Skip to main content
SpringerLink
Log in

Weak Laws of Large Numbers for sequences of random variables with infinite rth moments

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

Let \({(X_n;n \geq 1)}\) be a sequence of independent random variables with infinite rth absolute moments for some \({0 < r < 2}\). We investigate weak laws of large numbers for the weighted sum \({S_n = \sum_{j=1}^{m_n}c_{nj}X_j}\), where \({(c_{nj};1 \leq j \leq m_n,n \geq 1)}\) is an array of real numbers. As illustrative examples, we obtain a weak law of large numbers of extended Pareto–Zipf distributions and generalized Feller Game. Furthermore, these results are applied to study weak law of large numbers of moving average sums of a sequence of i.i.d. random variables.

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

Access this article

Log in via an institution

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. Adler A.: An exact weak law of large numbers. Bull. Inst. Math. Acad. Sin. (NS) 3, 417–422 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Bingham N., Goldie C., Teugels J.: Regular Variation. University Press, Cambridge (1989)

    MATH  Google Scholar 

  3. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, Wiley (New York, 1966).

  4. Griffin P.S., Kuelbs J.D.: Self-normalized laws of the iterated logatithm. Ann. Probab. 17, 1571–1601 (1989)

    Article  MathSciNet  Google Scholar 

  5. Matsumoto K., Nakata T.: Limit theorems for a generalized Feller game. J. Appl. Probab. 50, 54–63 (2013)

    Article  MathSciNet  Google Scholar 

  6. Nakata T.: Limit theorems for nonnegative independent random variables with truncation. Acta Math. Hungar. 145, 1–16 (2015)

    Article  MathSciNet  Google Scholar 

  7. Nakata T.: Weak laws of large numbers for weighted independent random variables with infinite mean. Statist. Probab. Lett. 109, 124–129 (2016)

    Article  MathSciNet  Google Scholar 

  8. Pipiras V., Taqqu M.S.: Long-Range Dependence and Self-Similarity. University Press, Cambridge (2017)

    Book  Google Scholar 

  9. Szewczak Z.S.: Marcinkiewicz laws with infinite moments. Acta Math. Hungar. 127, 64–84 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Da Nang University of Education, 459 Ton Duc Thang, Da Nang, Vietnam

    L. V. Dung & N. T. H. Yen

  2. Hanoi University of Science, 334 Nguyen Trai, Hanoi, Vietnam

    T. C. Son

Authors
  1. L. V. Dung
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. C. Son
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. T. H. Yen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. V. Dung.

Additional information

This research has been partially supported by Vietnams National Foundation for Science and Technology Development (grant no. 101.03-2017.24), the Vietnam National University, Hanoi (grant no. QG.16.09) and Da Nang Universitty of Education (grant no. T2017-TD-03-07).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dung, L.V., Son, T.C. & Yen, N.T.H. Weak Laws of Large Numbers for sequences of random variables with infinite rth moments. Acta Math. Hungar. 156, 408–423 (2018). https://doi.org/10.1007/s10474-018-0865-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-018-0865-0

Key words and phrases

Mathematics Subject Classification

Search

Navigation