The limit joint distributions of some statistics used in testing the quality of random number generators

TL;DR

This paper derives the joint distribution limits of generalized test statistics for random number generators under specific hypotheses, providing theoretical insights into their asymptotic behavior and independence.

Contribution

It introduces new results on the joint distribution limits and asymptotic independence of test statistics used in assessing random number generators.

Findings

Limit joint distribution derived for generalized statistics

An analogue of the Berry-Esseen inequality established

Conditions for asymptotic independence identified

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.