The extreme eigenvalues of two types of random matrices

• Main Author: Zhang, Zhixiang
• Other Authors: Pan Guangming
• Format: Thesis-Doctor of Philosophy
• Language: English
• Published: Nanyang Technological University 2021
• Subjects: Science::Mathematics::Statistics
• Online Access: https://hdl.handle.net/10356/150804
• Institution: Nanyang Technological University

The fluctuations of extreme eigenvalues of a large random matrix model is a central topic in random matrix theory, motivated by applications in principle component analysis, factor analysis, or signal detection problems. This thesis establishes asymptotic distributions for the largest eigenvalues of two types of matrices under the high dimensional setting where the dimension goes to infinity proportionally with the sample size. The first type is the spiked sample covariance matrix. We prove that the spiked eigenvalues converge in distribution to Gaussian distributions without the conventional assumption of the block-diagonal structure on the population covariance matrices. We also show that the sample spiked eigenvalues and linear spectral statistics are asymptotically independent for such a spiked sample covariance model. With these theoretical results, we propose a statistic by combining the largest eigenvalues and the linear spectral statistics together to test the equality of two population covariance matrices. The second type is the signal-plus-noise matrix. To be more specific, let S = R + X where R is the signal matrix and X is the noise matrix contains i.i.d. standardized entries. The signal matrix R is allowed to be full rank, which is rarely studied in literature compared with the low rank cases. Under a regularity condition of R that assures the square root behaviour of the spectral density near the edge, we prove that the largest eigenvalue of SS* has Tracy-Widom distribution under a tail condition on the entries of X. Moreover, such a condition is proved to be necessary and sufficient to assure the Tracy-Widom law.