Estimation of the Remainder Term in the Expansion of the Stable Law at α → 0
DOI:
https://doi.org/10.52575/2687-0959-2026-58-1-5-21Keywords:
Stable Law, Expansion of Stable Law, Probability Density, Distribution Function, Asymptotic Expansion, Bell PolynomialsAbstract
The paper studies the expansion of a stable law in the case of α → 0. Asymptotic expansions of the distribution function and probability density, as well as estimates of the remainder terms of these expansions, are obtained. Based on the estimates of the remainder terms, a criterion is introduced that allows one to determine the range of values of the parameter α within which the relative approximation error will not exceed the specified accuracy level ε. The presented calculations confirm the validity of the expressions obtained.
Funding
The work was carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation (project No. 123020700216 (FEUF-2023-0004))
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Saenko V.V. The Calculation of the Density and Distribution Functions of Strictly Stable Laws. Mathematics. 2020;8(5):775. Available from: https://www.mdpi.com/2227-7390/8/5/775.
Nolan J.P. Numerical calculation of stable densities and distribution functions. Communications in Statistics Stochastic Models. 1997;13(4):759–774. Available from: http://www.tandfonline.com/doi/abs/10.1080/15326349708807450.
Золотарев В.М. Одномерные устойчивые распределения. Москва: Наука. Глав. ред. физ.-мат. лит.; 1983. 304 с.
Cressie N. A note on the behaviour of the stable distributions for small index α Probability Theory and Related Fields. 1975;33(1):61–64. Available from: http://www.springerlink.com/index/VW66M4303X6J8800.pdf http://www.springerlink.com/index/10.1007/BF00539862.
Uchaikin V.V., Zolotarev V.M. Chance and stability. Stable Distributions and their Applications. Utrecht: VSP; 1999.
Nolan J.P. Univariate Stable Distributions. Springer Series in Operations Research and Financial Engineering. Cham: Springer International Publishing; 2020. Available from: http://link.springer.com/10.1007/978-3-030-52915-4.
Золотарев В.М. Преобразования Меллина – Стильтьеса в теории вероятностей. Теория вероятности и ее применения. 1957;2(4):444–469. Available from: http://mi.mathnet.ru/tvp4977.
Золотарев В.М.Общая теория перемножения независимых случайных величин.ДокладыАНСССР. 1962;142(4):788–791.
Градштейн И.С., Рыжик И.М. Таблицы интгералов, сумм, рядов и произведений. М.: Физматгиз; 1963. 1100 с.
Риордан Дж. Введение в комбинаторный анализ. Москва: Издательство иностранной литературы; 1963. 288 с.
Comtet L. Advanced Combinatorics. The Art of Finite and Infinite Expansions. D. Reidel Publishing Company; 1974.
Абрамовиц М., Стиган И. Справочник по специальным функциям. М.,1979: Наука. Глав. ред. физ.-мат. лит.; 1979. 832 с.
References
Saenko VV. The Calculation of the Density and Distribution Functions of Strictly Stable Laws. Mathematics. 2020;8(5):775. Available from: https://www.mdpi.com/2227-7390/8/5/775.
Nolan JP. Numerical calculation of stable densities and distribution functions. Communications in Statistics Stochastic Models. 1997;13(4):759–774. Available from: http://www.tandfonline.com/doi/abs/10.1080/15326349708807450.
Zolotarev VM., One-dimensional stable Distributions, Amer. Mat. Soc., Providence, RI, 1986.
Cressie N. A note on the behaviour of the stable distributions for small index α Probability Theory and Related Fields. 1975;33(1):61–64. Available from: http://www.springerlink.com/index/VW66M4303X6J8800.pdf http://www.springerlink.com/index/10.1007/BF00539862.
Uchaikin VV., Zolotarev VM. Chance and stability. Stable Distributions and their Applications. Utrecht: VSP; 1999.
Nolan JP. Univariate Stable Distributions. Springer Series in Operations Research and Financial Engineering. Cham: Springer International Publishing; 2020. Available from: http://link.springer.com/10.1007/978-3-030-52915-4.
Zolotarev VM. Mellin-Stieltjes Transforms in Probability Theory. Teoriya veroyatnostej i eyo primeneniya [Theory of Probability & Its Applications] 1957;2(4):433–460. Available from: http://mi.mathnet.ru/tvp4977.
Zolotarev VM. Obshchaya teoriya peremnozheniya nezavisimykh sluchainykh velichin [General theory of multiplication of independent random variables]. Doklady AN SSSR. 1962;142(4):788–791.
Gradshtein IS., Ryzhik IM. Tables Of Integrals, Series And Products. Academic Press; 2007.
Riordan J. An Introduction to Combinatorial Analysis. John Wiley & Sons, Inc, New York 1958.
Comtet L. Advanced Combinatorics. The Art of Finite and Infinite Expansions. D. Reidel Publishing Company; 1974.
Abramowitz M., Stegun IA. eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55. Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards. 1983.
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