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Complete f-moment convergence for extended negatively dependent random variables

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we study the complete f-moment convergence for arrays of rowwise extended negatively dependent (END) random variables. A general result on the complete moment convergence for arrays of rowwise END random variables is established, which generalizes and improves the corresponding ones of Wu et al. (Glasnik Matematički 49(69):447–466, 2014) and Hu et al. (Sankhya A 77(1):1–29, 2015). As applications, we present some new results on complete moment convergence for END random variables.

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References

  1. Adler, A., Rosalsky, A.: Some general strong laws for weighted sums of stochastically dominated random variables. Stoch. Anal. Appl. 5(1), 1–16 (1987). https://doi.org/10.1080/07362998708809104

    Article  MathSciNet  MATH  Google Scholar 

  2. Adler, A., Rosalsky, A., Taylor, R.L.: Strong laws of large numbers for weighted sums of random elements in normed linear spaces. Int. J. Math. Math. Sci. 12, 507–529 (1989). https://doi.org/10.1155/S0161171289000657

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, Y., Chen, A., Ng, K.W.: The strong law of large numbers for extended negatively dependent random variables. J. Appl. Probab. 47(4), 908–922 (2010). https://doi.org/10.1239/jap/1294170508

    Article  MathSciNet  MATH  Google Scholar 

  4. Chow, Y.S.: On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin. 16(3), 177–201 (1988)

    MathSciNet  MATH  Google Scholar 

  5. Hsu, P.-L., Robbins, H.: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33(2), 25 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hu, T.-C., Rosalsky, A., Wang, K.-L.: Complete convergence theorems for extended negatively dependent random variables. Sankhya A 77(1), 1–29 (2015). https://doi.org/10.1007/s13171-014-0058-z

    Article  MathSciNet  MATH  Google Scholar 

  7. Joag-Dev, K., Proschan, F.: Negative association of random variables with applications. Ann. Stat. 11(1), 286–295 (1983). https://doi.org/10.1214/aos/1176346079

    Article  MathSciNet  MATH  Google Scholar 

  8. Lehmann, E.L.: Some concepts of dependence. Ann. Math. Stat. 37(5), 1137–1153 (1966). https://doi.org/10.1214/aoms/1177699260

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu, L.: Precise large deviations for dependent random variables with heavy tails. Stat. Probab. Lett. 79(9), 1290–1298 (2009). https://doi.org/10.1016/j.spl.2009.02.001r

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu, L.: Necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails. Sci. China Math. 53(6), 1421–1434 (2010). https://doi.org/10.1007/s11425-010-4012-9r

    Article  MathSciNet  MATH  Google Scholar 

  11. Qiu, D.H., Chen, P.Y., Antonini, G.R., Volodin, A.: On the complete convergence for arrays of rowwise extended negatively dependent random variables. J. Korean Math. Soc. 50(2), 379–392 (2013). https://doi.org/10.4134/JKMS.2013.50.2.379

    Article  MathSciNet  MATH  Google Scholar 

  12. Shen, A.T.: Probability inequalities for END sequence and their applications. J. Inequal. Appl. 2011(1), 1–12 (2011). https://doi.org/10.1186/1029-242X-2011-98r

    Article  MathSciNet  Google Scholar 

  13. Shen, A.T.: On the strong convergence rate for weighted sums of arrays of rowwise negatively orthant dependent random variables. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 107(2), 257–271 (2013). https://doi.org/10.1007/s13398-012-0067-5r

    Article  MathSciNet  MATH  Google Scholar 

  14. Shen, A.T., Zhang, Y., Volodin, A.: Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables. Metrika 78(3), 295–311 (2015). https://doi.org/10.1007/s00184-014-0503-y

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, S.J., Wang, W.S.: Extended precise large deviations of random sums in the presence of END structure and consistent variation. J. Appl. Math. 2012, 12 (2012). https://doi.org/10.1155/2012/436531 (Article ID 436531)

  16. Wang, S.J., Wang, X.J.: Precise large deviations for random sums of END real-valued random variables with consistent variation. J. Math. Anal. Appl. 402(2), 660–667 (2013). https://doi.org/10.1016/j.jmaa.2013.02.002

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, X.J., Hu, T.-C., Volodin, A., Hu, S.H.: Complete convergence for weighted sums and arrays of rowwise extended negatively dependent random variables. Commun. Stat. Theory Methods 42(13), 2391–2401 (2013a). https://doi.org/10.1080/03610926.2011.609321

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, X.J., Wang, S.J., Hu, S.H., Ling, J.J., Wei, Y.F.: On complete convergence of weighted sums for arrays of rowwise extended negatively dependent random variables. Stochastics. Int. J. Probab. Stoch. Process. 85(6), 1060–1072 (2013). https://doi.org/10.1080/17442508.2012.736996

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, X.J., Li, X.Q., Hu, S.H., Wang, X.H.: On complete convergence for an extended negatively dependent sequence. Commun. Stat. Theory Methods 43(14), 2923–2937 (2014). https://doi.org/10.1080/03610926.2012.690489

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, X.J., Zheng, L.L., Xu, C., Hu, S.H.: Complete consistency for the estimator of nonparametric regression models based on extended negatively dependent errors. Statistics 49(2), 396–407 (2015). https://doi.org/10.1080/02331888.2014.888431

    Article  MathSciNet  MATH  Google Scholar 

  21. Wu, Q.Y.: Probability limit theory for mixing sequences. Science Press of China, Beijing (2006)

    Google Scholar 

  22. Wu, Y., Guan, M.: Convergence properties of the partial sums for sequences of END random variables. J. Korean Math. Soc. 49(6), 1097–1110 (2012). https://doi.org/10.4134/JKMS.2012.49.6.1097

    Article  MathSciNet  MATH  Google Scholar 

  23. Wu, Y., Ordonez Cabrera, M., Volodin, A.: Complete convergence and complete moment convergence for arrays of rowwise END random variables. Glasnik Matematički 49(69), 447–466 (2014). https://doi.org/10.3336/gm.49.2.16

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to the referee for offering some valuable suggestions which enabled them to improve the overall presentation.

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Authors and Affiliations

  1. School of Mathematical Sciences, Anhui University, Hefei, 230601, People’s Republic of China

    Yi Wu & Xuejun Wang

  2. Department of Mathematics, Tsing Hua University, Hsinchu, 30043, Taiwan, ROC

    Tien-Chung Hu

  3. Department of Mathematics and Statistics, University of Regina, Regina, SK, S4S 0A2, Canada

    Andrei Volodin

Authors
  1. Yi Wu
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  2. Xuejun Wang
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  3. Tien-Chung Hu
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  4. Andrei Volodin
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Corresponding author

Correspondence to Andrei Volodin.

Additional information

Supported by the National Natural Science Foundation of China (11671012), the Natural Science Foundation of Anhui Province (1508085J06), Ministry of Science and Technology, R.O.C. (MOST 105-2118-M-007-004-MY2), and the Natural Sciences and Engineering Research Council of Canada (NSERC).

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Wu, Y., Wang, X., Hu, TC. et al. Complete f-moment convergence for extended negatively dependent random variables. RACSAM 113, 333–351 (2019). https://doi.org/10.1007/s13398-017-0480-x

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  • DOI: https://doi.org/10.1007/s13398-017-0480-x

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