大维随机矩阵的谱分析

出版时间：2010-4  出版社：科学出版社  作者：Zhidong Bai Jack W. Silverstein 著  页数：551  

前言

The ongoing developments being made in large dimensional data analysis continue to generate great interest in random matrix theory in both theoret- ical investigations and applications in many disciplines. This has doubtlessly contributed to the significant demand for this monograph, resulting in its first printing being sold out. The authors have received many requests to publish a second edition of the book.   Since the publication of the first edition in 2006, many new results have been reported in the literature. However, due to limitations in space, we cannot include all new achievements in the second edition. In accordance with the needs of statistics and signal processing, we have added a new chapter on the limiting behavior of eigenvectors of large dimensional sample covariance matrices. To illustrate the application of RMT to wireless communications and statistical finance, we have added a chapter on these areas. Certain new developments are commented on throughout the book. Some typos and errors found in the first edition have been corrected.The authors would like to express their appreciation to Ms. Lii Hong for her help in the preparation of the second edition. They would also like to thank Professors Ying-Chang Liang, Zhaoben Fang, Baoxue Zhang, and Shurong Zheng, and Mr. Jiang Hu, for their valuable comments and suggestions. They also thank the copy editor, Mr. Hal Heinglein, for his careful reading, cor- rections, and helpful suggestions. The first author would like to acknowledge the support from grants NSFC 10871036, NUS R-155-000-079-112, and R- 155-000-096-720.

内容概要

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 Mareenko-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 f'mance. 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.

作者简介

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.

书籍目录

Preface to the Second Edition Preface to the First Edition 1 Introduction 　1.1 Large Dimensional Data Analysis 　1.2 Random Matrix Theory 　1.3 Methodologies 2 Wigner Matrices and Semicircular Law 　2.1 Semicircular Law by the Moment Method 　2.2 Generalizations to the Non-iid Case 　2.3 Semicircular Law by Stieltjes Transform 3 Sample Covariance Matrices and the Marcenko-Pastur Law 　3.1 M-P Law for the iid Case 　3.2 Generalization to the Non-iid Case 　3.3 Proof of Theorem 3.10 by the Stieltjes Transform 4 Product of Two Random Matrices 　4.1 Main:Results 　4.2 Some Graph Theory and Combinatorial Results 　4.3 Proof df'Theorem 4.1 　4.4 LSD of the F-Matrix 　4.5 Proof of TheoremS4:3 5 Limits of Extreme Eigenvalues 　5.1 Limit of Extreme Eigenvalues of the Wigner Matrix 　5.2 Limits,of Extreme Eigenvalues of the Sample Covariance Matrix 　5.3 Miscellanies 6 Spectrum Separation 　6.1 What Is Spectrum Separation? 　6.2 Proof of(1) 　6.3 Proof of(2) 　6.4 Proof of(3) 7 Semicircular Law for Hadamsrd Products 　7.1 Sparse Matrix and Hadamard Product 　7.2 Truncation and Normalization 　7.3 Proof of Theorem 7.1 by the Moment Approach 8 Convergence Rates of ESD 　8.1 Convergence Rates of the Expected ESD of Wigner Matrices 　8.2 Further Extensions 　8.3 Convergence Rates of the Expected ESD of Sample Covariance Matrices 　8.4 Some Elementary Calculus 　8.5 Rates of Convergence in Probability and Almost Surely9 CLT for Linear Spectral Statistics 　9.1 Motivation and Strategy 　9.2 CLT of LSS for the Wigner Matrix 　9.3 Convergence of the Process Mn-EMn 　9.4 Computation of tim Mean and Covauce Function of G(f) 　9.5 Application to Linear Spectral Statistics and Related Results 　9.6 Technical Lemmas 　9.7 CLT of the LSS for Sample Covariance Matrices 　9.8 Convergence of Stieltjes Transforms 　9.9 Convergence of Finite-Dimensional Distributions 　9.10 Tightness of Mi(z) 　9.11 Convergence of Mn2(Z) 　9.12 Some Derivations and Calculations 　9.13 CLT for the F-Matrix 　9.14 Proof of Theorem 9.14 　9.15 CLT for the LSS of a Large Dimensional Beta-Matrix 　9.16 Some Examples 　10 Eigenvectors of Sample Covariance Matrices 　10.1 Formulation and Conjectures 　10.2 A Necessary Condition for Property 5' 　10.3 Moments of Xp(Fsp) 　10.4 An Example of Weak Convergence 　10.5 Extension of (10.2.6) to Bn= T1/2SpT1/2 　10.6 Proof of Theorem 10.16 　10.7 Proof of Theorem 10.21 　10.8 Proof of Theorem 10.23 11 Circular Law 　11.1 The Problem and Difficulty 　11.2 A Theorem Establishing a Partial Answer to the Circular Law 　11.3 Lemmas on Integral Range Reduction 　11.4 Characterization of the Circular Law 　11.5 A Rough Rate on the Convergence of vn(x, z) 　11.6 Proofs of (11.2.3) and (11.2.4) 　11.7 Proof of Theorem 11.4 　11.8 Comments and Extensions 　11.9 Some Elementary Mathematics 　11.10 New Developments 12 Some Applications of RMT 　12.1 Wireless Communications 　12.2 ADDlication to Finance A Some Results in Linear Algebra 　A.1 Inverse Matrices and Resolvent 　A.2 Inequalities Involving Spectral Distributions 　A.3 Hadamard Product and Odot Product 　A.4 Extensions of Singular-Value Inequalities 　A.5 Perturbation Inequalities 　A.6 Rank Inequalities 　A.7 A Norm Inequality B Miscellanies 　B.1 Moment Convergence Theorem 　B.2 Stieltjes Transform 　B.3 Some Lemmas about Integrals of Stieltjes Transforms 　B.4 A Lemma on the Strong Law of Large Numbers 　B.5 A Lemma on Quadratic Forms Relevant Literature Index

章节摘录

插图：

编辑推荐

《大维随机矩阵的谱分析(英文版)(第2版)》是由科学出版社出版的。

用户评论 (总计2条)

•   白老师在随机矩阵和高维统计方面做了很多基础性的工作，这本书基本总结了随机矩阵近几十年的发展，主要是Bai以及其合作者的工作。特别强调了矩方法，Stiljies变换方法等
  ，内容很好，值得细细研读！
•   大维随机矩阵的谱分析，书的质量不错