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Applications of Mathematics

, Volume 64, Issue 1, pp 45–59 | Cite as

On the negative dependence in Hilbert spaces with applications

  • Nguyen Thi Thanh HienEmail author
  • Le Van Thanh
  • Vo Thi Hong Van
Article
  • 13 Downloads

Abstract

This paper introduces the notion of pairwise and coordinatewise negative dependence for random vectors in Hilbert spaces. Besides giving some classical inequalities, almost sure convergence and complete convergence theorems are established. Some limit theorems are extended to pairwise and coordinatewise negatively dependent random vectors taking values in Hilbert spaces. An illustrative example is also provided.

Keywords

negative dependence pairwise negative dependence Hilbert space law of large numbers 

MSC 2010

60B11 60B12 60F15 

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References

  1. [1]
    H. W. Block, T. H. Savits, M. Shaked: Some concepts of negative dependence. Ann. Probab. 10 (1982), 765–772.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J. Borcea, P. Brändén, T. M. Liggett: Negative dependence and the geometry of polynomials. J. Am. Math. Soc. 22 (2009), 521–567.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    R. M. Burton, A. R. Dabrowski, H. Dehling: An invariance principle for weakly associated random vectors. Ann. Probab. 9 (1981), 671–675.MathSciNetCrossRefGoogle Scholar
  4. [4]
    P. Chen, S. H. Sung: Complete convergence and strong laws of large numbers for weighted sums of negatively orthant dependent random variables. Acta Math. Hung. 148 (2016), 83–95.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    S. Csörg˝o, K. Tandori, V. Totik: On the strong law of large numbers for pairwise independent random variables. Acta Math. Hung. 42 (1983), 319–330.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    A. R. Dabrowski, H. Dehling: A Berry-Esséen theorem and a functional law of the iterated logarithm for weakly associated random vectors. Stochastic Processes Appl. 30 (1988), 277–289.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    N. Ebrahimi, M. Ghosh: Multivariate negative dependence. Commun. Stat., Theory Methods A10 (1981), 307–337.Google Scholar
  8. [8]
    N. Etemadi: An elementary proof of the strong law of large numbers. Z. Wahrscheinlichkeitstheor. Verw. Geb. 55 (1981), 119–122.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    J. Hájek, A. Rényi: Generalization of an inequality of Kolmogorov. Acta Math. Acad. Sci. Hung. 6 (1955), 281–283.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    N. T. T. Hien, L. V. Thanh: On the weak laws of large numbers for sums of negatively associated random vectors in Hilbert spaces. Stat. Probab. Lett. 107 (2015), 236–245.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    T.-C. Hu, S. H. Sung, A. Volodin: A note on the strong laws of large numbers for random variables. Acta Math. Hung. 150 (2016), 412–422.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    N. V. Huan, N. V. Quang, N. T. Thuan: Baum-Katz type theorems for coordinatewise negatively associated random vectors in Hilbert spaces. Acta Math. Hung. 144 (2014), 132–149.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    M.-H. Ko: Hájek-Rényi inequality for m-asymptotically almost negatively associated random vectors in Hilbert space and applications. J. Inequal. Appl. (2018), Paper No. 80, 9 pages.Google Scholar
  14. [14]
    M.-H. Ko, T.-S. Kim, K.-H. Han: A note on the almost sure convergence for dependent random variables in a Hilbert space. J. Theor. Probab. 22 (2009), 506–513.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    E. L. Lehmann: Some concepts of dependence. Ann. Math. Stat. 37 (1966), 1137–1153.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    D. Li, A. Rosalsky, A. I. Volodin: On the strong law of large numbers for sequences of pairwise negative quadrant dependent random variables. Bull. Inst. Math., Acad. Sin. (N. S. ) 1 (2006), 281–305.MathSciNetzbMATHGoogle Scholar
  17. [17]
    R. Li, W. Yang: Strong convergence of pairwise NQD random sequences. J. Math. Anal. Appl. 344 (2008), 741–747.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    P. Matu la: A note on the almost sure convergence of sums of negatively dependent random variables. Stat. Probab. Lett. 15 (1992), 209–213.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Y. Miao: Hájek-Rényi inequality for dependent random variables in Hilbert space and applications. Rev. Unión Mat. Argent. 53 (2012), 101–112.zbMATHGoogle Scholar
  20. [20]
    F. Móricz: Strong limit theorems for blockwise m-dependent and blockwise quasiorthogonal sequences of random variables. Proc. Am. Math. Soc. 101 (1987), 709–715.CrossRefzbMATHGoogle Scholar
  21. [21]
    F. Móricz, K.-L. Su, R. L. Taylor: Strong laws of large numbers for arrays of orthogonal random elements in Banach spaces. Acta Math. Hung. 65 (1994), 1–16.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    F. Móricz, R. L. Taylor: Strong laws of large numbers for arrays of orthogonal random variables. Math. Nachr. 141 (1989), 145–152.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    R. F. Patterson, R. L. Taylor: Strong laws of large numbers for negatively dependent random elements. Nonlinear Anal., Theory Methods Appl. 30 (1997), 4229–4235.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    R. Pemantle: Towards a theory of negative dependence. J. Math. Phys. 41 (2000), 1371–1390.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    A. Rosalsky, L. V. Thanh: On the strong law of large numbers for sequences of blockwise independent and blockwise p-orthogonal random elements in Rademacher type p Banach spaces. Probab. Math. Stat. 27 (2007), 205–222.MathSciNetzbMATHGoogle Scholar
  26. [26]
    A. Rosalsky, L. V. Thanh: Some strong laws of large numbers for blockwise martingale difference sequences in martingale type p Banach spaces. Acta Math. Sin., Engl. Ser. 28 (2012), 1385–1400.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    L. V. Thanh: On the almost sure convergence for dependent random vectors in Hilbert spaces. Acta Math. Hung. 139 (2013), 276–285.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Y. Wu, A. Rosalsky: Strong convergence for m-pairwise negatively quadrant dependent random variables. Glas. Mat., III. Ser. 50 (2015), 245–259.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    L.-X. Zhang: Strassen’s law of the iterated logarithm for negatively associated random vectors. Stochastic Processes Appl. 95 (2001), 311–328.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2019

Authors and Affiliations

  • Nguyen Thi Thanh Hien
    • 1
    Email author
  • Le Van Thanh
    • 1
  • Vo Thi Hong Van
    • 1
  1. 1.Department of MathematicsVinh UniversityVinh City, Nghe An ProvinceVietnam

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