Spectral Theory of Large Dimensional Random Matrices and Its Applications to Wireless Communications and Finance Statistics : Random Matrix Theory and Its Applications

The book contains three parts: Spectral theory of large dimensional random matrices; Applications to wireless communications; and Applications to finance. In the first part, we introduce some basic theorems of spectral analysis of large dimensional random matrices that are obtained under finite moment conditions, such as the limiting spectral distributions of Wigner matrix and that of large dimensional sample covariance matrix, limits of extreme eigenvalues, and the central limit theorems for linear spectral statistics. In the second part, we introduce some basic examples of applications of random matrix theory to wireless communications and in the third part, we present some examples of Applications to statistical finance. Readership: Graduate students and researchers in random matrices. Contents 10 Preface 8 1 Introduction 14 1.1 History of RMT and Current Development 14 1.1.1 A brief review of RMT 15 1.1.2 Spectral Analysis of Large Dimensional Random Matrices 16 1.1.3 Limits of Extreme Eigenvalues 17 1.1.4 Convergence Rate of ESD 17 1.1.5 Circular Law 18 1.1.6 CLT of Linear Spectral Statistics 18 1.1.7 Limiting Distributions of Extreme Eigenvalues and Spacings 19 1.2 Applications to Wireless Communications 19 1.3 Applications to Finance Statistics 20 2 Limiting Spectral Distributions 24 2.1 Semicircular Law 24 2.1.1 The iid Case 25 2.1.2 Independent but not Identically Distributed 31 2.2 Marcenko-Pastur Law 35 2.2.1 MP Law for iid Case 35 2.2.2 Generalization to the Non-iid Case 38 2.2.3 Proof of Theorem 2.11 by Stieltjes Transform 39 2.3 LSD of Products 40 2.3.1 Existence of the ESD of SnTn 41 2.3.2 Truncation of the ESD of Tn 42 2.3.3 Truncation, Centralization and Rescaling of the X-variables 43 2.3.4 Sketch of the Proof of Theorem 2.12 44 2.3.5 LSD of F Matrix 45 2.3.6 Sketch of the Proof of Theorem 2.14 49 2.3.7 When T is a Wigner Matrix 55 2.4 Hadamard Product 56 2.4.1 Truncation and Centralization 61 2.4.2 Outlines of Proof of the theorem 63 2.5 Circular Law 65 2.5.1 Failure of Techniques Dealing with Hermitian Matrices 66 2.5.2 Revisit of Stieltjes Transformation 68 2.5.3 A Partial Answer to the Circular Law 70 2.5.4 Comments and Extensions of Theorem 2.33 71 3 Extreme Eigenvalues 74 3.1 Wigner Matrix 75 3.2 Sample Covariance Matrix 77 3.2.1 Spectral Radius 79 3.3 Spectrum Separation 79 3.4 Tracy-Widom Law 86 3.4.1 TW Law for Wigner Matrix 86 3.4.2 TW Law for Sample Covariance Matrix 87 4 Central Limit Theorems of Linear Spectral Statistics 90 4.1 Motivation and Strategy 90 4.2 CLT of LSS for Wigner Matrix 92 4.2.1 Outlines of the Proof 94 4.3 CLT of LSS for Sample Covariance Matrices 103 4.4 F Matrix 111 4.4.1 Decomposition of Xnf 113 4.4.2 Limiting Distribution of X†nf 114 4.4.3 The Limiting Distribution of Xnf 116 5 Limiting Behavior of Eigenmatrix of Sample Covariance Matrix 122 5.1 Earlier Work by Silverstein 123 5.2 Further extension of Silverstein’s Work 125 5.3 Projecting the Eigenmatrix to a d-dimensional Space 130 5.3.1 Main Results 132 5.3.2 Sketch of Proof of Theorem 5.19 136 5.3.3 Proof of Corollary 5.23 145 6 Wireless Communications 146 6.1 Introduction 146 6.2 Channel Models 148 6.2.1 Basics of Wireless Communication Systems 148 6.2.2 Matrix Channel Models 149 6.2.3 Random Matrix Channels 150 6.2.4 Linearly Precoded Systems 152 6.3 Channel Capacity for MIMO Antenna Systems 156 6.3.1 Single-Input Single-Output Channels 156 6.3.2 MIMO Fading Channels 158 6.4 Limiting Capacity of Random MIMO Channels 164 6.4.1 CSI-Unknown Case 165 6.4.2 CSI-Known Case 166 6.5 Concluding Remarks 167 7 Limiting Performances of Linear and Iterative Receivers 168 7.1 Introduction 168 7.2 Linear Equalizers 169 7.2.1 ZF Equalizer 170 7.2.2 Matched Filter (MF) Equalizer 170 7.2.3 MMSE Equalizer 170 7.2.4 Suboptimal MMSE Equalizer 171 7.3 Limiting SINR Analysis for Linear Receivers 171 7.3.1 Random Matrix Channels 171 7.3.2 Linearly Precoded Systems 174 7.3.3 Asymptotic SINR Distribution 176 7.4 Iterative Receivers 178 7.4.1 MMSE-SIC 178 7.4.2 BI-GDFE 181 7.5 Limiting Performance of Iterative Receivers 182 7.5.1 MMSE-SIC Receiver 183 7.5.2 BI-GDFE Receiver 184 7.6 Numerical Results 186 7.7 Concluding Remarks 188 8 Application to Finance 190 8.1 Portfolio and Risk Management 190 8.1.1 Markowitz’s Portfolio Selection 190 8.1.2 Financial Correlations and Information Extracting 192 8.2 Factor Models 196 8.2.1 From PCA to Generalized Dynamic Factor Models 197 8.2.2 CAPM and APT 200 8.2.3 Determine the Number of Factors 201 8.3 Some Application in Finance of Factor Model 207 8.3.1 Inflation Forecasting 207 8.3.2 Leading and Coincident Index 209 8.3.3 Financial Crises Warning 211 References 214 Index 230 The book contains three parts: Spectral theory of large dimensional random matrices; Applications to wireless communications; and Applications to finance. In the first part, we introduce some basic theorems of spectral analysis of large dimensional random matrices that are obtained under finite moment conditions, such as the limiting spectral distributions of Wigner matrix and that of large dimensional sample covariance matrix, limits of extreme eigenvalues, and the central limit theorems for linear spectral statistics. In the second part, we introduce some basic examples of applications of random matrix theory to wireless communications and in the third part, we present some examples of Applications to statistical finance. Book jacket