  Hotelling's T-squared test is a classical tool to test if the normal mean of
a multivariate normal distribution is a specified one or the means of two
multivariate normal means are equal. When the population dimension is higher
than the sample size, the test is no longer applicable. Under this situation,
in this paper we revisit the tests proposed by Srivastava and Du (2008), who
revise the Hotelling's statistics by replacing Wishart matrices with their
diagonal matrices. They show the revised statistics are asymptotically normal.
We use the random matrix theory to examine their statistics again and find that
their discovery is just part of the big picture. In fact, we prove that their
statistics, decided by the Euclidean norm of the population correlation matrix,
can go to normal, mixing chi-squared distributions and a convolution of both.
Examples are provided to show the phase transition phenomenon between the
normal and mixing chi-squared distributions. The second contribution of ours is
a rigorous derivation of an asymptotic ratio-unbiased-estimator of the squared
Euclidean norm of the correlation matrix.
