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Lithuanian Mathematical Journal

, Volume 58, Issue 2, pp 127–140 | Cite as

Approximating by convolution of the normal and compound Poisson laws via Stein’s method

  • Vydas Čekanavičius
  • Palaniappan Vellaisamy
Article

Abstract

We apply Stein’s method to estimate the closeness of the mixtures of distributions to the convolution of the normal and Poisson laws. To derive the Stein operator, we use a moment generating function. Then we apply a perturbation technique to estimate the solution to the Stein equation.

Keywords

compound Poisson convolution normal distribution perturbation Stein’s method 

MSC

60F05 62E20 

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References

  1. 1.
    T. V. Arak and A. Yu. Zaĭtsev, Uniform limit theorems for sums of independent random variables, Proc. Steklov Inst. Math., 174:1–222, 1988.MathSciNetGoogle Scholar
  2. 2.
    A. D. Barbour, Asymptotic expansions in the Poisson limit theorem, Ann. Probab., 15(2):748–766, 1987.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. D. Barbour and L. H. Y. Chen (Eds.), An Introduction to Stein’s Method, Lect. Notes Ser., Inst. Math. Sci., Natl. Univ. Singap., Vol. 4, World Scientific, Singapore, 2005.Google Scholar
  4. 4.
    A. D. Barbour, L. H. Y. Chen, and W.-L. Loh, Compound Poisson approximation for nonnegative random variables vis Stein’s method, Ann. Probab., 20(4):1843–1866, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. D. Barbour and V. Čekanavičius, Total variation asymptotics for sums of independent integer random variables, Ann. Probab., 30(2):509–545, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. D. Barbour, V. Čekanavičius, and A. Xia, On Stein’s method and perturbations, ALEA, Lat. Am. J. Probab. Math. Stat., 3:31–53, 2007.MathSciNetzbMATHGoogle Scholar
  7. 7.
    A. D. Barbour and A. Xia, Poisson perturbations, ESAIM, Probab. Stat., 3:131–150, 1999.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    L. H. Y. Chen, L. Goldstein, and Q.-M. Shao, Normal Approximation by Stein’s Method, Springer, Heidelberg, 2011.CrossRefzbMATHGoogle Scholar
  9. 9.
    L. H. Y. Chen and A. Röllin, Approximating dependent rare events, Bernoulli, 19(4):1243–1267, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    H. L. Gan and A. Xia, Stein’s method for conditional Poisson approximation, Stat. Probab. Lett., 100:19–26, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    I. A. Ibragimov and Yu.V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen, 1971.zbMATHGoogle Scholar
  12. 12.
    S. K. Kattumannil, On Stein’s identity and its applications, Stat. Probab. Lett., 79(12):1444–1449, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. E. Koudou and C. Ley, Characterizations of GIG laws: A survey, Probab. Surveys, 11:161–176, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    C. Ley, G. Reinert, and Y. Swan, Stein’s method for comparison of univariate distributions, Probab. Surveys, 14:1–52, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    C. Ley and Y. Swan, Stein’s density approach and information inequalities, Electron. Commun. Probab., 18:7, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    N. Ross, Fundamentals of Stein’s method, Probab. Surveys, 8:210–293, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    N. S. Upadhye, V. Čekanavičius, and P. Vellaisamy, On Stein operators for discrete approximations, Bernoulli, 23(4A):2828–2859, 2017.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsVilnius UniversityVilniusLithuania
  2. 2.Department of MathematicsIndian Institute of Technology BombayPowaiIndia

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