Skip to main content
Log in

Strong Limit Theorems for Weighted Sums of Random Elements in Banach Spaces

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

We propose an approach to the strong law of large numbers for weighted sums of random elements, where the result of Jajte [4] will be extended to the Banach space setting. Some typical applications of the main results are given.

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. K. Alam and K. M. L. Saxena, ‘‘Positive dependence in multivariate distributions,’’ Comm. Stat. A-Theory Methods 10, 1183–1196 (1981).

    Article  MathSciNet  Google Scholar 

  2. J. Hoffmann-Jørgensen and G. Pisier, ‘‘The law of large numbers and the central limit theorem in Banach spaces,’’ Ann. Prob. 4, 587–599 (1976).

    Article  MathSciNet  Google Scholar 

  3. T. C. Hu, A. Rosalsky, and A. Volodin, ‘‘Complete convergence theorems for weighted row sums from arrays of random elements in Rademacher type \(p\) and martingale type \(p\) Banach spaces,’’ Stoch. Anal. Appl. 37, 1092–1106 (2019).

  4. R. Jajte, ‘‘On the strong law of large numbers,’’ Ann. Prob. 31, 409–412 (2003).

    Article  Google Scholar 

  5. B. Y. Jing and H. Y. Liang, ‘‘Strong limit theorems for weighted sums of negatively associated random variables,’’ J. Theor. Prob. 21, 890–909 (2008).

    Article  MathSciNet  Google Scholar 

  6. K. Joag-Dev and F. Proschan, ‘‘Negative association of random variables, with applications,’’ Ann. Stat. 11, 286–295 (1983).

    Article  MathSciNet  Google Scholar 

  7. Z. A. Lagodowski and P. Matuła, ‘‘On almost sure limiting behavior of weighted sums of random fields,’’ Acta Math. Hungar. 126, 16–22 (2010).

    Article  MathSciNet  Google Scholar 

  8. P. Matuła, ‘‘A note on the almost sure convergence of sums of negatively dependent random variables,’’ Stat. Prob. Lett. 15, 209–213 (1992).

    Article  MathSciNet  Google Scholar 

  9. Y. J. Meng and Z. Y. Lin, ‘‘Strong laws of large numbers for \(\tilde{\rho}\)-mixing random variables,’’ J. Math. Anal. Appl. 365, 711–717 (2010).

    Article  MathSciNet  Google Scholar 

  10. Y. Miao, J. Mu, and S. Zhang, ‘‘ Limit theorems for identically distributed martingale difference,’’ Comm. Stat. Theory Methods 49, 1435–1445 (2020).

    Article  MathSciNet  Google Scholar 

  11. G. Pisier, ‘‘Martingales with values in uniformly convex spaces,’’ Israel J. Math. 20, 326–350 (1975).

    Article  Google Scholar 

  12. G. Pisier, ‘‘Probabilistic methods in the geometry of Banach spaces. Probability and analysis,’’ Lect. Notes Math. 1206, 167–241 (1986).

    Article  MathSciNet  Google Scholar 

  13. Q. M. Shao, ‘‘A comparison theorem on moment inequalities between negatively associated and independent random variables,’’ J. Theor. Prob. 13, 343–356 (2000).

    Article  MathSciNet  Google Scholar 

  14. A. Rosalsky and 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,’’ Prob. Math. Stat. 27, 205–222 (2007).

  15. T. C. Son, T. M. Cuong, and L. V. Dung, ‘‘On the almost sure convergence for sums of negatively superadditive dependent random vectors in Hilbert spaces and its application,’’ J. Commun. Stat.—Theory Methods (2019). https://doi.org/10.1080/03610926.2019.1584304

  16. S. H. Sung, ‘‘On the strong law of large numbers for weighted sums of random variables,’’ Comput. Math. Appl. 62, 4277–4287 (2011).

    Article  MathSciNet  Google Scholar 

  17. Z. Wang, ‘‘On strong law of large numbers for dependent random variables,’’ J. Inequal. Appl. 2011, 279754 (2011).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Nguyen Van Huan or Andrei Volodin.

Additional information

(Submitted by A. M. Elizarov)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huan, N.V., Volodin, A. Strong Limit Theorems for Weighted Sums of Random Elements in Banach Spaces. Lobachevskii J Math 41, 996–1003 (2020). https://doi.org/10.1134/S1995080220060207

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080220060207

Navigation