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Some convergence properties for the maximum of partial sums of m-negatively associated random variables

  • Mengge Wang
  • Xuejun WangEmail author
Original Paper
  • 6 Downloads

Abstract

In this paper, we present the Spitzer type law of large numbers for the maximum of partial sums of m-negatively associated random variables \(\left\{ X_n,n\ge 1\right\} \) and the \(L_p\) convergence property under the Cesàro uniform integrability condition for the maximum of partial sums. In addition, we give some simulations to verify the convergence behavior of \(\frac{1}{\psi (n)}\sum _{i=1}^{n}X_i\) which is in accordance with our theoretical result. The main results obtained in this paper extend and improve the corresponding ones for negatively associated random variables.

Keywords

m-negatively associated random variables Spitzer type law of large numbers Cesàro uniform integrability \(L_p\) convergence 

Mathematics Subject Classification

60F15 

Notes

Acknowledgements

The authors are most grateful to the Editor and anonymous referee for carefully reading the manuscript and for valuable suggestions which helped in improving an earlier version of this paper.

References

  1. 1.
    Alam, K., Saxena, K.M.L.: Positive dependence in multivariate distributions. Commun. Stat. Theory Methods 10, 1183–1196 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chandra, T.K.: Uniform integrability in the Cesáro sense and the weak law of large numbers. Sankhyā Indian J. Stat. (Ser. A) 51, 309–317 (1989)zbMATHGoogle Scholar
  3. 3.
    Chen, P.Y., Hu, T.C., Liu, X., Volodin, A.: On complete convergence for arrays of rowwise negatively associated random variables. Theory Probab. Appl. 52, 323–328 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hu, T.C., Chiang, C.Y., Taylor, R.L.: On complete convergence for arrays of rowwise \(m\)-negatively associated random variables. Nonlinear Anal. 71, 1075–1081 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jajte, R.: On the strong law of large numbers. Ann. Probab. 31(1), 409–412 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Joag-Dev, K., Proschan, F.: Negative association of random variables with applications. Ann. Stat. 11, 286–295 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Liang, H.Y., Zhang, J.J.: Strong convergence for weighted sums of negatively associated arrays. Chin. Ann. Math. Ser. B 31B(2), 273–288 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Matula, P.: A note on the almost sure convergence of sums of negatively dependent random variables. Stat. Prob. Lett. 15, 209–213 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Shao, Q.M.: A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theor. Probab. 13, 343–356 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Shen, A.T.: On asymptotic approximation of inverse moments for a class of nonnegative random variables. Statistics 48(6), 1371–1379 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Shen, A.T., Zhang, Y., Xiao, B.Q., Volodin, A.: Moment inequalities for m-negatively associated random variables and their applications. Stat. Papers 58, 911–928 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Spitzer, F.: A combinatorial lemma and its application to probability theory. Trans. Am. Math. Soc. 82, 323–339 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sung, S.H.: On the strong convergence for weighted sums of random variables. Stat. Papers 52, 447–454 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wang, X.H., Hu, S.H.: Weak laws of large numbers for arrays of dependent random variables. Stoch. Int. J. Probab. Stoch. Processes 86(5), 759–775 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wang, X.J., Li, X.Q., Hu, S.H., Yang, W.Z.: Strong limit theorems for weighted sums of negatively associated random variables. Stoch. Anal. Appl. 29(1), 1–14 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wu, Y., Wang, X.J., Sung, S.H.: Complete moment convergence for arrays of rowwise negatively associated random variables and its application in nonparametric regression model. Probab. Eng. Inf. Sci. 32, 37–57 (2018)CrossRefGoogle Scholar
  17. 17.
    Wang, Z.Z.: On strong law of large numbers for dependent random variables. J. Inequ. Appl. 2011, Article ID 279754, 13 (2011)Google Scholar
  18. 18.
    Wu, Q.Y., Jiang, Y.Y.: A law of the iterated logarithm of partial sums for NA random variables. J. Korean Stat. Soc. 39(2), 199–206 (2010a)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wu, Q.Y., Jiang, Y.Y.: Chovers law of the iterated logarithm for negatively associated sequences. J. Syst. Sci. Complex. 23, 293–302 (2010b)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wu, Y.F., Hu, T.C., Volodin, A.: Complete convergence and complete moment convergence for weighted sums of \(m\)-NA random variables. J. Inequ. Appl. 2015, Article ID 200, 14 (2015)Google Scholar
  21. 21.
    Wu, Y.F.: On complete moment convergence for arrays of rowwise negatively associated random variables. RACSAM 108(2), 669–681 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zarei, H., Jabbari, H.: Complete convergence of weighted sums under negative dependence. Stat. Papers 52, 413–418 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesAnhui UniversityHefeiPeople’s Republic of China

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