The largest eigenvalue of large random matrices and its application

• Main Author: Han, Xiao
• Other Authors: Pan Guangming
• Format: Theses and Dissertations
• Language: English
• Published: 2016
• Subjects: DRNTU::Science
• Online Access: http://hdl.handle.net/10356/69019
• Institution: Nanyang Technological University

This thesis is concerned about the asymptotic behavior of the largest eigenvalues for some random matrices and their applications for high-dimensional data analysis. The first type of random matrix is the F-type matrix. More precisely, let $\bbA_p=\frac{\bbY\bbY^*}{m}$ and $\bbB_p=\frac{\bbX\bbX^*}{n}$ be two independent random matrices, where $\bbX=(X_{ij})_{p \times n}$, $\bbY=(Y_{ij})_{p \times m}$ and all entries are real (or complex) independent random variables with $\mathbf{E}X_{ij}=\mathbf{E}Y_{ij}=0$, $\mathbf{E}|X_{ij}|^2=\mathbf{E}|Y_{ij}|^2=1$. Denote the largest root of the determinantal equation $\det(\lambda \bbA_p-\bbB_p)=0$ by $\lambda_{1}$, which can be considered as a general version of the largest eigenvalue of the F matrix. We establish the Tracy-Widom Law for $\lambda_{1}$ under some mild conditions when both $\frac{p}{m}$ and $\frac{p}{n}$ tend to some positive constants as $p\rightarrow\infty$. More applications are given based on this limiting distribution of $\lambda_1$. The second type of random matrix is the sample covariance-type matrix. This type of random matrix is proposed to deal with the high-dimensional change point detection problem. By modifying the classic sample covariance matrix and then developing the corresponding asymptotic behavior of its largest eigenvalue, without any estimators, this thesis suggests an optimization approach that can figure out both the unknown number of change points and multiple change point positions simultaneously. Moreover, the largest eigenvalue of this matrix type can also be extended to other popular high-dimensional testing problems. Except for the theoretical results established for each random matrix type, various simulation studies are also provided in order to support these conclusions and to show the effectiveness of the proposed statistics.