# ON COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE FOR A CLASS OF RANDOM VARIABLES

• Wang, Xuejun ;
• Wu, Yi
• Published : 2017.05.01
• 22 18

#### Abstract

In this paper, the complete convergence and complete moment convergence for a class of random variables satisfying the Rosenthal type inequality are investigated. The sufficient and necessary conditions for the complete convergence and complete moment convergence are provided. As applications, the Baum-Katz type result and the Marcinkiewicz-Zygmund type strong law of large numbers for a class of random variables satisfying the Rosenthal type inequality are established. The results obtained in the paper extend the corresponding ones for some dependent random variables.

#### Keywords

complete convergence;complete moment convergence;stochastic domination;Rosenthal type inequality

#### References

1. Y. B. Wang, J. G. Yan, F. Y. Cheng, and C. Su, The strong law of large numbers and the law of iterated logarithm for product sums of NA and AANA random variables, Southeast Asian Bull. Math. 27 (2003), no. 2, 369-384.
2. Q. Y. Wu, Probability Limit Theory for Mixing Sequences, Science Press of China, Beijing, 2006.
3. W. Z. Yang, X. J. Wang, N. X. Ling, and S. H. Hu, On complete convergence of moving average process for AANA sequence, Discrete Dyn. Nat. Soc. 2012 (2012) Article ID 863931, 24 pages.
4. D. M. Yuan and J. An, Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications, Sci. China Ser. A 52 (2009), no. 9, 1887-1904. https://doi.org/10.1007/s11425-009-0154-z
5. D. M. Yuan and J. An, Laws of large numbers for Cesaro alpha-integrable random variables under dependence condition AANA or AQSI, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 6, 1103-1118. https://doi.org/10.1007/s10114-012-0033-3
6. X. C. Zhou, Complete moment convergence of moving average processes under ${\varphi}$-mixing assumptions, Statist. Probab. Lett. 80 (2010), no. 5-6, 285-292. https://doi.org/10.1016/j.spl.2009.10.018
7. K. Alam and K. M. L. Saxena, Positive dependence in multivariate distributions, Comm. Statist. Theory Methods 10 (1981), no. 12, 1183-1196. https://doi.org/10.1080/03610928108828102
8. Z. D. Bai and C. Su, The complete convergence for partial sums of i.i.d. random variables, Sci. Sinica Ser. A 28 (1985), no. 12, 1261-1277.
9. L. E. Baum and M. Katz, Convergence rates in the law of large numbers, Trans. Amer. Math. Soc. 120 (1965), no. 1, 108-123. https://doi.org/10.1090/S0002-9947-1965-0198524-1
10. T. K. Chandra and S. Ghosal, Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables, Acta Math. Hungar. 71 (1996), no. 4, 327-336. https://doi.org/10.1007/BF00114421
11. Z. Y. Chen, X. J. Wang, and S. H. Hu, Strong laws of large numbers for weighted sums of asymptotically almost negatively associated random variables, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 109 (2015), no. 1, 135-152.
12. Y. S. Chow, On the rate of moment complete convergence of sample sums and extremes, Bull. Inst. Math. Acad. Sinica 16 (1988), no. 3, 177-201.
13. P. Erdos, On a theorem of Hsu and Robbins, Ann. Math. Statist. 20 (1949), 286-291. https://doi.org/10.1214/aoms/1177730037
14. P. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proceedings of the National Academy of Sciences 33 (1947), 25-31. https://doi.org/10.1073/pnas.33.2.25
15. X. Hu, G. Fang, and D. Zhu, Strong convergence properties for asymptotically almost negatively associated sequence, Discrete Dyn. Nat. Soc. 2012 (2012), Article ID 562838, 8 pages.
16. K. Joag-Dev and F. Proschan, Negative association of random variables with applications, Ann. Statist. 11 (1983), no. 1, 286-295. https://doi.org/10.1214/aos/1176346079
17. A. T. Shen, On asymptotic approximation of inverse moments for a class of nonnegative random variables, Statistics 48 (2014), no. 6, 1371-1379. https://doi.org/10.1080/02331888.2013.801480
18. A. T. Shen and R. C. Wu, Strong convergence for sequences of asymptotically almost negatively associated random variables, Stochastic 86 (2014), no. 2, 291-303. https://doi.org/10.1080/17442508.2013.775289
19. A. T. Shen, R. C. Wu, Y. Chen, and Y. Zhou, Complete convergence of the maximum partial sums for arrays of rowwise of AANA random variables, Discrete Dyn. Nat. Soc. 2013 (2013), Article ID 741901, 7 pages.
20. 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. https://doi.org/10.1007/s00184-014-0503-y
21. S. H. Sung, Moment inequalities and complete moment convergence, J. Inequal. Appl. 2009 (2009), Article ID 271265, 14 papers.
22. X. J. Wang, S. H. Hu, and W. Z. Yang, Complete convergence for arrays of rowwise asymptotically almost negatively associated random variables, Discrete Dyn. Nat. Soc. 2011 (2011), Article ID 717126, 11 pages.
23. X. H. Wang, A. T. Shen, and X. Q. Li, A note on complete convergence of weighted sums for array of rowwise AANA random variables, J. Inequal. Appl. 2013 (2013), Article ID 359, 13 papers. https://doi.org/10.1186/1029-242X-2013-13

#### Cited by

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2. Sufficient and necessary conditions of complete convergence for asymptotically negatively associated random variables vol.2018, pp.1, 2018, https://doi.org/10.1186/s13660-018-1906-5
3. Equivalent conditions of the complete convergence for weighted sums of NSD random variables pp.1532-415X, 2018, https://doi.org/10.1080/03610926.2018.1500601
4. Some Types of Convergence for Negatively Dependent Random Variables under Sublinear Expectations vol.2019, pp.1607-887X, 2019, https://doi.org/10.1155/2019/9037258

#### Acknowledgement

Supported by : National Natural Science Foundation of China, Natural Science Foundation of Anhui Province