Asymptotic Distributions of Functions of the Eigenvalues of the Sample Covariance Matrix and Canonical Correlation Matrix in Multivariate Time Series

摘要

This paper discusses the asymptotic distributions of eigenvalues of sample covariance matrices of multivariate time series since the eigenvalues play a fundamental role in multivariate problems. Section 2 gives the limiting distribution of eigenvalues of sample covariance matrices for non-Gaussian linear vector processes. Further Section 3, derives the asymptotic expansions of certain functions of eigenvalues of covariance matrix for multivariate Gaussian stationary processes, and discuss their applications for time series principal component analysis. In Section 4 the author gives the asymptotic expansions of certain functions of canonical correlation matrix for multivariate Gaussian stationary processes, and discusses some asymptotic properties of a test statistic for canonical correlations.