Exponential probability inequalities for WNOD random variables and their applications

  • Aiting Shen
  • Mei Yao
  • Wenjuan Wang
  • Andrei Volodin
Original Paper
First Online:
Received:
Accepted:

Abstract

Some exponential probability inequalities for widely negative orthant dependent (WNOD, in short) random variables are established, which can be treated as very important roles to prove the strong law of large numbers among others in probability theory and mathematical statistics. By using the exponential probability inequalities, we study the complete convergence for arrays of rowwise WNOD random variables. As an application, the Marcinkiewicz–Zygmund type strong law of large numbers is obtained. In addition, the complete moment convergence for arrays of rowwise WNOD random variables is studied by using the exponential probability inequality and complete convergence that we established.

Keywords

Widely negative orthant dependent random variables Complete convergence Complete moment convergence Marcinkiewicz–Zygmund type strong law of large numbers 

Mathematics Subject Classification

60F15 60E05 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

The authors are most grateful to the Editor-in-Chief Manuel Lopez Pellicer and anonymous referee for careful reading of the manuscript and valuable suggestions which helped in improving an earlier version of this paper.

References

  1. 1.
    Asadian, N., Fakoor, V., Bozorgnia, A.: Rosenthal’s type inequalities for negatively orthant dependent random variables. JIRSS 5(1–2), 69–75 (2006)Google Scholar
  2. 2.
    Chen, P.Y., Hu, T.C., Liu, X., Volodin, A.: On complete convergence for arrays of row-wise negatively associated random variables. Theory Probab. Appl. 52(2), 323–328 (2008)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Chen, Y., Chen, A., Ng, K.W.: The strong law of large numbers for extend negatively dependent random variables. J. Appl. Probab. 47, 908–922 (2010)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Chen, Y., Wang, L., Wang, Y.B.: Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models. J. Math. Anal. Appl. 401, 114–129 (2013)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Chow, Y.S.: On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin. 16, 177–201 (1988)MATHGoogle Scholar
  6. 6.
    Fakoor, V., Azarnoosh, H.A.: Probability inequalities for sums of negatively dependent random variables. Pak. J. Stat. 21, 257–264 (2005)MathSciNetMATHGoogle Scholar
  7. 7.
    Hu, T.Z.: Negatively superadditive dependence of random variables with applications. Chin. J. Appl. Probab. Stat. 16, 133–144 (2000)MATHGoogle Scholar
  8. 8.
    Li, X.Q., Yang, W.Z., Hu, S.H., Wang, X.J.: The Bahadur representation for sample quantile under NOD sequence. J. Nonparametr. Stat. 23(1), 59–65 (2011)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Liu, L.: Precise large deviations for dependent random variables with heavy tails. Stat. Probab. Lett. 79(9), 1290–1298 (2009)CrossRefMATHGoogle Scholar
  10. 10.
    Liu, X.J., Gao, Q.W., Wang, Y.B.: A note on a dependent risk model with constant interest rate. Stat. Probab. Lett. 82, 707–712 (2012)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Nili Sani, H.R., Amini, M., Bozorgnia, A.: Strong laws for weighted sums of negative dependent random variables. Islam. Repub. Iran J. Sci. 16(3), 261–266 (2005)MathSciNetGoogle Scholar
  12. 12.
    Qiu, D.H., Chen, P.Y.: Complete and complete moment convergence for weighted sums of widely orthant dependent random variables. Acta Math. Sin. Engl. Ser. 30(9), 1539–1548 (2014)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Qiu, D.H., Chen, P.Y., Antonini, R.G., Volodin, A.: On the complete convergence for arrays of rowwise extended negatively dependent random variables. J. Korean Math. Soc. 50(2), 379–392 (2013)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Qiu, D.H., Hu, T.C.: Strong limit theorems for weighted sums of widely orthant dependent random variables. J. Math. Res. Appl. 34(1), 105–113 (2014)MathSciNetMATHGoogle Scholar
  15. 15.
    Shao, Q.M.: A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theor. Probab. 13(2), 343–356 (2000)CrossRefMATHGoogle Scholar
  16. 16.
    Shen, A.T.: Some strong limit theorems for arrays of rowwise negatively orthant-dependent random variables. J. Inequal. Appl. 2011, Article ID 93, 10 (2011)Google Scholar
  17. 17.
    Shen, A.T.: Probability inequalities for END sequence and their applications. J. Inequal. Appl. 2011, Article ID 98, 12 (2011)Google Scholar
  18. 18.
    Shen, A.T.: On strong convergence for weighted sums of a class of random variables. Abstr. Appl. Anal. 2013, Article ID 216236, 7 (2013)Google Scholar
  19. 19.
    Shen, A.T.: Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models. Abstr. Appl. Anal. 2013, Article ID 862602, 9 (2013)Google Scholar
  20. 20.
    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, 257–271 (2013)CrossRefMATHGoogle Scholar
  21. 21.
    Sung, S.H.: A note on the complete convergence for arrays of dependent random variables. J. Inequal. Appl. 2011, Article ID 76, 8 (2011)Google Scholar
  22. 22.
    Sung, S.H.: Complete \(q\) moment convergence for arrays of random variables. J. Inequal. Appl. 2013, Article ID 24, 11 (2013)Google Scholar
  23. 23.
    Wang, K.Y., Wang, Y.B., Gao, Q.W.: Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate. Methodol. Comput. Appl. Probab. 15(1), 109–124 (2013)CrossRefMathSciNetGoogle Scholar
  24. 24.
    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, 660–667 (2013)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Wang, X.J., Hu, S.H.: Complete convergence and complete moment convergence for martingale difference sequence. Acta Math. Sin. Engl. Ser. 30(1), 119–132 (2014)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Wang, X.J., Hu, S.H., Yang, W.Z., Ling, N.X.: Exponential inequalities and inverse moment for NOD sequence. Stat. Probab. Lett. 80(5–6), 452–461 (2010)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Wang, X.J., Hu, S.H., Yang, W.Z.: Complete convergence for arrays of rowwise negatively orthant dependent random variables. RACSAM 106(2), 235–245 (2012)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Wang, X.J., Wang, S.J., Hu, S.H., Ling, J.M., 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)MathSciNetMATHGoogle Scholar
  29. 29.
    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, 2391–2401 (2013)CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Wang, X.J., Deng, X., Zheng, L.L., Hu, S.H.: Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications. Statistics: J. Theor. Appl. Stat. 48(4), 834–850 (2014)CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Wang, X.J., Xu, C., Hu, T.C., Volodin, A., Hu, S.H.: On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST 23, 607–629 (2014)CrossRefMathSciNetMATHGoogle Scholar
  32. 32.
    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: J. Theor. Appl. Stat. 49(2), 396–407 (2015)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Wang, Y.B., Li, Y.W., Gao, Q.W.: On the exponential inequality for acceptable random variables. J. Inequal. Appl. 2011, Article ID 40, 1–10 (2011)Google Scholar
  34. 34.
    Wang, Y.B., Cheng, D.Y.: Basic renewal theorems for a random walk with widely dependent increments and their applications. J. Math. Anal. Appl. 384, 597–606 (2011)CrossRefMathSciNetMATHGoogle Scholar
  35. 35.
    Wang, Y.B., Cui, Z., Wang, K.Y., Ma, X.: Uniform asymptotics of the finite-time ruin probability for all times. J. Math. Anal. Appl. 390, 208–223 (2012)CrossRefMathSciNetMATHGoogle Scholar
  36. 36.
    Wu, Q.Y.: Probability Limit Theory for Mixing Sequences. Science Press of China, Beijing (2006)Google Scholar
  37. 37.
    Wu, Q.Y.: A strong limit theorem for weighted sums of sequences of negatively dependent random variables. J. Inequal. Appl. 2010, Article ID 383805, 1–8 (2010)Google Scholar
  38. 38.
    Wu, Q.Y.: A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables. J. Inequal. Appl. 2012, Article ID 50, 1–10 (2012)Google Scholar
  39. 39.
    Wu, Q.Y., Jiang, Y.Y.: The strong consistency of \(M\) estimator in a linear model for negatively dependent random samples. Commun. Stat. Theory Methods 40, 467–491 (2011)CrossRefMathSciNetMATHGoogle Scholar
  40. 40.
    Wu, Y.F., Cabrera, M.O., Volodin, A.: On limiting behavior for arrays of rowwise negatively orthant dependent random variables. J. Korean Stat. Soc. 42, 61–70 (2013)CrossRefMATHGoogle Scholar
  41. 41.
    Wu, Y.F., Zhu, D.J.: Convergence properties of partial sums for arrays of rowwise negatively orthant dependent random variables. J. Korean Stat. Soc. 39(2), 189–197 (2010)CrossRefMathSciNetMATHGoogle Scholar
  42. 42.
    Yang, W.Z., Hu, S.H., Wang, X.J.: Complete convergence for moving average process of martingale differences. Discret. Dyn. Nat. Soc. 2012, Article ID 128492, 1–16 (2012)Google Scholar
  43. 43.
    Yang, W.Z., Liu, T.T., Wang, X.J., Hu, S.H.: On the Bahadur representation of sample quantiles for widely orthant dependent sequences. Filomat 28(7), 1333–1343 (2014)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Italia 2015

Authors and Affiliations

  • Aiting Shen
    • 1
  • Mei Yao
    • 2
    • 3
  • Wenjuan Wang
    • 1
  • Andrei Volodin
    • 4
  1. 1.School of Mathematical SciencesAnhui UniversityHefeiPeople’s Republic of China
  2. 2.School of MathematicsHefei University of TechnologyHefeiPeople’s Republic of China
  3. 3.School of MathematicsShandong UniversityJinanPeople’s Republic of China
  4. 4.Department of Mathematics and StatisticsUniversity of ReginaReginaCanada

Personalised recommendations