Skip to main content
SpringerLink
Log in

On the rate of convergence in the global central limit theorem for random sums of uniformly strong mixing random variables

  • Published:
Lithuanian Mathematical Journal Aims and scope Submit manuscript

Abstract

We present upper bounds of the integral \( {\int}_{-\infty}^{\infty }{\left|x\right|}^l\left|\mathrm{P}\left\{{Z}_N<x\right\}-\varPhi (x)\right|\mathrm{d}x \) for 0  ⩽  l  ⩽  1 + δ, where 0 < δ  ⩽  1, Φ(x) is a standard normal distribution function, and \( {Z}_N={S}_N/\sqrt{\mathrm{E}{S}_N^2} \) is the normalized random sum with \( \mathrm{E}{S}_N^2>0\left({S}_N{X}_1+\dots +{X}_N\right) \) of centered random variables X1,X2, . . . satisfying the uniformly strong mixing condition. The number of summands N is a nonnegative integer-valued random variable independent of X1,X2, . . . .

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. A.V. Bulinskii, Limit Theorems under Weak Dependence Conditions, Moscow State Univ. Press, Moscow, 1989 (in Russian).

    Google Scholar 

  2. V. ˇCekanaviˇcius, Approximation Methods in Probability Theory, Universitext, Springer, Switzerland, 2016.

  3. P. Hall and S.S. Heyde, Martingale Limit Theory and Its Application, Academic Press, New York, 1980.

    MATH  Google Scholar 

  4. I.A. Ibragimov, Some limit theorems for stationary in the weak sense random processes, Dokl. Akad. Nauk SSSR, 125(4):711–714, 1959 (in Russian).

    MathSciNet  MATH  Google Scholar 

  5. I.A. Ibragimov, Some limit theorems for stationary processes, Teor. Veroyatn. Primen., 7(4):361–392, 1962 (in Russian). English transl.: Theory Probab. Appl., 7(4):349–382, 1962.

  6. U. Islak, Asymptotic normality of random sums of m-dependent random variables, Stat. Probab. Lett., 109:22–29, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  7. Z. Lin and C. Lu, Limit Theory for Mixing Dependent Random Variables, Math. Appl., Dordr., Vol. 378, Kluwer Academic/Science Press, Dordrecht/Beijing, 1996.

  8. B.L.S. Prakasa Rao, On the rate of convergence in the random central limit theorem for martingales, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., 22(12):1255–1260, 1974.

  9. B.L.S. Prakasa Rao, Remarks on the rate of convergence in the random central limit theorem for mixing sequences, Z. Wahrscheinlichkeitstheor. Verw. Geb., 31:157–160, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  10. B.L.S. Prakasa Rao and M. Sreehari, On the order of approximation in the random central limit theorem for m-dependent random variables, Probab. Math. Stat., 36(1):47–57, 2016.

    MathSciNet  MATH  Google Scholar 

  11. Y. Shang, A martingale central limit theorem with random indices, Azerb. J. Math., 1(2):109–114, 2011.

    MathSciNet  MATH  Google Scholar 

  12. Y. Shang, A central limit theorem for randomly indexed m-dependent random variables, Filomat, 26(4):713–717, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  13. Ch. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, in L.M. Le Cam, J. Neyman, and E.L. Scott (Eds.), Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Vol. 2: Probability Theory, Univ. California Press, Berkeley, CA, 1972, pp. 583–602.

    Google Scholar 

  14. J. Sunklodas, Approximation of distributions of sums of weakly dependent random variables by the normal distribution, in Yu.V. Prokhorov and V. Statuleviˇcius (Eds.), Limit Theorems of Probability Theory, Springer, Berlin, Heidelberg, New York, 113–165, p. 2000.

  15. J. Sunklodas, On the global central limit theorem for ϕ-mixing random variables, Lith. Math. J., 35(2):185–196, 1995.

    MathSciNet  MATH  Google Scholar 

  16. J. Sunklodas, On the rate of convergence in the global central limit theorem for random sums of independent random variables, Lith. Math. J., 57(2):244–258, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Sunklodas, On the rate of convergence in the central limit theorem for random sums of strongly mixing random variables, Lith. Math. J., 58(2):219–234, 2018.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Vilnius University, Akademijos 4, LT-08663, Vilnius, Lithuania

    Jonas Kazys Sunklodas

Authors
  1. Jonas Kazys Sunklodas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonas Kazys Sunklodas.

Additional information

Publisher’s note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sunklodas, J.K. On the rate of convergence in the global central limit theorem for random sums of uniformly strong mixing random variables. Lith Math J 60, 410–423 (2020). https://doi.org/10.1007/s10986-020-09483-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10986-020-09483-9

Keywords

MSC

Search

Navigation