The limit joint distributions of statistics of tests of the NIST package and their generalizations

The limiting joint distribution of statistics, that are generalizations of statistics of tests of the NIST package and other packages, is obtained under the following hypotheses $H_0$ and $H_1$. The hypothesis $H_0$ is that the test sequence consists of independent random variables with a given polynomial distribution, and the alternative hypothesis $H_1$ corresponds to a scheme of trials in which the distribution of the test sequence approaches its distribution at $H_0$. An example of the hypothesis $H_1$ is the Markov alternative of a special form. In the special case when $H_0$ corresponds to a sequence of independent Bernoulli trials with parameter $\frac12$ and when $H_1$ approaches $H_0$, the results obtained allow us to find the limiting joint distributions of statistics of the following nine tests of the NIST package: «Monobit Test» , «Frequency Test within a Block», «Runs Test», «Test for the Longest Run of Ones in a Block», «Binary Matrix Rank Test», «Non-overlapping Template Matching Test », «Linear Complexity Test», «Serial Test» and «Approximate Entropy Test», as well as their generalizations, under hypotheses $H_0$ and $H_1$.