New Theory of Large Dimensional Random Matrices and Its Applications to Wireless Communications and Finance Statistics

Preface of Alumni's 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 Silverstein's 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
6.2.2 Matrix Channel Models
6.2.3 Random Matrix Channels
6.2.4 Linearly Precoded Systems
6.3 Channel Capacity for MIMO Antenna Systems
6.3.1 Single-Input Single-Output Channels
6.3.2 MIMO Fading Channels
6.4 Limiting Capacity of Random MIMO Channels
6.4.1 CSI-Unknown Case
6.4.2 CSI-Known Case
6.5 Concluding Remarks

7 Limiting Performances of Linear and Iterative Receivers
7.1 Introduction
7.2 Linear Equalizers
7.2.1 ZF Equalizer
7.2.2 Matched Filter (MF) Equalizer
7.2.3 MMSE Equalizer
7.2.4 Suboptimal MMSE Equalizer
7.3 Limiting SINR Analysis for Linear Receivers
7.3.1 Random Matrix Channels
7.3.2 Linearly Precoded Systems
7.3.3 Asymptotic SINR Distribution
7.4 Iterative Receivers
7.4.1 MMSE-SIC Receiver
7.4.2 BI-GDFE Receiver
7.5 Limiting Performance of Iterative Receivers
7.5.1 MMSE-SIC Receiver
7.5.2 BI-GDFE Receiver
7.6 Numerical Results
7.7 Concluding Remarks
Applications to Finance Statistics

8.1 Portfolio and Risk Management
8.1.1 Markowitz's Portfolio Selection
8.1.2 Financial Correlations and Information Extracting
8.2 Factor Models
8.2.1 From PCA to Generalized Dynamic Factor Models
8.2.2 CAPM and APT
8.2.3 Determine the number of factors
8.3 Some Applications in Finance of Factor Model
8.3.1 Inflation Forecasting
8.3.2 Leading and Coincident Index
8.3.3 Financial Crises Warning
References
Index