Advertisement

Lithuanian Mathematical Journal

, Volume 49, Issue 2, pp 216–221 | Cite as

On the rate of convergence of L p norms in the CLT for poisson and gamma random variables

  • J. Sunklodas
Article

Abstract

In the paper, we present upper bounds of L p norms \( \Delta _{\mathbb{D}X,p} \) of order (\( \mathbb{D}\) X)-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X\( \mathbb{E}\) X)/ √\( \mathbb{D}\) X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter α > 0.

Keywords

central limit theorem Lp norms Poisson distribution gamma distribution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Balakrishnan and V.B. Nevzorov, A Primer on Statistical Distributions, Wiley, New Jersey, 2003.MATHCrossRefGoogle Scholar
  2. 2.
    R.V. Erickson, On an L p version of the Berry–Esséen theorem for independent and m-dependent random variables, Ann. Probab., 1(3):497–503, 1973.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    R.V. Erickson, L 1 bounds for asymptotic normality of m-dependent sums using Stein’s technique, Ann. Probab., 2:522–529, 1974.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    I.A. Ibragimov and Yu.V. Linnik, Independent and Stationary Connected Variables, Nauka, Moscow, 1965 (in Russian).Google Scholar
  5. 5.
    M. Loève, Probability Theory. I, 4th edition, Springer-Verlag, New York, Berlin, Heidelberg, 1977.MATHGoogle Scholar
  6. 6.
    E. Lukacs, Characteristic Functions, Nauka, Moscow, 1979 (in Russian).MATHGoogle Scholar
  7. 7.
    Ch. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, in Proc. Math. Statist. and Probab., Vol. 2, Univ. Calif. Press, Berkeley, CA, 1972, pp. 583–602.Google Scholar
  8. 8.
    J. Sunklodas, On the rate of convergence of L p norms in the central limit theorem for independent random variables, Lith. Math. J., 42(3):296–307, 2002.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    A.N. Tikhomirov, On the rate of convergence in the central limit theorem for weakly dependent variables, Theory Probab. Appl., 25:790–809, 1980.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsAkademijos 4, 08663 Vilnius Vilnius Gediminas Technical UniversityVilniusLithuania

Personalised recommendations