Spectral theory of large dimensional random matrices and its applications to wireless communications and finance statistics : random matrix theory and its applications

Cover image for Spectral theory of large dimensional random matrices and its applications to wireless communications and finance statistics : random matrix theory and its applications

Title:

Personal Author:

Bai, Zhidong author

Publication Information:

Singapore : World Scientific Publishing Co. Pte Ltd, 2014

Physical Description:

xi, 220 pages : illustrations ; 24 cm.

ISBN:

9789814579056

Subject Term:

Matrices

Matrices -- Mathematical models

Added Author:

Fang, Zhaoben

Liang, Ying-Chang

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010332333	QA188 B354 2014	Open Access Book	Book	Searching... Unknown

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.

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

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