Asymptotic distributions of functions of the eigenvalues of sample covariance matrix and canonical correlation matrix in multivariate time series

Let S = (1/n) [Sigma]t=1n X(t) X(t)', where X(1), ..., X(n) are p - 1 random vectors with mean zero. When X(t) (t = 1, ..., n) are independently and identically distributed (i.i.d.) as multivariate normal with mean vector 0 and covariance matrix [Sigma], many authors have investigated the asymptotic expansions for the distributions of various functions of the eigenvalues of S. In this paper, we will extend the above results to the case when {X(t)} is a Gaussian stationary process. Also we shall derive the asymptotic expansions for certain functions of the sample canonical correlations in multivariate time series. Applications of some of the results in signal processing are also discussed.