Asymptotic independence of statistics of tests of the NIST package and their generalizations

Let the hypothesis $H_0$ be that the tested sequence is a sequence of independent random variables with a known polynomial distribution, and let the simple alternative hypothesis $H_1$ correspond to the scheme of series in which the distribution of the tested sequence approaches its distribution under $H_0$. Necessary and sufficient conditions are obtained for the asymptotic independence of statistics that are generalizations of the NIST and other packages' test statistics under given hypotheses $H_0$ and $H_1$. In the particular case where $H_0$ corresponds to a sequence of independent Bernoulli trials with parameter $\frac12$ and where $H_1$ approaches $H_0$, a test for the asymptotic independence of multivariate statistics is obtained, the components of which are the statistics of the following nine tests of the NIST STS: «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 the hypotheses $H_0$ and $H_1$.