Skip to main content
Log in

On the \( L^1 \)-Convergence for Double Arrays of Blockwise M-Pairwise Negatively Dependent Random Variables

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

This paper introduces the concept of double arrays of blockwise M-pairwise negatively dependent random variables and establishes \(L^1\) convergence for double arrays of blockwise M-pairwise negatively dependent random variables. Our proof is completely different from those of the aforementioned papers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Lehmann, E.: Some concepts of dependence. Ann. Math. Stat. 37(5), 1137–1153 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anh, V.T.V., Anh, N.T.N., Hien, N.T.T., Tu, N.N.: Convergence in mean for double arrays of \(m\)-pairwise negatively dependent random variables. Bulletin Malays. Math. Sci. Soc. 45(4), 1507–1520 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  3. Stadtmüller, U., Thành, L.V.: On the strong limit theorems for double arrays of blockwise \(M\)-dependent random variables. Acta Math. Sinica Engl. Series 27(10), 1923–1934 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Huan, N.V., Quang, N.V., Volodin, A.I.: Strong laws for blockwise martingale difference arrays in banach spaces. Lobachevskii J. Math. 31(4), 326–335 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Rosalsky, A., Thành, L.V.: A note on the stochastic domination condition and uniform integrability with applications to the strong law of large numbers. Statist. Probab. Lett. 178, 109181 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. Czerebak-Mrozowicz, E.B., Klesov, O.I., Rychlik, Z.: Marcinkiewicz-type strong law of large numbers for pairwise independent random fields. Probab. Math. Stat. 22(1), 127–139 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Hong, D.H., Hwang, S.Y.: Marcinkiewicz-type strong law of large numbers for double arrays of pairwise independent random variables. Int. J. Math. Math. Sci. 22(1), 171–177 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hong, D.H., Volodin, A.I.: Marcinkiewicz-type law of large numbers for double arrays. J. Korean Math. Soc. 36(6), 1133–1143 (1999)

    MathSciNet  MATH  Google Scholar 

  9. Rosalsky, A., Thành, L.V.: Weak laws of large numbers for double sums of independent random elements in Rademacher type \(p\) and stable type \(p\) Banach spaces. Nonlinear Anal. Theory Methods Appl. 71(12), 1065–1074 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Thành, L.V.: On the \(L_p\)-convergence for multidimensional arrays of random variables. Int. J. Math. Math. Sci. 2005(8), 1317–1320 (2005)

    Article  MATH  Google Scholar 

  11. Klesov, O.: Limit Theorems for Multi-indexed Sums of Random Variables vol. 71, (2014)

  12. Rosalsky, A., Thành, L.V.: On almost sure and mean convergence of normed double sums of Banach space valued random elements. Stoch. Anal. Appl. 25(4), 895–911 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Móricz, F.: The Kronecker lemmas for multiple series and some application. Acta Mathematica Academiae Scientiarum Hungarica 37(1), 39–50 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  14. Anh, V.T.N.: On the strong laws of large numbers for sequences of blockwise pairwise and coordinatewise negatively dependent random vectors in hilbert spaces. Lobachevskii J. Math. 42(13), 3077–3087 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hien, N.T.T., Thành, L.V., Van, V.T.H.: On the negative dependence in Hilbert spaces with applications. Appl. Math. 64(1), 45–59 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Thi Ngoc Anh Vu.

Ethics declarations

Conflict of interest

The author reports there are no competing interests to declare.

Additional information

Communicated by See Keong Lee.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vu, T.N.A. On the \( L^1 \)-Convergence for Double Arrays of Blockwise M-Pairwise Negatively Dependent Random Variables. Bull. Malays. Math. Sci. Soc. 46, 172 (2023). https://doi.org/10.1007/s40840-023-01558-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40840-023-01558-3

Keywords

Mathematics Subject Classification

Navigation