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Maurer's universal statistical test can widely detect non-randomness of givensequences. Coron proposed an improved test, and further Yamamoto and Liuproposed a new test based on Coron's test. These tests use normal distributionsas their reference distributions, but the soundness has not been theoreticallydiscussed so far. Additionally, Yamamoto and Liu's test uses an experimentalvalue as the variance of its reference distribution. In this paper, wetheoretically derive the variance of the reference distribution of Yamamoto andLiu's test and prove that the true reference distribution of Coron's testconverges to a normal distribution in some sense. We can apply the proof to theother tests with small changes.

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