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Lithuanian Mathematical Journal

, Volume 57, Issue 1, pp 38–58 | Cite as

Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions

Article

Abstract

We improve bounds of accuracy of the normal approximation to the distribution of a sum of independent random variables under relaxed moment conditions, in particular, under the absence of moments of orders higher than the second. We extend these results to Poisson binomial, binomial, and Poisson random sums. Under the same conditions, we obtain bounds for the accuracy of approximation of the distributions of mixed Poisson random sums by the corresponding limit law. In particular, we construct these bounds for the accuracy of approximation of the distributions of geometric, negative binomial, and Poisson-inverse gamma (Sichel) random sums by the Laplace, variance gamma, and Student distributions, respectively.

Keywords

central limit theorem normal distribution convergence rate estimate Lindeberg condition uniform distance Poisson binomial distribution Poisson binomial random sum binomial random sum Poisson random sum mixed Poisson random sum geometric random sum negative binomial random sum Poisson-inverse gamma random sum Laplace distribution variance gamma distribution Student distribution absolute constant 

MSC

60F05 60E15 

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Hangzhou Dianzi UniversityHangzhouChina
  2. 2.Lomonosov Moscow State University, Leninskie Gory, GSP-1MoscowRussia
  3. 3.Institute for Informatics Problems, Federal Research Center “Informatics and Control”MoscowRussia

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