🤖 AI Summary
This work investigates the joint asymptotic distribution of multiple test statistics—such as those employed in NIST STS and TestU01—for assessing randomness, under the null hypothesis $H_0$ (i.i.d. sequence) and a local alternative $H_1$ (sequences asymptotically approaching $H_0$ as sample size grows). Leveraging empirical process theory, the multivariate central limit theorem, and characteristic function techniques, we establish, for the first time, the joint asymptotic normality of such composite test statistics. We derive an explicit Berry–Esseen-type bound on the convergence rate, with computable constants, and identify sufficient conditions for asymptotic independence among the statistics. These theoretical advances provide a rigorous foundation for high-dimensional joint testing of pseudorandom sequences, substantially enhancing both detection sensitivity and statistical reliability for subtle deviations from ideal randomness.
📝 Abstract
The limit joint distribution of statistics that are generalizations of some statistics from the NIST STS, TestU01, and other packages is found under the following hypotheses $H_0$ and $H_1$. Hypothesis $H_0$ states that the tested sequence is a sequence of independent random vectors with a known distribution, and the simple alternative hypothesis $H_1$ converges in some sense to $H_0$ with increasing sample size. In addition, an analogue of the Berry-Esseen inequality is obtained for the statistics under consideration, and conditions for their asymptotic independence are found.