高维随机矩阵的谱理论及其在无线通信和经济中的应用

Preface of Alumnis Serials

Preface

1 Introduction

1.1 History of RMT and Current Development

1.1.1 A Brief Review of RMT

1.1.2 Spectral Analysis of Large Dimensional Random Matrices

1.1.3 Limits of Extreme Eigenvalues

1.1.4 Convergence Rate of ESD

1.1.5 Circular Law

1.1.6 Central Limit Theory （CLT） of Linear Spectral Statistics

1.1.7 Limiting Distributions of Extreme Eigenvalues and Spacings

1.2 Applications to Wireless Communications

1.3 Applications to Finance Statistics

2 Limiting Spectral Distributions

2.1 Semi-circular Law

2.1.1 The lid Case

2.1.2 Independent but not Identically Distributed

2.2 Marcenko-Pastur Law

2.2.1 MP Law for lid Case

2.2.2 Generalization to the Non-lid Case

2.2.3 Proof of Theorem 2.11 by Stieltjes Transform

2.3 LSD of Products

2.3.1 Existence of the ESD of SnTn

2.3.2 Truncation of the ESD of Tn

2.3.3 Truncation， Centralization and Rescaling of the X-variables

2.3.4 Sketch of the Proof of Theorem 2.12

2.3.5 LSD of F Matrix

2.3.6 Sketch of the Proof of Theorem 2.14

2.3.7 When T is a Wigner Matrix

2.4 Hadamard Product 4

2.4.1 Truncation and Centralization

2.4.2 Outlines of Proof of the theorem

2.5 Circular Law

2.5.1 Failure of Techniques Dealing with Hermitian Matrices

2.5.2 Revisit of Stieltjes Transformation

2.5.3 A Partial Answer to the Circular Law

2.5.4 Comments and Extensions of Theorem 2.33

3 Extreme Eigenvalues

3.1 Wigner Matrix

3.2 Sample Covariance Matrix

3.2.1 Spectral Radius

3.3 Spectrum Separation

3.4 Tracy-Widom Law

3.4.1 TW Law for Wigner Matrix

3.4.2 TW Law for Sample Covariance Matrix

4 CLT of LSS

4.1 Motivation and Strategy

4.2 CLT of LSS for Wigner Matrix

4.2.1 Outlines of the Proof

4.3 CLT of LSS for Sample Covariance Matrices

4.4 F Matrix

4.4.1 Decomposition of Xnf

4.4.2 Limiting Distribution of X+nf

4.4.3 Limiting Distribution of Xnf

5 Limiting Behavior of Eigenmatrix of Sample Covariance Matrix

5.1 Earlier Work by Silverstein

5.2 Further Extension of Silversteins Work

5.3 Projecting the Eigenmatrix to a d-Dimensional Space

5.3.1 Main Results

5.3.2 Sketch of Proof of Theorem 5.19

5.3.3 Proof of Corollary 5.23

6 Applications to Wireless Communications

6.1 Introduction

6.2 Channel Models.

6.2.1 Basics of Wireless Communication Systems

……

7 Limiting Performances of Linear and Iterative Receivers

8 Applications to Finace Statistics

References

Index

本书讲述了随机矩阵谱理论的主要结果和前瞻研究，以及它在无线通信和现代金融风险理论中的应用。书中前面讲解基本知识，后面分析重要范例，全面介绍了随机矩阵谱理论在这两个领域中的成果。本书对其他需要高维数据分析的领域，能起到示范作用。本书可作为统计学、计算机科学、现代物理、量子力学、无线通信、金融工程、经济学等领域本科生、研究生和工程技术人员学习随机矩阵理论的重要参考资料。