Abstract
We consider random vectors that consist of numbers of colored balls belonging to a fixed value set in a multi-color urn scheme without return. We proved, that under certain conditions, random vectors, consisting of centered and normed elements of these vectors, converge in distribution to a random vector made up of independent Gaussian random variables with means 0 and variances of 1. We also obtained limit theorems for the functions of these random vectors. Applications of these theorems to estimate probabilities of type I errors of the \(\chi^{2}\)-test and to estimate probabilities of type I errors and type II errors of some statistical tests are given.
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REFERENCES
V. F. Kolchin, Random Graphs (Cambridge Univ. Press, Cambridge, 1999).
V. F. Kolchin, ‘‘A certain class of limit theorems for conditional distributions,’’ Litovsk. Math. Sb. 8, 53–63 (1968).
I. A. Ibragimov and Y. V. Linnik, ‘‘Independent and stationary connected random variables,’’ Math. Rev. 48, 1287–1330 (1971).
D. E. Chikrin, A. N. Chuprunov, and P. A. Kokunin, ‘‘Limit theorems for a number of particles from a fixed set of cells,’’ Lobachevskii J. Math. 40 (5), 496–501 (2019).
A. Rényi, Probability Theory (Elsevier, New York, 1970).
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(Submitted by A. I. Volodin)
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Abdushukurov, F.A., Chuprunov, A.N. On the Number of Colored Balls from a Fixed Set in a Multi-Color Urn Scheme without Return. Lobachevskii J Math 42, 280–286 (2021). https://doi.org/10.1134/S1995080221020037
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DOI: https://doi.org/10.1134/S1995080221020037