On the Number of Colored Balls from a Fixed Set in a Multi-Color Urn Scheme without Return

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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