Skip to main content
SpringerLink
Log in

Estimate for the accuracy of the Poisson approximation for the number of empty cells in an equiprobable scheme for group allocation of particles, and applications

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

The properties of the distribution of the number of empty cells are analyzed for a natural generalization of an equiprobable scheme for group allocation of particles. An error estimate is obtained for the Chen-Stein method of Poisson approximation for the distribution of the number of empty cells in this scheme. This estimate is used to derive sufficient conditions for the distribution of the number of empty cells to converge to the convolutions of the Poisson distribution and two-point distributions. On the basis of these results, asymptotic properties of the solution set of a perturbed system of linear Boolean equations are studied (in the case of consistent increase in the number of unknowns and the number of equations).

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. V. F. Kolchin, B. A. Sevastyanov, and V. P. Chistyakov, Random Allocations (Nauka, Moscow, 1976; J. Wiley & Sons, New York, 1978).

    MATH  Google Scholar 

  2. A. A. Markoff, Wahrscheinlichkeitsrechnung (Teubner, Leipzig, 1912).

    MATH  Google Scholar 

  3. B. A. Sevastyanov, “Limit theorems in a scheme for allocation of particles in cells,” Teor. Veroyatn. Primen. 11(4), 696–700 (1966) [Theory Probab. Appl. 11, 614–619 (1966)].

    MathSciNet  Google Scholar 

  4. V. A. Vatutin and V. G. Mikhailov, “Limit theorems for the number of empty cells in an equiprobable scheme for group allocation of particles,” Teor. Veroyatn. Primen. 27(4), 684–692 (1982) [Theory Probab. Appl. 27, 734–743 (1983)].

    MathSciNet  MATH  Google Scholar 

  5. A. D. Barbour, L. Holst, and S. Janson, Poisson Approximation (Clarendon Press, Oxford, 1992).

    MATH  Google Scholar 

  6. V. P. Chistyakov, “Discrete limit distributions in the problem of balls falling in cells with arbitrary probabilities,” Mat. Zametki 1(1), 9–16 (1967) [Math. Notes 1, 6–11 (1967)].

    MathSciNet  Google Scholar 

  7. V. N. Sachkov, “Random coverings and systems of functional equations,” Intellekt. Sist. 2, 297–313 (1997).

    Google Scholar 

  8. V. N. Sachkov, “Random nonuniform coverings and functional equations,” in Proceedings in Discrete Mathematics (Fizmatlit, Moscow, 2002), Vol. 5, pp. 205–218 [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    V. G. Mikhailov

Authors
  1. V. G. Mikhailov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © V.G. Mikhailov, 2013, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2013, Vol. 282, pp. 165–180.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mikhailov, V.G. Estimate for the accuracy of the Poisson approximation for the number of empty cells in an equiprobable scheme for group allocation of particles, and applications. Proc. Steklov Inst. Math. 282, 157–171 (2013). https://doi.org/10.1134/S008154381306014X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S008154381306014X

Keywords

Search

Navigation