Random matrix theory: From mathematical physics to high dimensional statistics and time series analysis

Random matrix serves as one of the key tools in understanding the eigen-structure of large dimensional matrices. The application ranges from the estimation and inference of the high dimensional covariance matrices, the noise reduction of rectangular matrices to the understanding of separable matrices and even matrices having correlation both in rows and columns. Assuming that we can observe a p by n data matrix, where log p is comparable to log n, we derive the convergent limits and distributions for the eigenvalues and eigenvectors for a few random matrix models related to the above problems in the study of high dimensional statistics. This part is based on a few papers jointly with Zhigang Bao (HKUST), Fan Yang (UCLA) and Ke Wang (HKUST), where we employ the dynamic approach developed by Laszlo Erdos and Horng-Tzer Yau. Non-stationary time series is important in understanding the temporal correlation of data. Assuming that only one time series is observed, we develop a methodology to estimate the underlying high dimensional covariance and precision matrices. Based on our methodology, we can infer the covariance and precision matrices using the strategy of bootstrapping. This part is based on two papers jointly with Professor Zhou Zhou (UofT). It is notable that, we apply Stein's method to prove the Gaussian approximation, which is essentially the same as the Green function comparison strategy for proving the universality for random matrix models.Ph.D