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A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints

Author

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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, GSP-1, 1-52 Leninskiye Gory, Moscow 119991, Russia
    Federal Research Center “Informatics and Control” of the Russian Academy of Sciences, Moscow 119333, Russia
    Moscow Center for Fundamental and Applied Mathematics, Moscow 119991, Russia)

  • Mikhail Tselishchev

    (Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, GSP-1, 1-52 Leninskiye Gory, Moscow 119991, Russia
    Moscow Center for Fundamental and Applied Mathematics, Moscow 119991, Russia)

Abstract

We introduce a generalized stationary renewal distribution (also called the equilibrium transform) for arbitrary distributions with finite nonzero first moment and study its properties. In particular, we prove an optimal moment-type inequality for the Kantorovich distance between a distribution and its equilibrium transform. Using the introduced transform and Stein’s method, we investigate the rate of convergence in the Rényi theorem for the distributions of geometric sums of independent random variables with identical nonzero means and finite second moments without any constraints on their supports. We derive an upper bound for the Kantorovich distance between the normalized geometric random sum and the exponential distribution which has exact order of smallness as the expectation of the geometric number of summands tends to infinity. Moreover, we introduce the so-called asymptotically best constant and present its lower bound yielding the one for the Kantorovich distance under consideration. As a concluding remark, we provide an extension of the obtained estimates of the accuracy of the exponential approximation to non-geometric random sums of independent random variables with non-identical nonzero means.

Suggested Citation

  • Irina Shevtsova & Mikhail Tselishchev, 2020. "A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints," Mathematics, MDPI, vol. 8(4), pages 1-21, April.
  • Handle: RePEc:gam:jmathe:v:8:y:2020:i:4:p:577-:d:344886
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    Cited by:

    1. Victor Korolev, 2022. "Bounds for the Rate of Convergence in the Generalized Rényi Theorem," Mathematics, MDPI, vol. 10(22), pages 1-16, November.

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