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

Alexander Bulinski () and Nikolay Slepov
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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

Mathematics, 2022, vol. 10, issue 24, 1-37

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.

Keywords: probability metrics; Stein method; geometric sums; generalization of the Rényi theorem; generalized transformation of equilibrium for probability measures and its inverse; generalized gamma distribution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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