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

The following sections are included:

• Dedication
• Preface
• Contents

The following sections are included:

• History of RMT and Current Development
  • A brief review of RMT
  • Spectral Analysis of Large Dimensional Random Matrices
  • Limits of Extreme Eigenvalues
  • Convergence Rate of ESD
  • Circular Law
  • CLT of Linear Spectral Statistics
  • Limiting Distributions of Extreme Eigenvalues and Spacings
• Applications to Wireless Communications
• Applications to Finance Statistics

The following sections are included:

• Semicircular Law
  • The iid Case
  • Independent but not Identically Distributed
• Marčenko-Pastur Law
  • MP Law for iid Case
  • Generalization to the Non-iid Case
  • Proof of Theorem 2.11 by Stieltjes Transform
• LSD of Products
  • Existence of the ESD of S_nT_n
  • Truncation of the ESD of T_n
  • Truncation, Centralization and Rescaling of the X-variables
  • Sketch of the Proof of Theorem 2.12
  • LSD of F Matrix
  • Sketch of the Proof of Theorem 2.14
  • When T is a Wigner Matrix
• Hadamard Product
  • Truncation and Centralization
  • Outlines of Proof of the theorem
• Circular Law
  • Failure of Techniques Dealing with Hermitian Matrices
  • Revisit of Stieltjes Transformation
  • A Partial Answer to the Circular Law
  • Comments and Extensions of Theorem 2.33

In multivariate analysis, many statistics involved with a random matrix can be written as functions of integrals with respect to the ESD of the random matrix. When the LSD is known, one may want to apply the Helly-Bray Theorem to find approximate values of the statistics. However, the integrands are usually unbounded. For instance, the determinant of a random positive definite matrix can be written as an exponential function of the integral of log x with respect to the ESD where the integrand is unbounded both from below and above. Thus, one cannot use the LSD and Helly-Bray Theorem to find approximate values of the statistics. This would render the LSD useless. Fortunately, in most cases, the supports of the LSD's are compact intervals. Still, this does not mean that the Helly-Bray Theorem is applicable unless one can prove that the extreme eigenvalues of the random matrix remain in certain bounded intervals…

The following sections are included:

• Motivation and Strategy
• CLT of LSS for Wigner Matrix
  • Outlines of the Proof
• CLT of LSS for Sample Covariance Matrices
• F Matrix
  • Decomposition of X_nf
  • Limiting Distribution of X^†_nf
  • The Limiting Distribution of X_nf

Let X_n = (X_ij) be an p × n matrix of i.i.d. complex random variables and let T_n be an p × p nonnegative definite Hermitian matrix with a square root . In this chapter, we shall consider the matrix . If T_n is nonrandom, then A_n can be considered as a sample covariance matrix of a sample drawn from a population with distribution of T^1/2_nX_.,1, where X_.,1 = (X₁₁,…,X_n1)'. If T_n is an inverse of another sample covariance matrix, then the multivariate F matrix can be considered as a special case of the matrix A_n…

The following sections are included:

• Introduction
• Channel Models
  • Basics of Wireless Communication Systems
  • Matrix Channel Models
  • Random Matrix Channels
  • Linearly Precoded Systems
• Channel Capacity for MIMO Antenna Systems
  • Single-Input Single-Output Channels
  • MIMO Fading Channels
• Limiting Capacity of Random MIMO Channels
  • CSI-Unknown Case
  • CSI-Known Case
• Concluding Remarks

The following sections are included:

• Introduction
• Linear Equalizers
  • ZF Equalizer
  • Matched Filter (MF) Equalizer
  • MMSE Equalizer
  • Suboptimal MMSE Equalizer
• Limiting SINR Analysis for Linear Receivers
  • Random Matrix Channels
  • Linearly Precoded Systems
  • Asymptotic SINR Distribution
• Iterative Receivers
  • MMSE-SIC
  • BI-GDFE
• Limiting Performance of Iterative Receivers
  • MMSE-SIC Receiver
  • BI-GDFE Receiver
• Numerical Results
• Concluding Remarks

It is a popular understanding that the financial environment today is riskier than it was used to. From the second half of the 20th century the financial environment has changed. The price indices have gone up and the volatility of foreign exchange rates, interest rates and commodity prices all has been increasing. All firms and financial institutes are facing uncertainty. The markets for risk management products have grew dramatically since 1980's. Risk management becomes a key and basic technique for all market participants. Risk should be carefully measured before well managed. Var(Value at risk) matrix and credit matrix become popular tools in Banks and Fund management company. People using correlation matrix and factor models to work on internal or external measure for financial risk.

The following sections are included:

• References
• Index