The reference distributions of Maurer's universal statistical test and its improved tests

TL;DR

This paper provides a theoretical analysis of the reference distributions used in Maurer's universal statistical test and its improved variants, clarifying their statistical properties and convergence behaviors.

Contribution

It derives the variance of Yamamoto and Liu's test and proves the convergence of Coron's test distribution to a normal distribution, enhancing the theoretical understanding of these tests.

Findings

Derived the variance of Yamamoto and Liu's test

Proved the convergence of Coron's test distribution to normal

Applicable to other similar tests with minor modifications

Abstract

Maurer's universal statistical test can widely detect non-randomness of given sequences. Coron proposed an improved test, and further Yamamoto and Liu proposed a new test based on Coron's test. These tests use normal distributions as their reference distributions, but the soundness has not been theoretically discussed so far. Additionally, Yamamoto and Liu's test uses an experimental value as the variance of its reference distribution. In this paper, we theoretically derive the variance of the reference distribution of Yamamoto and Liu's test and prove that the true reference distribution of Coron's test converges to a normal distribution in some sense. We can apply the proof to the other tests with small changes.