Spectral Analysis of Large Dimensional Random Matrices-Mathematics Monograph Series 2

The aim of the book is to introduce basic concepts, main results, and widely applied mathematical tools in the spectral analysis of large dimensional random matrices. In it we will introduce many of the fundamental results, such as the semicircular law of Wigner matrices, the Marchenko-Pastur law, the limiting spectral distribution of the multivariate F matrix, limits of extremal eigenvalues, spectrum separation theorems, convergence rates of empirical spectral distributions, central limit theorems of linear spectral statistics and the partial solution of the famous circular law. While deriving the main results, the book will simultaneously emphasize the ideas and methodologies of the fundamental mathematical tools, among them being, truncation techniques, matrix transformations, moment convergence theorems, and the Stieltjes transform. Thus, its treatment is especially fitting to the needs of mathematics and statistic graduate students, and beginning researchers, who can learn the basic methodologies and ideas to solve problems in this area. It may also serve as a detailed handbook on results of large dimensional random matrices for practical users.
CONTENTS
1. Introduction
2. Wigner Matrices and Semicircular Law
3. Sample Covariance Matrices, Marcenko-Pastur Law
4. Product of Two Random Matrices
5. Limits of Extreme Eigenvalues
6. Spectrum Sepration
7. Semicircle Law for Hadamard Products
8. Convergence Rates of ESD
9. CLTfor Linear Spectral Statistics
10. Circular Law
11. Appendix A. Some Results in Linear Algebra
12. Appendix B. Moment Convergence Theorem and Stielfjes Transform
References
Index