# zbMATH — the first resource for mathematics

On complete moment convergence for weighted sums of negatively superadditive dependent random variables. (English) Zbl 07250667
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 L. E. Baum and M. Katz [Trans. Am. Math. Soc. 120, 108–123 (1965; Zbl 0142.14802)] and Y. S. Chow [Bull. Inst. Math., Acad. Sin. 16, No. 3, 177–201 (1988; Zbl 0655.60028)] 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.
##### MSC:
 60F15 Strong limit theorems
Full Text:
##### References:
 [1] Alam, K.; Saxena, K. M. L., Positive dependence in multivariate distributions, Commun. Stat., Theory Methods A10 (1981), 1183-1196 [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 [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 [4] Baum, L. E.; Katz, M., Convergence rates in the law of large numbers, Trans. Am. Math. Soc. 120 (1965), 108-123 [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 [6] Chow, Y. S., On the rate of moment convergence of sample sums and extremes, Bull. Inst. Math., Acad. Sin. 16 (1988), 177-201 [7] Christofides, T. C.; Vaggelatou, E., A connection between supermodular ordering and \hbox{positive/negative} association, J. Multivariate Anal. 88 (2004), 138-151 [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 [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 [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 [11] Erdős, P., On a theorem of Hsu and Robbins, Ann. Math. Stat. 20 (1949), 286-291 [12] Gut, A., Probability: A Graduate Course, Springer Texts in Statistics. Springer, New York (2005) [13] Hsu, P. L.; Robbins, H., Complete convergence and the law of large numbers, Proc. Natl. Acad. Sci. USA 33 (1947), 25-31 [14] Hu, T., Negatively superadditive dependence of random variables with applications, Chin. J. Appl. Probab. Stat. 16 (2000), 133-144 [15] Joag-Dev, K.; Proschan, F., Negative association of random variables, with applications, Ann. Stat. 11 (1983), 286-295 [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 [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 [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 [19] Shen, A.; Xue, M.; Volodin, A., Complete moment convergence for arrays of rowwise NSD random variables, Stochastics 88 (2016), 606-621 [20] Shen, A.; Zhang, Y.; Volodin, A., Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables, Metrika 78 (2015), 295-311 [21] Sung, S. H., Moment inequalities and complete moment convergence, J. Inequal. Appl. 2009 (2009), Article ID 271265, 14 pages [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 [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 [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 [25] Wu, Q., Probability Limit Theory for Mixing Sequences, Science Press of China, Beijing (2006) [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 [27] Zhang, Y., On strong limit theorems for negatively superadditive dependent random variables, Filomat 29 (2015), 1541-1547 [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 [29] Zhou, X., Complete moment convergence of moving average processes under $$\varphi$$-mixing assumptions, Stat. Probab. Lett. 80 (2010), 285-292
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.