大维随机矩阵的谱分析

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 dfTheorem 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 Surely

9 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

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 fmance. 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.