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On the negative dependence in Hilbert spaces with applications

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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.

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References

  1. H. W. Block, T. H. Savits, M. Shaked: Some concepts of negative dependence. Ann. Probab. 10 (1982), 765–772.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Borcea, P. Brändén, T. M. Liggett: Negative dependence and the geometry of polynomials. J. Am. Math. Soc. 22 (2009), 521–567.

    Article  MathSciNet  MATH  Google Scholar 

  3. R. M. Burton, A. R. Dabrowski, H. Dehling: An invariance principle for weakly associated random vectors. Ann. Probab. 9 (1981), 671–675.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  7. N. Ebrahimi, M. Ghosh: Multivariate negative dependence. Commun. Stat., Theory Methods A10 (1981), 307–337.

    Google Scholar 

  8. N. Etemadi: An elementary proof of the strong law of large numbers. Z. Wahrscheinlichkeitstheor. Verw. Geb. 55 (1981), 119–122.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Hájek, A. Rényi: Generalization of an inequality of Kolmogorov. Acta Math. Acad. Sci. Hung. 6 (1955), 281–283.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  15. E. L. Lehmann: Some concepts of dependence. Ann. Math. Stat. 37 (1966), 1137–1153.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  17. R. Li, W. Yang: Strong convergence of pairwise NQD random sequences. J. Math. Anal. Appl. 344 (2008), 741–747.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  19. Y. Miao: Hájek-Rényi inequality for dependent random variables in Hilbert space and applications. Rev. Unión Mat. Argent. 53 (2012), 101–112.

    MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  22. F. Móricz, R. L. Taylor: Strong laws of large numbers for arrays of orthogonal random variables. Math. Nachr. 141 (1989), 145–152.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  24. R. Pemantle: Towards a theory of negative dependence. J. Math. Phys. 41 (2000), 1371–1390.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  27. L. V. Thanh: On the almost sure convergence for dependent random vectors in Hilbert spaces. Acta Math. Hung. 139 (2013), 276–285.

    Article  MathSciNet  MATH  Google Scholar 

  28. Y. Wu, A. Rosalsky: Strong convergence for m-pairwise negatively quadrant dependent random variables. Glas. Mat., III. Ser. 50 (2015), 245–259.

    Article  MathSciNet  MATH  Google Scholar 

  29. L.-X. Zhang: Strassen’s law of the iterated logarithm for negatively associated random vectors. Stochastic Processes Appl. 95 (2001), 311–328.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Vinh University, 182 Le Duan Street, Vinh City, Nghe An Province, Vietnam

    Nguyen Thi Thanh Hien, Le Van Thanh & Vo Thi Hong Van

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  1. Nguyen Thi Thanh Hien
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  2. Le Van Thanh
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  3. Vo Thi Hong Van
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Correspondence to Nguyen Thi Thanh Hien.

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Hien, N.T.T., Thanh, L.V. & Van, V.T.H. On the negative dependence in Hilbert spaces with applications. Appl Math 64, 45–59 (2019). https://doi.org/10.21136/AM.2018.0060-18

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  • DOI: https://doi.org/10.21136/AM.2018.0060-18

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