Limit joint distribution of «Approximate $\phi$-entropy test» statistics

Consider a sequence of independent polynomial trials with $s$ outcomes. «Approximate Entropy Test» is one of the tools used to test the hypothesis of the equiprobability of outcomes. In 2000 A.L. Rukhin proposed a more general «Approximate $\phi$-Entropy Test» and found the limiting distribution of its statistic $T^{\phi}$. In the present paper a generalization of this result is obtained in a simpler way. In addition, the limiting joint distribution $(T^{\phi_1}, \ldots, T^{\phi_r}$) is obtained in the situation with equiprobable outcomes. As a corollary, in the case $s=2$ the limiting joint distribution of the statistics of two NIST package tests «Approximate Entropy Test» and «Serial Test» is found for a sequence of independent Bernoulli trials with parameter $\frac12$.