Skip to main content
SpringerLink
Log in

Lower-Bound Estimates for Poisson-Type Approximations

  • Published:
Lithuanian Mathematical Journal Aims and scope Submit manuscript

Abstract

Let S = w 1 S 1 + w 2 S 2 + ⋯ + w N S N . Here S j is a sum of identically distributed random variables with weight w j > 0. We consider the cases where S j is a sum of independent random variables, the sum of independent lattice variables, or has the Markov binomial distribution. Apart from the general case, we investigate the case of symmetric random variables. Distribution of S is approximated by a compound Poisson distribution, by a second-order asymptotic expansion, and by a signed exponential measure. Lower bounds for the accuracy of approximations in uniform metric are established.

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. T. V. Arak and A. Yu. Zaitsev, Uniform limit theorems for sums of independent random variables, Proc. Steklov Inst. Math., 174(1)(1988).

  2. A. D. Barbour and P. Hall, On the rate of Poisson convergence, Math. Proc. Cambridge Philos. Soc., 95, 473–480 (1984).

    MathSciNet  Google Scholar 

  3. A. D. Barbour and O. Chryssaphinou, Compound Poisson approximation: a user's guide, Ann. Appl. Probab., 11, 964–1002 (2001).

    MathSciNet  Google Scholar 

  4. A. D. Barbour and V. Cekanavicius, Total variation asymptotics for sums of independent integer random variables, Ann. Probab., 30(3), 509–545 (2002).

    MathSciNet  Google Scholar 

  5. J. G. Booth, P. Hall, and A. T. A. Wood, On the validity of Edgeworth and saddlepoint approximations, J. Multivariate Anal., 51, 121–138 (1994).

    Article  MathSciNet  Google Scholar 

  6. V. Cekanavicius, Lower bound in the Bergstrom identity, Liet. Matem. Rink., 28(1), 153–169 (1988).

    MATH  MathSciNet  Google Scholar 

  7. V. Cekanavicius, Estimates in total variation for convolutions of compound distributions, J. London Math. Soc. (2), 58, 748–760 (1998).

    MathSciNet  Google Scholar 

  8. V. Cekanavicius, Infinitely divisible approximations for discrete nonlattice variables, Adv. Appl. Probab., 35, 982–1006 (2003).

    MATH  MathSciNet  Google Scholar 

  9. V. Cekanavicius and M. Mikalauskas, Signed Poisson approximations for Markov chains, Stochastic. Proc. Appl., 82, 205–227 (1999).

    MathSciNet  Google Scholar 

  10. V. Cekanavicius and M. Mikalauskas Large deviations for the Markov binomial distribution, Lith. Math. J., 41(4), 307–318 (2001).

    MathSciNet  Google Scholar 

  11. P. Franken, Approximation des Verteilungen von Summen unabhangiger nichtnegativer ganzzahliger Zufallsgroßen durch Poissonsche Verteilungen, Math. Nachr., 27, 303–340 (1964).

    MATH  MathSciNet  Google Scholar 

  12. J. Gani, On the probability generating function of the sum of Markov-Bernoulli random variables, J. Appl. Probab., 19A, 321–326 (1982).

    MATH  MathSciNet  Google Scholar 

  13. C. Hipp, Approximation of aggregate claims distributions by compound Poisson distributions, Insurance: Math. and Economics., 4, 227–232 (1985).

    MATH  MathSciNet  Google Scholar 

  14. C. Hipp, Improved approximations for the aggregate claims distribution in the individual model, ASTIN Bull., 16, 89–100 (1986).

    Google Scholar 

  15. P. Kornya, Distribution of aggregate claims in the Individual Risk Theory model, Society of Actuaries: Transactions, 35, 823–858 (1983).

    Google Scholar 

  16. J. Kruopis, Precision of approximations of the generalized Binomial distribution by convolutions of Poisson measures, Lith. Math. J., 26(1), 37–49 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  17. L. Le Cam, On the distribution of sums of independent random variables, in: J. Neyman and L. Le Cam (Eds.), Bernoulli, Bayes, Laplace, Anniversary volume, Springer, Berlin (1965), pp. 179–202.

    Google Scholar 

  18. V. K. Mackevicius, A lower bound for the convergence rate in the central limit theorem, Theory Probab. Appl., 28, 596–601 (1983).

    MathSciNet  Google Scholar 

  19. E. L. Presman, Approximation of binomial distributions by infinitely divisible ones, Theory Probab. Appl., 28, 393–403 (1983).

    MathSciNet  Google Scholar 

  20. E. L. Presman, The variation distance between the distribution of a sum of independent Bernoulli variables and the Poisson law, Theory Probab. Appl., 30, 417–422 (1985).

    MathSciNet  Google Scholar 

  21. Yu. V. Prokhorov, On the sums of identically distributed random variables, Doklady Akad. Nauk SSSR, 105, 645–647 (1955).

    Google Scholar 

  22. B. Roos, Kerstan's method in the multivariate Poisson approximation: an expansion in the exponent, Theory Probab. Appl., 47, 358–363 (2002).

    MATH  MathSciNet  Google Scholar 

  23. B. Roos, On Hipp's compound Poisson approximations via concentration functions, Bernoulli, 11(3), 533–557 (2005).

    MATH  MathSciNet  Google Scholar 

  24. J. Siaulys and V. Cekanavicius, Approximation of distributions of integer-valued additive functions by discrete charges. I, Lith. Math. J., 28(4), 392–401 (1988).

    MathSciNet  Google Scholar 

  25. R. J. Serfling, A general Poisson approximation theorem, Ann. Probab., 3, 726–731 (1975).

    MATH  MathSciNet  Google Scholar 

  26. R. F. Serfozo, Compound Poisson approximation for sums of random variables, Ann. Probab., 14, 1391–1398 (1986). Correction Ann. Probab., 16, 429 (1988).

    MATH  MathSciNet  Google Scholar 

  27. Y. H. Wang, Approximating kth order two-state Markov chains, J. Appl. Probab., 29, 861–868 (1992).

    MATH  MathSciNet  Google Scholar 

  28. A. Yu. Zaitsev, On the accuracy of approximation of distributions of sums of independent random variables — which are nonzero with a small probability — by means of accompanying laws, Theory Probab. Appl., 28, 657–669 (1983).

    Google Scholar 

  29. A. Yu. Zaitsev, On the uniform approximation of distributions of sums of independent random variables, Theory Probab. Appl., 32, 40–47 (1987).

    Article  Google Scholar 

  30. A. Yu. Zaitsev, A multidimensional variant of Kolmogorov's second uniform limit theorem, Theory Probab. Appl., 34, 108–128 (1989).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Vilnius University, Naugarduko 24, LT-03225, Vilnius, Lithuania

    V. Cekanavicius & A. Elijio

Authors
  1. V. Cekanavicius
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Elijio
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 501–524, October–December, 2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cekanavicius, V., Elijio, A. Lower-Bound Estimates for Poisson-Type Approximations. Lith Math J 45, 405–423 (2005). https://doi.org/10.1007/s10986-006-0004-4

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10986-006-0004-4

Keywords

Search

Navigation