On a randomized approach to the construction of tests for the equiprobability of outcomes of a multinomial scheme

We consider the most powerful test and the $\chi^2$ test for verifying equiprobability of the outcomes of multinomial scheme in the case where the alternative distribution is obtained by averaging all non-equiprobable multinomial distributions with $N$ outcomes over the symmetric $(N-1)$-dimensional Dirichlet distribution. Limit theorems on the convergence of the distributions of the test statistics under both of the hypotheses are proved.