Central limit theorem for the spiked eigenvalues of separable sample covariance matrices - Zhang, Bo (2017)
This thesis is concerned about the central limit theorems for the spiked eigenvalues of separable sample covariance matrices and their applications. The first problem is to test a p-dimensional time series model with unit root. We establish both the convergence in probability and the asymptotic joint distribution of the first k largest eigenvalues of separable sample covariance matrices. Then we give two new unit root tests for high-dimensional time series as applications. We also provide some simulation results about the two tests. Then we extend our theoretical results to the more general case. We study the separable sample covariance matrix with two different kinds of population covariance matrices and each of them has some extremely large eigenvalues. We prove the central limit theorems of the largest eigenvalues for the two cases and give two examples in time series data.