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Title: On complete moment convergence for weighted sums of negatively superadditive dependent random variables (English)
Author: Huang, Haiwu
Author: Lu, Xuewen
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 65
Issue: 4
Year: 2020
Pages: 355-377
Summary lang: English
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Category: math
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Summary: In this work, the complete moment convergence and complete convergence for weighted sums of negatively superadditive dependent (NSD) random variables are studied, and some equivalent conditions of these strong convergences are established. These main results generalize and improve the corresponding theorems of Baum and Katz (1965) and Chow (1988) to weighted sums of NSD random variables without the assumption of identical distribution. As an application, a Marcinkiewicz-Zygmund-type strong law of large numbers for weighted sums of NSD random variables is obtained. (English)
Keyword: NSD random variables
Keyword: complete moment convergence
Keyword: weighted sum
Keyword: equivalent conditions
MSC: 60F15
idZBL: 07250667
idMR: MR4134139
DOI: 10.21136/AM.2020.0255-18
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Date available: 2020-09-07T09:44:30Z
Last updated: 2020-11-18
Stable URL: http://hdl.handle.net/10338.dmlcz/148338
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Reference: [1] Alam, K., Saxena, K. M. L.: Positive dependence in multivariate distributions.Commun. Stat., Theory Methods A10 (1981), 1183-1196. Zbl 0471.62045, MR 0623526, 10.1080/03610928108828102
Reference: [2] Amini, M., Bozorgnia, A., Naderi, H., Volodin, A.: On complete convergence of moving average processes for NSD sequences.Sib. Adv. Math. 25 (2015), 11-20. Zbl 1328.60082, MR 3490729, 10.3103/S1055134415010022
Reference: [3] Bai, Z., Su, C.: The complete convergence for partial sums of i.i.d. random variables.Sci. Sin., Ser. A 28 (1985), 1261-1277. Zbl 0554.60039, MR 0851970
Reference: [4] Baum, L. E., Katz, M.: Convergence rates in the law of large numbers.Trans. Am. Math. Soc. 120 (1965), 108-123. Zbl 0142.14802, MR 0198524, 10.1090/S0002-9947-1965-0198524-1
Reference: [5] Chen, P. Y., Wang, D. C.: Complete moment convergence for sequence of identically distributed $\varphi$-mixing random variables.Acta Math. Sin., Engl. Ser. 26 (2010), 679-690. Zbl 1205.60062, MR 2591647, 10.1007/s10114-010-7625-6
Reference: [6] Chow, Y. S.: On the rate of moment convergence of sample sums and extremes.Bull. Inst. Math., Acad. Sin. 16 (1988), 177-201. Zbl 0655.60028, MR 1089491
Reference: [7] Christofides, T. C., Vaggelatou, E.: A connection between supermodular ordering and \hbox{positive/negative} association.J. Multivariate Anal. 88 (2004), 138-151. Zbl 1034.60016, MR 2021866, 10.1016/S0047-259X(03)00064-2
Reference: [8] Deng, X., Wang, X., Wu, Y., Ding, Y.: Complete moment convergence and complete convergence for weighted sums of NSD random variables.Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 110 (2016), 97-120. Zbl 1334.60037, MR 3462077, 10.1007/s13398-015-0225-7
Reference: [9] Eghbal, N., Amini, M., Bozorgnia, A.: Some maximal inequalities for quadratic forms of negative superadditive dependence random variables.Stat. Probab. Lett. 80 (2010), 587-591. Zbl 1187.60020, MR 2595134, 10.1016/j.spl.2009.12.014
Reference: [10] Eghbal, N., Amini, M., Bozorgnia, A.: On the Kolmogorov inequalities for quadratic forms of dependent uniformly bounded random variables.Stat. Probab. Lett. 81 (2011), 1112-1120. Zbl 1228.60039, MR 2803752, 10.1016/j.spl.2011.03.005
Reference: [11] Erdős, P.: On a theorem of Hsu and Robbins.Ann. Math. Stat. 20 (1949), 286-291. Zbl 0033.29001, MR 0030714, 10.1214/aoms/1177730037
Reference: [12] Gut, A.: Probability: A Graduate Course.Springer Texts in Statistics. Springer, New York (2005). Zbl 1076.60001, MR 2125120, 10.1007/978-1-4614-4708-5
Reference: [13] Hsu, P. L., Robbins, H.: Complete convergence and the law of large numbers.Proc. Natl. Acad. Sci. USA 33 (1947), 25-31. Zbl 0030.20101, MR 0019852, 10.1073/pnas.33.2.25
Reference: [14] Hu, T.: Negatively superadditive dependence of random variables with applications.Chin. J. Appl. Probab. Stat. 16 (2000), 133-144. Zbl 1050.60502, MR 1812714
Reference: [15] Joag-Dev, K., Proschan, F.: Negative association of random variables, with applications.Ann. Stat. 11 (1983), 286-295. Zbl 0508.62041, MR 0684886, 10.1214/aos/1176346079
Reference: [16] Kemperman, J. H. B.: On the FKG-inequality for measures on a partially ordered space.Nederl. Akad. Wet., Proc., Ser. A 80 (1977), 313-331. Zbl 0384.28012, MR 0467867, 10.1016/1385-7258(77)90027-0
Reference: [17] Naderi, H., Amini, M., Bozorgnia, A.: On the rate of complete convergence for weighted sums of NSD random variables and an application.Appl. Math., Ser. B (Engl. Ed.) 32 (2017), 270-280. Zbl 1399.60053, MR 3694062, 10.1007/s11766-017-3437-0
Reference: [18] Shen, Y., Wang, X. J., Yang, W. Z., Hu, S. H.: Almost sure convergence theorem and strong stability for weighted sums of NSD random variables.Acta Math. Sin., Engl. Ser. 29 (2013), 743-756. Zbl 1263.60025, MR 3029287, 10.1007/s10114-012-1723-6
Reference: [19] Shen, A., Xue, M., Volodin, A.: Complete moment convergence for arrays of rowwise NSD random variables.Stochastics 88 (2016), 606-621. Zbl 1337.60038, MR 3473853, 10.1080/17442508.2015.1110153
Reference: [20] Shen, A., Zhang, Y., Volodin, A.: Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables.Metrika 78 (2015), 295-311. Zbl 1333.60022, MR 3320899, 10.1007/s00184-014-0503-y
Reference: [21] Sung, S. H.: Moment inequalities and complete moment convergence.J. Inequal. Appl. 2009 (2009), Article ID 271265, 14 pages. Zbl 1180.60019, MR 2551753, 10.1155/2009/271265
Reference: [22] Wang, X., Deng, X., Zheng, L., Hu, S.: Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications.Statistics 48 (2014), 834-850. Zbl 1319.60063, MR 3234065, 10.1080/02331888.2013.800066
Reference: [23] Wang, X., Shen, A., Chen, Z., Hu, S.: Complete convergence for weighted sums of NSD random variables and its application in the EV regression model.TEST 24 (2015), 166-184. Zbl 1316.60042, MR 3314578, 10.1007/s11749-014-0402-6
Reference: [24] Wang, X., Wu, Y.: On complete convergence and complete moment convergence for a class of random variables.J. Korean Math. Soc. 54 (2017), 877-896. Zbl 1366.60068, MR 3640914, 10.4134/JKMS.j160293
Reference: [25] Wu, Q.: Probability Limit Theory for Mixing Sequences.Science Press of China, Beijing (2006).
Reference: [26] Wu, Y.: On complete moment convergence for arrays of rowwise negatively associated random variables.Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 108 (2014), 669-681. Zbl 1296.60078, MR 3249968, 10.1007/s13398-013-0133-7
Reference: [27] Zhang, Y.: On strong limit theorems for negatively superadditive dependent random variables.Filomat 29 (2015), 1541-1547. Zbl 06749122, MR 3373155, 10.2298/FIL1507541Z
Reference: [28] Zheng, L., Wang, X., Yang, W.: On the strong convergence for weighted sums of negatively superadditive dependent random variables.Filomat 31 (2017), 295-308. MR 3628840, 10.2298/FIL1702295Z
Reference: [29] Zhou, X.: Complete moment convergence of moving average processes under $\varphi$-mixing assumptions.Stat. Probab. Lett. 80 (2010), 285-292. Zbl 1186.60031, MR 2593564, 10.1016/j.spl.2009.10.018
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