Limit joint distribution of the statistics of «Monobit test», «Frequency Test within a Block» and «Test for the Longest Run of Ones in a Block»

For a sequence consisting of independent random variables having a Bernoulli distribution with the parameter $p = \frac12$ the limit joint distribution of the statistics $T_1, T_2, T_3$ of the following three tests of the NIST package is obtained: «Monobit Test», «Frequency Test within a Block» and «Test for the Longest Run of Ones in a Block». It is proved that the covariance matrix $C$ of the limit distribution of the vector $(T_1, T_2, T_3)$ satisfies the relations $C_{12}=C_{21}=C_{13}=C_{31}=0$, $C_{23}=C_{32} \ge 0$. For arbitrary $p$ necessary and sufficient conditions for asymptotic uncorrelatedness and\slash or asymptotic independence of these statistics are obtained. The limit behavior of the vector $(T_1, T_2, T_3)$ is described for a wide class of values $p \neq \frac12$.