Skip to main content

An Improved Multivariate Version of Kolmogorov’s Second Uniform Limit Theorem

The aim of the present work is to show that the results obtained earlier on approximation of distributions of sums of independent summands by infinitely divisible laws may be transferred to estimation of the closeness of distributions on convex polyhedra.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    T. V. Arak and A. Yu. Zaitsev, “Uniform limit theorems for sums of independent random variables,” Proc. Steklov Inst. Math., 174 (1988).

  2. 2.

    A. A. Borovkov and A. I. Sakhanenko, “Estimates for the convergence rate in the invariance principle for Banach spaces,” Teor. Veroyatn. Primen., 25, 734–744 (1980).

    MathSciNet  MATH  Google Scholar 

  3. 3.

    F. Götze and A. Yu. Zaitsev, “Estimates for the closeness of convolutions of probability distributions on convex polyhedra,” Zap. Nauchn. Semin. POMI, 474, 108–117 (2018); English transl. J. Math. Sci., 251, No. 1, 67–73 (2020).

  4. 4.

    I. A. Ibragimov and E. L. Presman, “The rate of convergence of the distributions of sums of independent random variables to accompanying laws,” Teor. Veroyatn. Primen., 18, 753–766 (1973).

    MathSciNet  Google Scholar 

  5. 5.

    A. N. Kolmogorov, “Two uniform limit theorems for sums of independent random variables,” Teor. Veroyatn. Primen., 1, 426–436 (1956).

    MATH  Google Scholar 

  6. 6.

    A. N. Kolmogorov, “Approximation of distributions of sums of independent terms by infinitely divisible distributions,” Trudy Moskov. Matem. Ob., 12, 437–451 (1963).

    MathSciNet  MATH  Google Scholar 

  7. 7.

    L. Le Cam, “On the distribution of sums of independent random variables,” in: Proc. Internat. Res. Sem. Statist. Lab., Univ. California, Berkeley (1965), pp. 179–202.

  8. 8.

    B. A. Rogozin, “An estimate of the concentration functions,” Teor. Veroyatn. Primen., 6, 103–105 (1961).

    MathSciNet  Google Scholar 

  9. 9.

    B. A. Rogozin, “On the increase of dispersion of sums of independent random variables,” Teor. Veroyatn. Primen., 6, 106–108 (1961).

    MathSciNet  Google Scholar 

  10. 10.

    A. Yu. Zaitsev, “Several remarks on approximation of distributions of sums of independent terms,” Zap. Nauchn. Semin. LOMI, 136, 48–57 (1984); English transl. J. Math. Sci., 33, No. 1 (1986).

  11. 11.

    A. Yu. Zaitsev, “A multidimensional version of Kolmogorov’s second uniform limit theorem,” Teor. Veroyatn. Primen., 34, 128–151 (1989).

    Google Scholar 

  12. 12.

    A. Yu. Zaitsev, “A class of nonuniform estimates in multidimensional limit theorems,” Zap. Nauchn. Semin. POMI, 184, 92–105 (1990); English transl. J. Math. Sci., 68, No. 4 (1994).

  13. 13.

    A. Yu. Zaitsev and T. V. Arak, “The rate of convergence in Kolmogorov’s second uniform limit theorem,” Teor. Veroyatn. Primen., 28, 333–353 (1984).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to F. Götze.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 486, 2019, pp. 71–85.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Götze, F., Zaitsev, A.Y. & Zaporozhets, D. An Improved Multivariate Version of Kolmogorov’s Second Uniform Limit Theorem. J Math Sci 258, 782–792 (2021). https://doi.org/10.1007/s10958-021-05594-x

Download citation