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On the Accuracy of the Generalized Gamma Approximation to Generalized Negative Binomial Random Sums

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  • Irina Shevtsova

    (Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
    Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
    Federal Research Center ”Computer Science and Control” of the Russian Academy of Sciences, 119333 Moscow, Russia
    Moscow Center for Fundamental and Applied Mathematics, 119991 Moscow, Russia)

  • Mikhail Tselishchev

    (Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia)

Abstract

We investigate the proximity in terms of zeta-structured metrics of generalized negative binomial random sums to generalized gamma distribution with the corresponding parameters, extending thus the zeta-structured estimates of the rate of convergence in the Rényi theorem. In particular, we derive upper bounds for the Kantorovich and the Kolmogorov metrics in the law of large numbers for negative binomial random sums of i.i.d. random variables with nonzero first moments and finite second moments. Our method is based on the representation of the generalized negative binomial distribution with the shape and exponent power parameters no greater than one as a mixed geometric law and the infinite divisibility of the negative binomial distribution.

Suggested Citation

  • Irina Shevtsova & Mikhail Tselishchev, 2021. "On the Accuracy of the Generalized Gamma Approximation to Generalized Negative Binomial Random Sums," Mathematics, MDPI, vol. 9(13), pages 1-8, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:13:p:1571-:d:588188
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