Table of Contents
1 Introduction 1
1.1 History of RMT and Current Development 1
1.1.1 A brief review of RMT 2
1.1.2 Spectral Analysis of Large Dimensional Random Matrices 3
1.1.3 Limits of Extreme Eigenvalues 4
1.1.4 Convergence Rate of ESD 4
1.1.5 Circular Law 5
1.1.6 CLT of Linear Spectral Statistics 5
1.1.7 Limiting Distributions of Extreme Eigenvalues and Spacings 6
1.2 Applications to Wireless Communications 6
1.3 Applications to Finance Statistics 7
2 Limiting Spectral Distributions 11
2.1 Semicircular Law 11
2.1.1 The iid Case 12
2.1.2 Independent but not Identically Distributed 18
2.2 Marcenko-Pastur Law 22
2.2.1 MP Law for iid Case 22
2.2.2 Generalization to the Non-iid Case 25
2.2.3 Proof of Theorem 2. LI by Stieltjes Transform 26
2.3 LSD of Products 27
2.3.1 Existence of the ESD of SnTn 28
2.3.2 Truncation of the ESD of Tn 29
2.3.3 Truncation. Centralization and Rescaling of the X-variables 30
2.3.4 Sketch of the Proof of Theorem 2.12 31
2.3.5 LSD of F Matrix 32
2.3.6 Sketch of the Proof of Theorem 2.14 36
2.3.7 When T is a Wigner Matrix 42
2.4 Hadamard Product 43
2.4.1 Truncation and Centralization 48
2.4.2 Outlines of Proof of the theorem 50
2.5 Circular Law 52
2.5.1 Failure of Techniques Dealing with Hermitian Matrices 53
2.5.2 Revisit of Stieltjes Transformation 55
2.5.3 A Partial Answer to the Circular Law 57
2.5.4 Comments and Extensions of Theorem 2.33 58
3 Extreme Eigenvalues 61
3.1 Wigner Matrix 62
3.2 Sample Covariance Matrix 64
3.2.1 Spectral Radius 66
3.3 Spectrum Separation 66
3.4 Tracy-Widom Law 73
3.4.1 TW Law for Wigner Matrix 73
3.4.2 TW Law for Sample Covariance Matrix 74
4 Central Limit Theorems of Linear Spectral Statistics 77
4.1 Motivation and Strategy 77
4.2 CLT of LSS for Wigner Matrix 79
4.2.1 Outlines of the Proof 81
4.3 CLT of LSS for Sample Covariance Matrices 90
4.4 F Matrix 98
4.4.1 Decomposition of Xnf 109
4.4.2 Limiting Distribution of X†nf 101
4.4.3 The Limiting Distribution of Xnf 103
5 Limiting Behavior of Eigenmatrix of Sample Covariance Matrix 109
5.1 Earlier Work by Silverstein 110
5.2 Further extension of Silverstcin's Work 112
5.3 Projecting the Eigenmatrix to a d-dimensional Space 117
5.3.1 Main Results 119
5.3.2 Sketch of Proof of Theorem 5.19 123
5.3.3 Proof of Corollary 5.23 132
6 Wireless Communications 133
6.1 Introduction 133
6.2 Channel Models 135
6.2.1 Basics of Wireless Communication Systems 135
6.2.2 Matrix Channel Models 136
6.2.3 Random Matrix Channels 137
6.2.4 Linearly Precoded Systems 139
6.3 Channel Capacity for MIMO Antenna Systems 143
6.3.1 Single-Input Single-Output Channels 143
6.3.2 MIMO Fading Channels 145
6.4 Limiting Capacity of Random MIMO Channels 151
6.4.1 CSI-Unknown Case 152
6.4.2 CSI-Known Case 153
6.5 Concluding Remarks 154
7 Limiting Performances of Linear and Iterative Receivers 155
7.1 Introduction 155
7.2 Linear Equalizers 156
7.2.1 ZF Equalizer 157
7.2.2 Matched Filter (MF) Equalizer 157
7.2.3 MMSE Equalizer 157
7.2.4 Suboptimal MMSE Equalizer 158
7.3 Limiting SINR Analysis for Linear Receivers 158
7.3.1 Random Matrix Channels 158
7.3.2 Linearly Precoded Systems 161
7.3.3 Asymptotic SINR Distribution 163
7.4 Iterative Receivers 165
7.4.1 MMSE-SIC 165
7.4.2 BI-GDFE 168
7.5 Limiting Performance of Iterative Receivers 169
7.5.1 MMSE-SIC Receiver 170
7.5.2 BI-GDFE Receiver 171
7.6 Numerical Results 173
7.7 Concluding Remarks 175
8 Application to Finance 177
8.1 Portfolio and Risk Management 177
84.1 Markowitz's Portfolio Selection 177
8.1.2 Financial Correlations and Information Extracting 179
8.2 Factor Models 183
8.2.1 From PCA to Generalized Dynamic Factor Models 184
8.2.2 CAPM and APT 187
8.2.3 Determine the Number of Factors 188
8.3 Some Application in Finance of Factor Model 194
8.3.1 Inflation Forecasting 194
8.3.2 Leading and Coincident Index 196
8.3.3 Financial Crises Warning 198
References 201
Index 217
