Spectral Analysis of Large Dimensional Random Matrices

  • Zhidong Bai
  • Jack W. Silverstein

Part of the Springer Series in Statistics book series (SSS)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Zhidong Bai, Jack W. Silverstein
    Pages 1-14
  3. Zhidong Bai, Jack W. Silverstein
    Pages 15-38
  4. Zhidong Bai, Jack W. Silverstein
    Pages 39-58
  5. Zhidong Bai, Jack W. Silverstein
    Pages 59-89
  6. Zhidong Bai, Jack W. Silverstein
    Pages 91-118
  7. Zhidong Bai, Jack W. Silverstein
    Pages 119-163
  8. Zhidong Bai, Jack W. Silverstein
    Pages 165-180
  9. Zhidong Bai, Jack W. Silverstein
    Pages 181-221
  10. Zhidong Bai, Jack W. Silverstein
    Pages 223-329
  11. Zhidong Bai, Jack W. Silverstein
    Pages 331-390
  12. Zhidong Bai, Jack W. Silverstein
    Pages 391-431
  13. Zhidong Bai, Jack W. Silverstein
    Pages 433-468
  14. Back Matter
    Pages 469-551

About this book

Introduction

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. The core of the book focuses on results established under moment conditions on random variables using probabilistic methods, and is thus easily applicable to statistics and other areas of science. The book introduces fundamental results, most of them investigated by the authors, such as the semicircular law of Wigner matrices, the Marcenko-Pastur law, the limiting spectral distribution of the multivariate F matrix, limits of extreme eigenvalues, spectrum separation theorems, convergence rates of empirical distributions, central limit theorems of linear spectral statistics, and the partial solution of the famous circular law. While deriving the main results, the book simultaneously emphasizes the ideas and methodologies of the fundamental mathematical tools, among them being: truncation techniques, matrix identities, moment convergence theorems, and the Stieltjes transform. Its treatment is especially fitting to the needs of mathematics and statistics graduate students and beginning researchers, having a basic knowledge of matrix theory and an understanding of probability theory at the graduate level, who desire to learn the concepts and tools in solving problems in this area. It can also serve as a detailed handbook on results of large dimensional random matrices for practical users.

This second edition includes two additional chapters, one on the authors' results on the limiting behavior of eigenvectors of sample covariance matrices, another on applications to wireless communications and finance. While attempting to bring this edition up-to-date on recent work, it also provides summaries of other areas which are typically considered part of the general field of random matrix theory.

Zhidong Bai is a professor of the School of Mathematics and Statistics at Northeast Normal University and Department of Statistics and Applied Probability at National University of Singapore. He is a Fellow of the Third World Academy of Sciences and a Fellow of the Institute of Mathematical Statistics.

Jack W. Silverstein is a professor in the Department of Mathematics at North Carolina State University. He is a Fellow of the Institute of Mathematical Statistics.

 

Keywords

Eigenvalue Eigenvector Fitting Matrix Matrix Theory Random variable

Authors and affiliations

  • Zhidong Bai
    • 1
  • Jack W. Silverstein
    • 2
  1. 1.Dept. Statistics & Applied ProbabilityNational University of SingaporeSingaporeSingapore
  2. 2.Dept. Mathematics, Biomathematics ProgramNorth Carolina State UniversityRaleighUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4419-0661-8
  • Copyright Information Springer Science+Business Media, LLC 2010
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4419-0660-1
  • Online ISBN 978-1-4419-0661-8
  • Series Print ISSN 0172-7397
  • About this book