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Sharp Estimates for Proximity of Geometric and Related Sums Distributions to Limit Lawsстатья
Статья опубликована в высокорейтинговом журнале
Статья опубликована в журнале из списка Web of Science и/или Scopus- Авторы: Bulinski A., Slepov N.
- Журнал: Mathematics
- Том: 10
- Год издания: 2022
- Издательство: MDPI
- Местоположение издательства: Basel, Switzerland
- Первая страница: 1
- Последняя страница: 37
- Номер статьи: 4747
- DOI: 10.3390/math10244747
- Аннотация: The convergence rate in the famous Rényi theorem is studied by means of the Steinmethod refinement. Namely, it is demonstrated that the new estimate of the convergence rate of thenormalized geometric sums to exponential law involving the ideal probability metric of the secondorder is sharp. Some recent results concerning the convergence rates in Kolmogorov and Kantorovichmetrics are extended as well. In contrast to many previous works, there are no assumptions thatthe 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 isprovided. The convergence rate of the appropriately normalized random sums of random summandsto the generalized gamma distribution is estimated. Here, the number of summands follows thegeneralized negative binomial law. The sharp estimates of the proximity of random sums of randomsummands distributions to the limit law are established for independent summands and for the modelof exchangeable ones. The inverse to the equilibrium transformation of the probability measures isintroduced, and in this way a new approximation of the Pareto distributions by exponential lawsis proposed. The integral probability metrics and the techniques of integration with respect to signmeasures are essentially employed.
- Добавил в систему: Булинский Александр Вадимович