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Sharp Estimates for Proximity of Geometric and Related Sums Distributions to Limit Laws

Author

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  • Alexander Bulinski

    (Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, Leninskie Gory 1, 119991 Moscow, Russia)

  • Nikolay Slepov

    (Department of Higher Mathematics, Moscow Institute of Physics and Technology, National Research University, 9 Instituskiy per., Dolgoprudny, 141701 Moscow, Russia)

Abstract

The convergence rate in the famous Rényi theorem is studied by means of the Stein method refinement. Namely, it is demonstrated that the new estimate of the convergence rate of the normalized geometric sums to exponential law involving the ideal probability metric of the second order is sharp. Some recent results concerning the convergence rates in Kolmogorov and Kantorovich metrics are extended as well. In contrast to many previous works, there are no assumptions that the summands of geometric sums are positive and have the same distribution. For the first time, an analogue of the Rényi theorem is established for the model of exchangeable random variables. Also within this model, a sharp estimate of convergence rate to a specified mixture of distributions is provided. The convergence rate of the appropriately normalized random sums of random summands to the generalized gamma distribution is estimated. Here, the number of summands follows the generalized negative binomial law. The sharp estimates of the proximity of random sums of random summands distributions to the limit law are established for independent summands and for the model of exchangeable ones. The inverse to the equilibrium transformation of the probability measures is introduced, and in this way a new approximation of the Pareto distributions by exponential laws is proposed. The integral probability metrics and the techniques of integration with respect to sign measures are essentially employed.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:24:p:4747-:d:1003032
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    References listed on IDEAS

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    1. Irina Shevtsova & Mikhail Tselishchev, 2020. "On the Accuracy of the Exponential Approximation to Random Sums of Alternating Random Variables," Mathematics, MDPI, vol. 8(11), pages 1-11, November.
    2. Alison L. Gibbs & Francis Edward Su, 2002. "On Choosing and Bounding Probability Metrics," International Statistical Review, International Statistical Institute, vol. 70(3), pages 419-435, December.
    3. 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).
    4. Gaunt, Robert E. & Walton, Neil, 2020. "Stein’s method for the single server queue in heavy traffic," Statistics & Probability Letters, Elsevier, vol. 156(C).
    5. Robert E. Gaunt, 2020. "Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I," Journal of Theoretical Probability, Springer, vol. 33(1), pages 465-505, March.
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    Cited by:

    1. Gerd Christoph & Vladimir V. Ulyanov, 2023. "Second Order Chebyshev–Edgeworth-Type Approximations for Statistics Based on Random Size Samples," Mathematics, MDPI, vol. 11(8), pages 1-18, April.
    2. Alexander N. Tikhomirov & Vladimir V. Ulyanov, 2023. "On the Special Issue “Limit Theorems of Probability Theory”," Mathematics, MDPI, vol. 11(17), pages 1-4, August.

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