-545.   DOI
14 S. H. Sung, A Strong limit theorem for weighted sums of negatively dependent random variables, Comm. Statist. Theory Methods 44 (2015), no. 2, 428-439.   DOI
15 A. Volodin, On the Kolmogorov exponential inequality for negatively dependent random variables, Pakistan J. Statist. 18 (2002), no. 2, 249-253.
16 Q. Y. Wu, Complete convergence for weighted sums of sequences of negatively dependent random variables, J. Probab. Stat. 2011 (2011), Article ID 202015, 16 pages.
17 X. J. Wang, S. H. Hu, and W. Z. Yang, Complete convergence for arrays of rowwise negatively orthant dependent random variables, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM 106 (2012), no. 2, 235-245.
18 X. J. Wang, S. H. Hu, W. Z. Yang, and N. X. Ling, Exponential inequalities and inverse moment for NOD sequence, Statist. Probab. Lett. 80 (2010), no. 5-6, 452-461.   DOI
19 X. J. Wang and Z. Y. Si, Complete consistency of the estimator of nonparametric regression model under ND sequence, Statist. Papers 56 (2015), no. 3, 585-596.   DOI
20 Q. Y. Wu, Probability Limit Theory for Mixing Sequences, Science Press of China, Beijing, 2006.
21 Q. Y. Wu and Y. Y. Jiang, The strong consistency of M estimator in a linear model for negatively dependent random samples, Comm. Statist. Theory Methods 40 (2011), no. 3, 467-491.   DOI
22 Y. F. Wu and A. Volodin, Complete moment convergence of weighted sums for arrays of negatively dependent random variables and its applications, Comm. Statist. Theory Methods 45 (2016), no. 11, 3185-3195.   DOI
23 Y. C. Yi, D. Hu, and P. Y. Chen, On complete convergence for Stout's type weighted sums of NOD sequence, Appl. Math. J. Chinese Univ. Ser. B 30 (2015), no. 3, 340-346.   DOI
24 H. Zarei and H. Jabbari, Complete convergence of weighted sums under negative dependence, Statist. Papers 52 (2011), no. 2, 413-418.   DOI
25 P. Y. Chen and S. H. Sung, Complete convergence and strong laws of large numbers for weighted sums of negatively orthant dependent random variables, Acta Math. Hungar. 148 (2016), no. 1, 83-95.   DOI
26 N. Asadian, V. Fakoor, and A. Bozorgnia, Rosenthal's type inequalities for negatively orthant dependent random variables, J. Iranian Statist. Soc. 5 (2006), no. 1-2, 66-75.
27 L. E. Baum and M. Katz, Convergence rates in the law of large numbers, Trans. Amer. Math. Soc. 120 (1965), 108-123.   DOI
28 A. Bozorgnia, R. F. Patterson, and R. L. Taylor, Limit theorems for dependent random variables, in: Proceedings of the First World Congress of Nonlinear Analysts'92(II), 1639-1650, Walter de Grutyer, Berlin, 1996.