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Bounds for convergence rate in laws of large numbers for mixed Poisson random sums

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  • Korolev, Victor
  • Zeifman, Alexander

Abstract

In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev ζ-metric is used. The obtained results extend the known convergence rate estimates for geometric random sums (in the famous Rényi theorem) to a considerably wider class of random indices with mixed Poisson distributions including, e.g., those with the (generalized) negative binomial distribution.

Suggested Citation

  • Korolev, Victor & Zeifman, Alexander, 2021. "Bounds for convergence rate in laws of large numbers for mixed Poisson random sums," Statistics & Probability Letters, Elsevier, vol. 168(C).
    • Handle: RePEc:eee:stapro:v:168:y:2021:i:c:s0167715220302212
    DOI: 10.1016/j.spl.2020.108918
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    References listed on IDEAS

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    1. Willmot, Gord, 1986. "Mixed Compound Poisson Distributions," ASTIN Bulletin, Cambridge University Press, vol. 16(S1), pages 59-79, April.
    2. Karlis, Dimitris, 2005. "EM Algorithm for Mixed Poisson and Other Discrete Distributions," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 3-24, May.
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    5. Grandell, Jan, 2000. "Simple approximations of ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 26(2-3), pages 157-173, May.
    6. Korolev, V.Yu. & Chertok, A.V. & Korchagin, A.Yu. & Zeifman, A.I., 2015. "Modeling high-frequency order flow imbalance by functional limit theorems for two-sided risk processes," Applied Mathematics and Computation, Elsevier, vol. 253(C), pages 224-241.
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    Cited by:

    1. Luca Pratelli & Pietro Rigo, 2021. "Convergence in Total Variation of Random Sums," Mathematics, MDPI, vol. 9(2), pages 1-11, January.
    2. Leonid Hanin & Lyudmila Pavlova, 2023. "A Rényi-Type Limit Theorem on Random Sums and the Accuracy of Likelihood-Based Classification of Random Sequences with Application to Genomics," Mathematics, MDPI, vol. 11(20), pages 1-19, October.
    3. Alexander Bulinski & Nikolay Slepov, 2022. "Sharp Estimates for Proximity of Geometric and Related Sums Distributions to Limit Laws," Mathematics, MDPI, vol. 10(24), pages 1-37, December.
    4. 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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