  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.
