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 ESD
1.2.4 Circular Law
1.2.5 CLT of Linear Spectral Statisticslinear 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
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 of Combinatorics
2.1.3 Semicircular Law for iid Case
2.2 Generalizations to the Non-iid Case
2.2.1 Proof of Theorem 2.9
2.3 Semicircular Law by Stieltjes Transform
2.3.1 Stieltjes Transform of Semicircular Law
2.3.2 Proof of Theorem 2.9
3 Sample Covariance Matrices, Marˇcenko-Pastur Law
3.1 MP Law for iid Case
3.1.1 Moments of the MP Law
3.1.2 Some Lemmas on Graph Theory and Combinatorics
3.1.3 MP Law for iid Case
3.2 Generalization to the Non-iid Case
3.3 Proof of Theorem 3.9 by Stieltjes Transform
3.3.1 Stieltjes Transform of MP Law
3.3.2 Proof of Theorem 3.9
4 Product of Two Random Matrices
4.1 Some Graph Theory and Combinatoric Results
4.2 Existence of the ESD of SnTn
4.2.1 Truncation of the ESD of Tn
4.2.2 Truncation, Centralization and Rescaling of the X-variables
4.2.3 Proof of Theorem 4.3
4.3 LSD of F matrix
4.3.1 General Formula for the Product
4.3.2 LSD of Multivariate F Matrices
4.4 Proof of Theorem 4.5
4.4.1 Truncation and Centralization
4.4.2 Proof by 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 Sample Covariance Matrix
5.2.1 Proof of Theorem 5.10
5.2.2 The Proof of Theorem 5.11
5.2.3 Necessity of the Conditions
5.3 Miscellanies
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 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 Eigenvalues
6.4.3 Dependence on y
6.4.4 Completing the Proof of (3)
7 Semicircle Law for Hadamard Products
7.1 Renormalized Sample Covariance Matrix
7.2 Sparse Matrix and Hadamard Product
7.3 Proof of Theorem 7.4
7.3.1 Truncation and Centralization
7.4 Proof of Theorem 7.4 by Moment Approach
8 Convergence Rates of ESD
8.1 Some Lemmas About Integrals of Stieltjes Transforms
8.2 Convergence Rates of Expected ESD of Wigner Matrices
8.2.1 Lemmas on Truncation, Centralization and Rescaling
8.2.2 Proof of Theorem 8.6
8.2.3 Some Lemmas of Preliminary Calculation
8.3 Further Extensions
8.4 Convergence Rates of Expected ESD of Sample Covariance Matrices
8.4.1 Assumptions and Results
8.4.2 Truncation and Centralization
8.4.3 Proof of Theorem 8.16
8.5 Some Elementary Calculus
8.5.1 Increment of M-P Density
8.5.2 Integral of Tail Probability
8.5.3 Bounds of Stieltjes Transforms of M-P Law
8.5.4 Bounds for ~bn
8.5.5 Integrals of Squared Absolute Values of Stieltjes Transforms
8.5.6 Higher Central Moments of Stieltjes Transforms
8.5.7 Integral of δ
8.6 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 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 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 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 M2n(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)
10 Circular Law
10.1 The Problem and Difficulty
10.1.1 Failure of Techniques Dealing with Hermitian Matrices
10.1.2 Revisit of Stieltjes Transformation
10.2 A Theorem Establishing a Partial Answer to the Circular Law
10.3 Lemmas on Integral Range Reduction
10.4 Characterization of the Circular Law
10.5 A Rough Rate on the Convergence of vn(x,z)
10.5.1 Truncation and Centralization
10.5.2 A Convergence Rate of the Stieltjes Transform of vn(·, z)
10.6 Proofs of (10.2.3) and (10.2.4)
10.7 Proof of Theorem 10.3
10.8 Comments and Extensions
10.8.1 Relaxation of Conditions Assumed in Theorem 10.3
10.9 Some Elementary Mathematics
11 Appendix A. Some Results in Linear Algebra
11.1 Inverse Matrices and Resolvent
11.1.1 Inverse Matrix Formula
11.1.2 Holing a Matrix
11.1.3 Trace of Inverse Matrix
11.1.4 Difference of Traces of a Matrix A and Its Major Submatrices
11.1.5 Inverse Matrix of Complex Matrices
11.2 Inequalities Involving Spectral Distributions
11.2.1 Singular Value Inequalities
11.3 Hadamard Product and Odot Product
11.4 Extensions of Singular Value Inequalities
11.4.1 Definitions and Properties
11.4.2 Graph-Associated Multiple Matrices
11.4.3 Fundamental Theorem on Graph-Associated MM
11.5 Perturbation Inequalities
11.6 Rank Inequalities
11.7 A Norm Inequality
12 Appendix B. Moment Convergence Theorem and Stieltjes Transform
12.1 Moment Convergence Theorem
12.2 Stieltjes Transform
12.2.1 Preliminary Properties
12.2.2 Inequalities of Distance between Distributions in Terms of Their Stieltjes Transforms
12.2.3 Lemmas Concerning Levy Distance
References
Index |
