A note on asymptotics of classical likelihood ratio tests for high-dimensional normal distributions.

For random samples of size n obtained from m -variate normal distributions, we investigate the classical likelihood ratio tests for their means and covariance matrices in the high-dimensional case. Many researchers analyze the test statistics for the case of n going to infinity and m keeping fixed. In the high-dimensional setting, the objective of this paper is to prove that the likelihood ratio test statistics converge in distribution to normal distributions when both m and n go to infinity with m / n → y ∈ (0 , 1 ]. We obtain this conclusion very intuitively by using Lyapunov's central limit theorem. [ABSTRACT FROM AUTHOR]