高维随机矩阵的谱理论及其在无线通信和金融统计中的应用=Spectral Theory of Large Dimensional Random Matrices and Applications to Wireless Communications and Finance Statistics(“十一五”国家重点图书)

高维随机矩阵的谱理论及其在无线通信和金融统计中的应用=Spectral Theory of Large Dimensional Random Matrices and Applications to Wireless Communications and Finance Statistics(“十一五”国家重点图书)
作者:白志东 方兆本 梁应敞(著)






During the past three to four decades, computer science has been developed rapidly and computing facilities have been adopted in almost every discipline. At the same time, data sets collected from experiments or natural phenomena have become larger and larger in both size and dimension. As a result, the computers need to deal with these extremely huge data sets. For example, in the biological sciences, a DNA sequence can be as long as several billions. In finance research, the number of different stocks can be as large as tens of thousands. In wireless communications, the number of users supported by each base station can be several hundreds. In image processing, the pixels of a picture may be several thousands.
  On the other hand, however, in statistics, classical limit theorems have been found to be seriously inadequate in aiding in the analysis of large dimensional data. All these are challenging the applicability of classical statistics. Nowa-days, an urgent need to statistics is to create new limiting theories that are applicable to large dimensional data analysis. Therefore, since last decade, the large dimensional data analysis has become a very hot topic in statistics and various disciplines where statistics is applicable.
  Currently, the spectral analysis of large dimensional random matrices (simply Random Matrix Theory (RMT)) is the only systematic theory that can be applied to many problems of large dimensional data analysis. The RMT dates back to the early development of Quantum Mechanics in the 1940's and 50'S. In an attempt to explain the complex organizational structure of heavy nuclei, E. Wigner, Professor of Mathematical Physics at Princeton University, argued that one should not compute energy levels from Schrodinger's equation. Instead, one should imagine the complex nuclei system as a black box described by n x n Hamiltonian matrices with elements drawn from a probability distribution with only mild constraints dictated by symmetry considerations. Under these assumptions and some mild conditions imposed on the probability measure in the space of matrices, one can find the joint probability density of the n eigenvalues. Based on this consideration, Wigner established the well-known semi-circular law. Since then, RMT has been developed into an active research area in mathematical physics and probability.
  Due to the need of large dimensional data analysis, the number of researchers and publications on RMT has been growing rapidly.
  The purpose of this monograph is to introduce the basic concepts and results of RMT and some applications to wireless communications and finance statistics. The readers of this book would be graduate students and researchers who are interested in RMT and/or its applications to their own research areas. As for the theorems in RMT, we only provide an outline of their proofs. The detailed proofs are referred to the book Spectral analysis of large dimensional random matrices by Bai, Z. D. and Silverstein, J. W. (2006). As for the applications to wireless communications and finance statistics, we are more emphasizing the problem formulation to illustrate how the RMT is applied to, rather than the detailed mathematical derivations and proofs.
  Special thanks go to Mr. Liuzhi Yin who contributed to the book by providing editing and extensive literature review, and to Ms. Yiyang Pei for proof-reading.

Changchun, China
Hefei, China

  Zhidong Bai
   Zhaoben Fang
Yingchang Liang
       April 2008



Preface of Alumni's Serials


1 Introduction
1.1 History of RMT and Current Development
1.2 Applications to Wireless Communications
1.3 Applications to Finance Statistics

2 Limiting Spectral Distributions
 2.1 Semi-circular Law
 2.2 Marcenko-Pastur Law
 2.3 LSD of Products
 2.4 Hadamard Product
 2.5 Circular Law
3 Extreme Eigenvalues 
3.1 Wigner Matrix 
3.2 Sample Covariance Matrix
3.3 Spectrum Separation..
3.4 rIyacyWidom Law 

4 CLT of LSS
4.1 Motivation and Strategy
4.2 CLT of LSS for Wigner Matrix
4.3 CLT of LSS for Sample Covariance Matrices
4.4 F Matrix
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
6 Applications to Wireless Communications
6.1 Introduction
6.2 Channel Models
6.3 Channel Capacity for MIMO Antenna Systems
6.4 Limiting Capacity of Random MIMO Channels
6.5 Concluding Remarks

7 Limiting Performances of Linear and Iterative Receivers
7.1 Introduction.
7.2 Linear Equalizers

7.3 Limiting SINR Analysis for Linear Receivers
7.4 Iterative Receivers
7.5Limiting Performance of Iterative Receivers
7.6Numerical Results
7.7 Concluding Remarks

8 Applications to Finance Statistics
8.1 Portfolio and Risk Management
8.2 Factor Models
8.3 Some Applications in Finance of Factor Model

Copyright 2011 中国科学技术大学出版社