SpringerLink
Log in

Some Mean Convergence Theorems for the Maximum of Normed Double Sums of Banach Space Valued Random Elements

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this correspondence, we establish mean convergence theorems for the maximum of normed double sums of Banach space valued random elements. Most of the results pertain to random elements which are M-dependent. We expand and improve a number of particular cases in the literature on mean convergence of random elements in Banach spaces. One of the main contributions of the paper is to simplify and improve a recent result of Li, Presnell, and Rosalsky [Journal of Mathematical Inequalities, 16, 117–126 (2022)]. A new maximal inequality for double sums of M-dependent random elements is proved which may be of independent interest. The sharpness of the results is illustrated by four examples.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chen, P. Y., Wang, D. C.: Lr convergence for B-valued random elements. Acta Mathematica Sinica, English Series, 28(4), 857–868 (2012)

    Article  MathSciNet  Google Scholar 

  2. Gnedenko, B. V.: The Theory of Probability. Fourth printing. Translated from the Russian by G. Yankovskiĭ. Mir Publisher, Moscow, 1978

    Google Scholar 

  3. Hoffmann-Jørgensen, J., Pisier, G.: The law of large numbers and the central limit theorem in Banach spaces. Annals of Probability, 4(4), 587–599 (1976)

    Article  MathSciNet  Google Scholar 

  4. Hu, T.-C., Ordóñez Cabrera, M., Volodin, A.: Convergence of randomly weighted sums of Banach space valued random elements and uniform integrability concerning the random weights. Statistics and Probability Letters, 51(2), 155–164 (2001)

    Article  MathSciNet  Google Scholar 

  5. Korzeniowski, A.: On Marcinkiewicz SLLN in Banach spaces. Annals of Probability, 12(1), 279–280 (1984)

    Article  MathSciNet  Google Scholar 

  6. Li, D., Presnell, B., Rosalsky, A.: Some degenerate mean convergence theorems for Banach space valued random elements. Journal of Mathematical Inequalities, 16(1), 117–126 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Ordóñez Cabrera, M., Rosalsky, A., Ünver, M., et al.: A new type of compact uniform integrability with application to degenerate mean convergence of weighted sums of Banach space valued random elements. Journal of Mathematical Analysis and Applications, 487(1), 123975 (2020)

    Article  MathSciNet  Google Scholar 

  9. Parker, R., Rosalsky, A.: Strong laws of large numbers for double sums of Banach space valued random elements. Acta Mathematica Sinica, English Series, 35(5), 583–596 (2019)

    Article  MathSciNet  Google Scholar 

  10. Quang, N. V., Thanh, L. V., Tien, N. D.: Almost sure convergence for double arrays of blockwise M-dependent random elements in Banach spaces. Georgian Mathematical Journal, 18(4), 777–800 (2011)

    Article  MathSciNet  Google Scholar 

  11. Rosalsky, A., Thanh, L. V.: Strong and weak laws of large numbers for double sums of independent random elements in Rademacher type p Banach spaces. Stochastic Analysis and Applications, 24(6), 1097–1117 (2006)

    Article  MathSciNet  Google Scholar 

  12. Rosalsky, A., Thanh, L. V.: A note on the stochastic domination condition and uniform integrability with applications to the strong law of large numbers. Statistics and Probability Letters, 178, 109181 (2021)

    Article  MathSciNet  Google Scholar 

  13. Rosalsky, A., Thành, L. V., Thuy, N. T.: On the laws of large numbers for double arrays of independent random elements in Banach spaces. Acta Mathematica Sinica, English Series, 30(8), 1353–1364 (2014)

    Article  MathSciNet  Google Scholar 

  14. Wang, X. C., Rao, M. B.: Convergence in the pth-mean and some weak laws of large numbers for weighted sums of random elements in separable normed linear spaces. Journal of Multivariate Analysis, 15(1), 124–134 (1984)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, University of Florida, Gainesville, FL, 32611-8545, USA

    Andrew Rosalsky

  2. Department of Mathematics, Vinh University, Vinh, Nghe An, Viet Nam

    Lê Vǎn Thành

  3. High School for Gifted Students, Vinh University, Vinh, Nghe An, Viet Nam

    Nguyen Thi Thuy

Authors
  1. Andrew Rosalsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lê Vǎn Thành
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nguyen Thi Thuy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lê Vǎn Thành.

Ethics declarations

Conflict of Interest The authors declare no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rosalsky, A., Thành, L.V. & Thuy, N.T. Some Mean Convergence Theorems for the Maximum of Normed Double Sums of Banach Space Valued Random Elements. Acta. Math. Sin.-English Ser. (2024). https://doi.org/10.1007/s10114-024-2669-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10114-024-2669-1

Keywords

MR(2010) Subject Classification

Search

Navigation