Preface to the Second Edition...vii Preface to the First Edition... ix

Transcription

1 Contents Preface to the Second Edition...vii Preface to the First Edition ix 1 Introduction Large Dimensional Data Analysis Random Matrix Theory Spectral Analysis of Large Dimensional Random Matrices Limits of Extreme Eigenvalues Convergence Rate of the ESD Circular Law CLT of Linear Spectral Statistics Limiting Distributions of Extreme Eigenvalues and Spacings Methodologies Moment Method Stieltjes Transform Orthogonal Polynomial Decomposition Free Probability Wigner Matrices and Semicircular Law Semicircular Law by the Moment Method Moments of the Semicircular Law Some Lemmas in Combinatorics Semicircular Law for the iid Case Generalizations to the Non-iid Case Proof of Theorem Semicircular Law by the Stieltjes Transform Stieltjes Transform of the Semicircular Law Proof of Theorem xi

2 xii Contents 3 Sample Covariance Matrices and the Marčenko-Pastur Law M-P Law for the iid Case Moments of the M-P Law Some Lemmas on Graph Theory and Combinatorics M-P Law for the iid Case Generalization to the Non-iid Case Proof of Theorem 3.10 by the Stieltjes Transform Stieltjes Transform of the M-P Law Proof of Theorem Product of Two Random Matrices Main Results Some Graph Theory and Combinatorial Results Proof of Theorem Truncation of the ESD of T n Truncation, Centralization, and Rescaling of the X-variables Completing the Proof LSD of the F-Matrix Generating Function for the LSD of S n T n Completing the Proof of Theorem Proof of Theorem Truncation and Centralization Proof by the Stieltjes Transform Limits of Extreme Eigenvalues Limit of Extreme Eigenvalues of the Wigner Matrix Sufficiency of Conditions of Theorem Necessity of Conditions of Theorem Limits of Extreme Eigenvalues of the Sample Covariance Matrix Proof of Theorem Proof of Theorem Necessity of the Conditions Miscellanies Spectral Radius of a Nonsymmetric Matrix TW Law for the Wigner Matrix TW Law for a Sample Covariance Matrix Spectrum Separation What Is Spectrum Separation? Mathematical Tools Proof of (1) Truncation and Some Simple Facts A Preliminary Convergence Rate

3 Contents xiii Convergence of s n Es n Convergence of the Expected Value Completing the Proof Proof of (2) Proof of (3) Convergence of a Random Quadratic Form spread of eigenvaluesspread of Eigenvalues Dependence on y Completing the Proof of (3) Semicircular Law for Hadamard Products Sparse Matrix and Hadamard Product Truncation and Normalization Truncation and Centralization Proof of Theorem 7.1 by the Moment Approach Convergence Rates of ESD Convergence Rates of the Expected ESD of Wigner Matrices Lemmas on Truncation, Centralization, and Rescaling Proof of Theorem Some Lemmas on Preliminary Calculation Further Extensions Convergence Rates of the Expected ESD of Sample Covariance Matrices Assumptions and Results Truncation and Centralization Proof of Theorem Some Elementary Calculus Increment of M-P Density Integral of Tail Probability Bounds of Stieltjes Transforms of the M-P Law Bounds for b n Integrals of Squared Absolute Values of Stieltjes Transforms Higher Central Moments of Stieltjes Transforms Integral of δ Rates of Convergence in Probability and Almost Surely CLT for Linear Spectral Statistics Motivation and Strategy CLT of LSS for the Wigner Matrix Strategy of the Proof Truncation and Renormalization Mean Function of M n Proof of the Nonrandom Part of (9.2.13) for j = l, r.. 238

4 xiv Contents 9.3 Convergence of the Process M n EM n Finite-Dimensional Convergence of M n EM n Limit of S Completion of the Proof of (9.2.13) for j = l, r Tightness of the Process M n (z) EM n (z) Computation of the Mean and Covariance Function of G(f) Mean Function Covariance Function Application to Linear Spectral Statistics and Related Results Tchebychev Polynomials Technical Lemmas CLT of the LSS for Sample Covariance Matrices Truncation Convergence of Stieltjes Transforms Convergence of Finite-Dimensional Distributions Tightness of Mn 1 (z) Convergence of Mn(z) Some Derivations and Calculations Verification of (9.8.8) Verification of (9.8.9) Derivation of Quantities in Example (1.1) Verification of Quantities in Jonsson s Results Verification of (9.7.8) and (9.7.9) CLT for the F-Matrix CLT for LSS of the F-Matrix Proof of Theorem Lemmas Proof of Theorem CLT for the LSS of a Large Dimensional Beta-Matrix Some Examples Eigenvectors of Sample Covariance Matrices Formulation and Conjectures Haar Measure and Haar Matrices Universality A Necessary Condition for Property Moments of X p (F Sp ) Proof of (10.3.1) (10.3.2) Proof of (b) Proof of (10.3.2) (10.3.1) Proof of (c) An Example of Weak Convergence Converting to D[0, ) A New Condition for Weak Convergence

5 Contents xv Completing the Proof Extension of (10.2.6) to B n = T 1/2 S p T 1/ First-Order Limit CLT of Linear Functionals of B p Proof of Theorem Proof of Theorem An Intermediate Lemma Convergence of the Finite-Dimensional Distributions Tightness of Mn 1(z) and Convergence of M2 n (z) Proof of Theorem Circular Law The Problem and Difficulty Failure of Techniques Dealing with Hermitian Matrices Revisiting Stieltjes Transformation A Theorem Establishing a Partial Answer to the Circular Law Lemmas on Integral Range Reduction Characterization of the Circular Law A Rough Rate on the Convergence of ν n (x, z) Truncation and Centralization A Convergence Rate of the Stieltjes Transform of ν n (, z) Proofs of (11.2.3) and (11.2.4) Proof of Theorem Comments and Extensions Relaxation of Conditions Assumed in Theorem Some Elementary Mathematics New Developments Some Applications of RMT Wireless Communications Channel Models random matrix channelrandom Matrix Channels Linearly Precoded Systems Channel Capacity for MIMO Antenna Systems Limiting Capacity of Random MIMO Channels A General DS-CDMA Model Application to Finance A Review of Portfolio and Risk Management Enhancement to a Plug-in Portfolio A Some Results in Linear Algebra A.1 Inverse Matrices and Resolvent A.1.1 Inverse Matrix Formula A.1.2 Holing a Matrix

6 xvi Contents A.1.3 Trace of an Inverse Matrix A.1.4 Difference of Traces of a Matrix A and Its Major Submatrices A.1.5 Inverse Matrix of Complex Matrices A.2 Inequalities Involving Spectral Distributions A.2.1 Singular-Value Inequalities A.3 Hadamard Product and Odot Product A.4 Extensions of Singular-Value Inequalities A.4.1 Definitions and Properties A.4.2 Graph-Associated Multiple Matrices A.4.3 Fundamental Theorem on Graph-Associated MMs A.5 Perturbation Inequalities A.6 Rank Inequalities A.7 A Norm Inequality B Miscellanies B.1 Moment Convergence Theorem B.2 Stieltjes Transform B.2.1 Preliminary Properties B.2.2 Inequalities of Distance between Distributions in Terms of Their Stieltjes Transforms B.2.3 Lemmas Concerning Levy Distance B.3 Some Lemmas about Integrals of Stieltjes Transforms B.4 A Lemma on the Strong Law of Large Numbers B.5 A Lemma on Quadratic Forms Relevant Literature Index

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