The aim of the book is to introduce basic concepts,main results,and widely applied mathematical tools in the spectral analysis of large dimensional ran dom matrices.The core of the book focuses on results established under moment conditions on random variales using probabilistic methods,and is thus easily applicable to statistice and other areas of science.The book introduces fundamental results,most of them invertigated by the authors,such as the semicircular law of Wigner matrices,the Marcenko-Pastur law,the limiting soectral distribution of the multivariate F-matrix,limits of extreme eigenvalues,spectrum separation theorems,convergence rates of emporocal distributions,central limit theorems if linera soectral statistics,and the partial solution of the famous circular law.While deriving the main results,the book simultaneously emphasizes the ideas and methdologies of the fundamental mathematical tools,among them being:truncation techniques,matrix identities,moment convergence theorems,and the Stieltjes tra
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目录
Preface to the Second Edition Preface to the First Edition 1 Introduction 1.1 Large Dimensional Data Analysis 1.2 Random Matrix Theory 1.2.1 Spectral Analysis of Large Dimensional Random Matrices 1.2.2 Limits of Extreme Eigenvalues 1.2.3 Convergence Rate of the ESD 1.2.4 Circular Law 1.2.5 CLT of Linear Spectral Statistics 1.2.6 Limiting Distributions of Extreme Eigenvalues and Spacings 1.3 Methodologies 1.3.1 Moment Method 1.3.2 Stieltjes Transform 1.3.3 Orthogonal Polynomial Decomposition 1.3.4 Free Probability 2 Wigner Matrices and Semicircular Law 2.1 Semicircular Law by the Moment Method 2.1.1 Moments of the Semicircular Law 2.1.2 Some Lemmas in Combinatorics 2.1.3 Semicircular Law for the iid Case 2.2 Generalizations to the Non-iid Case 2.2.1 Proof of Theorem 2.9 2.3 Semicircular Law by the Stieltjes Transform 2.3.1 Stieltjes Transform of the Semicircular Law 2.3.2 Proof of Theorem 2.9 3 Sample Covariance Matrices and the Marcenko-Pastur Law 3.1 M-P Law for the iid Case 3.1.1 Moments of the M-P Law 3.1.2 Some Lemmas on Graph Theory and Combinatorics 3.1.3 M-P Law for the iid Case 3.2 Generalization to the Non-iid Case 3.3 Proof of Theorem 3.10 by the Stieltjes Transform 3.3.1 Stieltjes Transform of the M-P Law 3.3.2 Proof of Theorem 3.10 4 Product of Two Random Matrices 4.1 Main Results 4.2 Some Graph Theory and Combinatorial Results 4.3 Proof of Theorem 4.1 4.3.1 Truncation of the ESD of Tn 4.3.2 Truncation, Centralization, and Rescaling of the X-variables 4.3.3 Completing the Proof 4.4 LSD of the F-Matrix 4.4.1 Generating Function for the LSD of SnTn 4.4.2 Completing the Proof of Theorem 4.10 4.5 Proof of Theorem 4.3 4.5.1 Truncation and Centralization 4.5.2 Proof by the Stieltjes Transform 5 Limits of Extreme Eigenvalues 5.1 Limit of Extreme Eigenvalues of the Wigner Matrix 5.1.1 Sufficiency of Conditions of Theorem 5.1 5.1.2 Necessity of Conditions of Theorem 5.1 5.2 Limits of Extreme Eigenvalues of the Sample Covariance Matrix 5.2.1 Proof of Theorem 5.10 5.2.2 Proof of Theorem 5.11 5.2.3 Necessity of the Conditions 5.3 Miscellanies 5.3.1 Spectral Radius of a Nonsymmetric Matrix 5.3.2 TW Law for the Wigner Matrix 5.3.3 TW Law for a Sample Covariance Matrix 6 Spectrum Separation 6.1 What Is Spectrum Separation? 6.1.1 Mathematical Tools 6.2 Proof of (1) 6.2.1 Truncation and Some Simple Facts 6.2.2 A Preliminary Convergence Rate 6.2.3 Convergence of sn-Esn 6.2.4 Convergence of the Expected Value 6.2.5 Completing the Proof 6.3 Proof of (2) 6.4 Proof of (3) 6.4.1 Convergence of a Random Quadratic Form 6.4.2 spread of eigenvaluesSpread of Eigenvalues 6.4.3 Dependence on y 6.4.4 Completing the Proof of (3) 7 Semicircular Law for Hadamard Products 7.1 Sparse Matrix and Hadamard Product 7.2 Truncation and Normalization 7.2.1 Truncation and Centralization 7.3 Proof of Theorem 7.1 by the Moment Approach 8 Convergence Rates of ESD 8.1 Convergence Rates of the Expected ESD of Wigner Matrices 8.1.1 Lemmas on Truncation, Centralization, and Rescaling 8.1.2 Proof of Theorem 8.2 8.1.3 Some Lemmas on Preliminary Calculation 8.2 Further Extensions 8.3 Convergence Rates of the Expected ESD of Sample Covariance Matrices 8.3.1 Assumptions and Results 8.3.2 Truncation and Centralization 8.3.3 Proof of Theorem 8.10 8.4 Some Elementary Calculus 8.4.1 Increment of M-P Density 8.4.2 Integral of Tail Probability 8.4.3 Bounds of Stieltjes Transforms of the M-P Law 8.4.4 Bounds for bn 8.4.5 Integrals of Squared Absolute Values of Stieltjes Transforms 8.4.6 Higher Central Moments of Stieltjes Transforms 8.4.7 Integral of δ 8.5 Rates of Convergence in Probability and Almost Surely 9 CLT for Linear Spectral Statistics 9.1 Motivation and Strategy 9.2 CLT of LSS for the Wigner Matrix 9.2.1 Strategy of the Proof 9.2.2 Truncation and Renormalization 9.2.3 Mean Function of Mn 9.2.4 Proof of the Nonrandom Part of (9.2.13) for j=l, r 9.3 Convergence of the Process Mn-EMn 9.3.1 Finite-Dimensional Convergence of Mn-EMn 9.3.2 Limit of S1 9.3.3 Completion of the Proof of (9.2.13) for j=l, r 9.3.4 Tightness of the Process Mn(z)-EMn(z) 9.4 Computation of the Mean and Covariance Function of G(f) 9.4.1 Mean Function 9.4.2 Covariance Function 9.5 Application to Linear Spectral Statistics and Related Results 9.5.1 Tchebychev Polynomials 9.6 Technical Lemmas 9.7 CLT of the LSS for Sample Covariance Matrices 9.7.1 Truncation 9.8 Convergence of Stieltjes Transforms 9.9 Convergence of Finite-Dimensional Distributions 9.10 Tightness of M1n(z) 9.11 Convergence of 2n(z) 9.12 Some Derivations and Calculations 9.12.1 Verification of (9.8.8) 9.12.2 Verification of (9.8.9) 9.12.3 Derivation of Quantities in Example (1.1) 9.12.4 Verification of Quantities in Jonsson's Results 9.12.5 Verification of (9.7.8) and (9.7.9) 9.13 CLT for the F-Matrix 9.13.1 CLT for LSS of the F-Matrix 9.14 Proof of Theorem 9.14 9.14.1 Lemmas 9.14.2 Proof of Theorem 9.14 9.15 CLT for the LSS of a Large Dimensional Beta-Matrix 9.16 Some Examples 10 Eigenvectors of Sample Covariance Matrices 10.1 Formulation and Conjectures 10.1.1 Haar Measure and Haar Matrices 10.1.2 Universality 10.2 A Necessary Condition for Property 5' 10.3 Moments of Xp(FSp) 10.3.1 Proof of (10.3.1) ⇒ (10.3.2) 10.3.2 Proof of (b) 10.3.3 Proof of (10.3.2) ⇒ (10.3.1) 10.3.4 Proof of (c) 10.4 An Example of Weak Convergence 10.4.1 Converting to D[0,∞) 10.4.2 A New Condition for Weak Convergence 10.4.3 Completing the Proof 10.5 Extension of (10.2.6) to Bn=T1/2SpT1/2 10.5.1 First-Order Limit 10.5.2 CLT of Linear Functionals of Bp 10.6 Proof of Theorem 10.16 10.7 Proof of Theorem 10.21 10.7.1 An Intermediate Lemma 10.7.2 Convergence of the Finite-Dimensional Distributions 10.7.3 Tightness of M1n(z) and Convergence of M2n(z) 10.8 Proof of Theorem 10.23 11 Circular Law 11.1 The Problem and Difficulty 11.1.1 Failure of Techniques Dealing with Hermitian Matrices 11.1.2 Revisiting Stieltjes Transformation 11.2 A Theorem Establishing a Partial Answer to the Circular Law 11.3 Lemmas on Integral Range Reduction 11.4 Characterization of the Circular Law 11.5 A Rough Rate on the Convergence of νn(x, z) 11.5.1 Truncation and Centralization 11.5.2 A Convergence Rate of the Stieltjes Transform of νn(·, z) 11.6 Proofs of (11.2.3) and (11.2.4) 11.7 Proof of Theorem 11.4 11.8 Comments and Extensions 11.8.1 Relaxation of Conditions Assumed in Theorem 11.4 11.9 Some Elementary Mathematics 11.10 New Developments 12 Some Applications of RMT 12.1 Wireless Communications 12.1.1 Channel Models 12.1.2 random matrix channelRandom Matrix Channels 12.1.3 Linearly Precoded Systems 12.1.4 Channel Capacity for MIMO Antenna Systems 12.1.5 Limiting Capacity of Random MIMO Channels 12.1.6 A General DS-CDMA Model 12.2 Application to Finance 12.2.1 A Review of Portfolio and Risk Management 12.2.2 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 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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