Spectral Analysis of Large Dimensional Random Matrices

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.

List of contents

Introduction.- Wigner matrices and semicircular law.- Sample covariance matrices and the Marcenko-Pastur law.- Product of two random matrices.- Limits of extreme eigenvalues.- Spectrum separation.- Semicircle law for Hadamard products.- Convergence rates of ESD.- CLT for linear spectral statistics.- Eigenvectors of sample covariance matrices.- Circular law.- Some applications of RMT.

Summary

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.

Additional text

From the reviews of the second edition:
“This book provides an excellent introduction to basic results and methodologies developed on large-dimensional random matrices. The targeted readers are graduate students and researchers who are interested in spectral aspects of RMT and large-dimensional data analysis. The book can also serve as a reference text for practical users. … The book is organized and written very well, with a wide collection of useful historical notes and references. … In summary, the book is going to be a classic in the field of RMT.”­­­ (Wenbo V. Li, Mathematical Reviews, Issue 2011 d)
“The aim and scope of this edition is to provide upper-level undergraduate students, graduate students, and research workers the understandings and working knowledge of spectral analysis of large-dimensional random matrices … . it deepens the understanding of applications of random matrices and its applications in finance and engineering. … an important contribution, providing up-to-date coverage on the general field of random matrix theory in a systematic and logical manner. … Both graduate students and researchers in this area will find this book handy and helpful.” (Technometrics, Vol. 54 (1), February, 2012)

Report

From the reviews of the second edition:
"This book provides an excellent introduction to basic results and methodologies developed on large-dimensional random matrices. The targeted readers are graduate students and researchers who are interested in spectral aspects of RMT and large-dimensional data analysis. The book can also serve as a reference text for practical users. ... The book is organized and written very well, with a wide collection of useful historical notes and references. ... In summary, the book is going to be a classic in the field of RMT." (Wenbo V. Li, Mathematical Reviews, Issue 2011 d)
"The aim and scope of this edition is to provide upper-level undergraduate students, graduate students, and research workers the understandings and working knowledge of spectral analysis of large-dimensional random matrices ... . it deepens the understanding of applications of random matrices and its applications in finance and engineering. ... an important contribution, providing up-to-date coverage on the general field of random matrix theory in a systematic and logical manner. ... Both graduate students and researchers in this area will find this book handy and helpful." (Technometrics, Vol. 54 (1), February, 2012)

Product details

• Authors: Zhidon Bai, Zhidong Bai, Jack W Silverstein, Jack W. Silverstein
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 14.01.2010
• EAN: 9781441906601
• ISBN: 978-1-4419-0660-1
• No. of pages: 552
• Dimensions: 164 mm x 34 mm x 243 mm
• Weight: 958 g
• Illustrations: XVI, 552 p.
• Series: Springer Series in Statistics; Springer Texts in Statistics; Springer Texts in Statistics; Springer Series in Statistics; Springer Statistics
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics