Uniform convergence rate estimates for the integral balance index

A. A. Kudryavtsev^ab, O. V. Shestakov<sup>abc

a</sup> Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, M. V. Lomo- nosov Moscow State University, 1-52 Leninskie Gory, GSP-1, Moscow 119991, Russian Federation
^b Moscow Center for Fundamental and Applied Mathematics, M. V. Lomonosov Moscow State University, 1 Leninskie Gory, GSP-1, Moscow 119991, Russian Federation
^c Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation

Abstract: The paper considers the Bayesian balance model in which the rate of weak convergence of the normalized integral balance index to the digamma distribution is studied in terms of a uniform metric. The integral factors negatively and positively influencing the functioning of the system and their ratio, the integral balance index of the system, are considered. It is assumed that the number of factors is not known a priori and is described by a mixed Poisson distribution with a structural generalized gamma distribution. The rate of weak convergence to the digamma distribution in the described scheme is studied. As an auxiliary statement, the rate of weak convergence of a normalized random sum with an index having a generalized negative binomial distribution to a limiting generalized gamma distribution is estimated. The results of the work may be in demand in the study of models used to describe processes with distributions having an unlimited nonnegative support.

Keywords: digamma distribution, mixed distributions, balance index, random summation, weak convergence, estimates of convergence rate.

Received: 15.01.2024

DOI: 10.14357/19922264240105