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 Teor. Veroyatnost. i Primenen., 2018, Volume 63, Issue 1, Pages 89–116 (Mi tvp5143)

Convergence rate estimates in the global CLT for compound mixed Poisson distributions

I. G. Shevtsovaabc

a Zhejiang Sci-tech University
b Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, Moscow
c Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Abstract: Using the estimates of the accuracy of the normal approximation to distributions of Poisson-binomial random sums from [I. G. Shevtsova, Theory Probab. Appl., 62 (2018), pp. 278–294], we obtain moment-type estimates of the rate of convergence in the central limit theorem for Poisson and mixed Poisson random sums in the uniform and mean metrics. As corollaries, we provide estimates of the accuracy of the approximation to distributions of negative binomial random sums by the normal law (with the growth of the shape parameter) and by the variance-gamma mixture of the normal law (as the “success probability” tends to zero); in particular, we present estimates of the accuracy of the Laplace approximation to distributions of geometric random sums.

Keywords: Poisson random sum, geometric random sum, compound Poisson distribution, central limit theorem (CLT), convergence rate estimate, normal approximation, Laplace distribution, Berry–Esseen inequality, asymptotically exact constant.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation ÌÄ-2116.2017.1 Russian Foundation for Basic Research 15-07-02984-à16-31-60110-ìîë_à_äê The work was supported by the Grant of the President of Russia (project MD-2116.2017.1) and by the Russian Foundation for Basic Research (projects 15-07-02984-a and 16-31-60110-mol_a_dk).

DOI: https://doi.org/10.4213/tvp5143

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English version:
Theory of Probability and its Applications, 2018, 63:1, 72–93

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Accepted:22.06.2017

Citation: I. G. Shevtsova, “Convergence rate estimates in the global CLT for compound mixed Poisson distributions”, Teor. Veroyatnost. i Primenen., 63:1 (2018), 89–116; Theory Probab. Appl., 63:1 (2018), 72–93

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp5143
• https://doi.org/10.4213/tvp5143
• http://mi.mathnet.ru/eng/tvp/v63/i1/p89

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This publication is cited in the following articles:
1. Oliveira G. Barreto-Souza W. Silva R.W.C., “Convergence and Inference For Mixed Poisson Random Sums”, Metrika
2. N. A. Slepov, “Skorost skhodimosti raspredelenii geometricheskikh summ k zakonu Laplasa”, Teoriya veroyatn. i ee primen., 66:1 (2021), 149–174
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