On the power of tests of multidimensional discrete uniformity used for statistical analysis of random number generators

In this paper, we obtained the asymptotic power values for the statistical tests of multidimensional discrete uniformity under conditions of contiguous convergence of alternatives. Two versions of the test are considered, namely, with overlapping blocks (included in the $NIST ~SP ~800-22$ test suit) and with non-overlapping blocks. The null hypothesis $H_{0}$ is related to the so-called pure randomness of the observed sequence, i. e. independence and the same uniform distribution of its elements. An alternative $H_{1}$ is assumed to be a Markov chain of some arbitrary fixed finite order.