Spectral Analysis Of Large Dimensional Random Matrices Book

Preface to the Second Edition vii

Preface to the First Edition ix

1 Introduction 1

1.1 Large Dimensional Data Analysis 1

1.2 Random Matrix Theory 4

1.2 1 Spectral Analysis of Large Dimensional Random Matrices 4

1.2.2 Limits of Extreme Eigenvalues 6

1.2.3 Convergence Rate of the ESD 6

1.2.4 Circular Law 7

1.2.5 CLT of Linear Spectral Statistics 8

1.2.6 Limiting Distributions of Extreme Eigenvalues and Spacings 9

1.3 Methodologies 9

1.3.1 Moment Method 9

1.3.2 Stieltjes Transform 10

1.3.3 Orthogonal Polynomial Decomposition 11

1.3.4 Free Probability 13

2 Wigner Matrices and Semicircular Law 15

2.1 Semicircular Law by the Moment Method 16

2.1.1 Moments of the Semicircular Law 16

2.1.2 Some Lemmas in Combinatorics 16

2.1.3 Semicircular Law for the iid Case 20

2.2 Generalizations to the Non-iid Case 26

2.2.1 Proof of Theorem 2.9 26

2.3 Semicircular Law by the Stieltjes Transform 31

2.3.1 Stieltjes Transform of the Semicircular Law 31

2.3.2 Proof of Theorem 2.9 33

3 Sample Covariance Matrices and the Marčenko-Pastur Law 39

3.1 M-P Law for the iid Case 40

3.1.1 Moments of the M-P Law 40

3.1.2 Some Lemmas on Graph Theory and Combinatorics 41

3.1.3 M-P Law for the iid Case 47

3.2 Generalization to the Non-iid Case 51

3.3 Proof of Theorem 3.10 by the Stieltjes Transform 52

3.3.1 Stieltjes Transform of the M-P Law 52

3.3.2 Proof of Theorem 3.10 53

4 Product of two Random Matrices 59

4.1 Main Results 60

4.2 Some Graph Theory and Combinatorial Results 61

4.3 Proof of Theorem 4.1 68

4.3.1 Truncation of the ESD of T_n 68

4.3.2 Truncation, Centralization, and Rescaling of the X-variables 70

4.3.3 Completingthe Proof 71

4.4 LSD of the F-Matrix 75

4.4.1 Generating Function for the LSD of S_n T_n 75

4.4.2 Completing the Proof of Theorem 4.10 77

4.5 Proof of Theorem 4.3 80

4.5.1 Truncation and Centralization 80

4.5.2 Proof by the Stieltjes Transform 82

5 Limits of Extreme Eigenvalues 91

5.1 Limit of Extreme Eigenvalues of the Wigner Matrix 92

5.1.1 Sufficiency of Conditions of Theorem 5.1 93

5.1.2 Necessity of Conditions of Theorem 5.1 101

5.2 Limits of Extreme Eigenvalues of the Sample Covariance Matrix 105

5.2.1 Proof of Theorem 5.10 106

5.2.2 Proof of Theorem 5.11 113

5.2.3 Necessity of the Conditions 113

5.3 Miscellanies 114

5.3.1 Spectral Radius of a Nonsymmetric Matrix 114

5.3.2 TW Law for the Wigner Matrix 115

5.3.3 TW Law for a Sample Covariance Matrix 117

6 Spectrum Separation 119

6.1 What is Spectrum Separation? 119

6.1.1 Mathematical Tools 126

6.2 Proof of (1) 128

6.2.1 Truncation and Some Simple Facts 128

6.2.2 A Preliminary Convergence Rate 129

6.2.3 Convergence of S_n - Es_n 139

6.2.4 Convergence of the Expected Value 144

6.2.5 Completing the Proof 148

6.3 Proof of (2) 149

6.4 Proof of (3) 151

6.4.1 Convergence of a Random Quadratic Form 151

6.4.3 Dependence on y 157

6.4.4 Completing the Proof of (3) 160

7 Semicircular Law for Hadamard Products 165

7.1 Sparse Matrix and Hadamard Product 165

7.2 Truncation and Normalization 168

7.2.1 Truncation and Centralization 169

7.3 Proof.of Theorem 7.1 by the Moment Approach 172

8 Convergence Rates of ESD 181

8.1 Convergence Rates of the Expected ESD of Wigner Matrices 181

8.1.1 Lemmas on Truncation, Centralization, and Rescaling 182

8.1.2 Proof of Theorem 8.2 185

8.1.3 Some Lemmas on Preliminary Calculation 189

8.2 Further Extensions 194

8.3 Convergence Rates of the Expected ESD of Sample Covariance Matrices 195

8.3.1 Assumptions and Results 195

8.3.2 Truncation and Centralization 197

8.3.3 Proof of Theorem 8.10 198

8.4 Some Elementary Calculus 204

8.4.1 Increment of M-P Density 204

8.4.2 Integral of Tail Probability 206

8.4.3 Bounds of Stieltjes Transforms of the M-P Law 207

8.4.4 Bounds for <$$> 209

8.4.5 Integrals of Squared Absolute Values of Stieltjes Transforms 212

8.4.6 Higher Central Moments of Stieltjes Transforms 213

8.4.7 Integral of δ 217

8.5 Rates of Convergence in Probability and Almost Surely 219

9 CLT for Linear Spectral Statistics 223

9.1 Motivation and Strategy 223

9.2 CLT of LSS for the Wigner Matrix 227

9.2.1 Strategy of the Proof 229

9.2.2 Truncation and Renormalization 231

9.2.3 Mean Function of M_n 232

9.2.4 Proof of the Nonrandom Part of (9.2.13) for j = l, r 238

9.3 Convergence of the Process M_n - EM_n 239

9.3.1 Finite-Dimensional Convergence of M_n - EM_n 239

9.3.2 Limit of S₁ 242

9.3.3 Completion of the Proof of (9.2.13) for j = l, r 250

9.3.4 Tightness of the Process M_n(z) - EM_n(z) 251

9.4 Computation of the Mean and Covariance Function of G(f) 252

9.4.1 Mean Function 252

9.4.2 Covariance Function 254

9.5 Application to Linear Spectral Statistics and Related Results 256

9.5.1 Tchebychev Polynomials 256

9.6 Technical Lemmas 257

9.7 CLT of the LSS for Sample Covariance Matrices 259

9.7.1 Truncation 261

9.8 Convergence of Stieltjes Transforms 263

9.9 Convergence of Finite-Dimensional Distributions 269

9.10 Tightness of <$$> 280

9.11 Convergence of <$$> 286

9.12 Some Derivations and Calculations 292

9.12.1 Verification of (9.8.8) 292

9.12.2 Verification of (9.8.9) 295

9.12.3 Derivation of Quantities in Example (1.1) 296

9.12.4 Verification of Quantities in Jonsson's Results 298

9.12.5 Verification of (9.7.8) and (9.7.9) 300

9.13 CLT for the F-Matrix 304

9.13.1 CLT for LSS of the F-Matrix 306

9.14 Proof of Theorem 9.14 308

9.14.1 Lemmas 308

9.14.2 Proof of Theorem 9.14 318

9.15 CLT for the LSS of a Large Dimensional Beta-Matrix 325

9.16 Some Examples 326

10 Eigenvectors of Sample Covariance Matrices 331

10.1 Formulation and Conjectures 332

10.1.1 Haar Measure and Haar Matrices 332

10.1.2 Universality 335

10.2 A Necessary Condition for Property 5' 336

10.3 Moments of <$$> 339

10.3.1 Proof of (10.3.1) → (10.3.2) 340

10.3.2 Proof of (b) 341

10.3.3 Proof of (10.3.2) → (10.3.1) 341

10.3.4 Proof of (c) 349

10.4 An Example of Weak Convergence 349

10.4.1 Converting to D[0, ∞) 350

10.4.2 A New Condition for Weak Convergence 357

10.4.3 Completing the Proof 362

10.5 Extension of (10.2.6) to <$$> 366

10.5.1 First-Order Limit 366

10.5.2 CLT of Linear Functional of B_p 367

10.6 Proof of Theorem 10.16 368

10.7 Proof of Theorem 10.21 372

10.7.1 An Intermediate Lemma 372

10.7.2 Convergence of the Finite-Dimensional Distributions 373

10.7.3 Tightness of <$$> and Convergence of <$$> 385

10.8 Proof of Theorem 10.23 388

11 Circular Law 391

11.1 The Problem and Difficulty 391

11.1.1 Failure of Techniques Dealing with Hermitian Matrices 392

11.1.2 Revisiting Stieltjes Transformation 393

11.2 A Theorem Establishing a Partial Answer to the Circular Law 396

11.3 Lemmas on Integral Range Reduction 397

11.4 Characterization of the Circular Law 401

11.5 A Rough Rate on the Convergence of v_n(x, z) 409

11.5.1 Truncation and Centralization 409

11.5.2 A Convergence Rate of the Stieltjes Transform of v_n ( , z) 411

11.6 Proofs of (11.2.3) and (11.2.4) 420

11.7 Proof of Theorem 11.4 424

11.8 Comments and Extensions 425

11.8.1 Relaxation of Conditions Assumed in Theorem 11.4 425

11.9 Some Elementary Mathematics 428

11.10 New Developments 430

12 Some Applications of RMT 433

12.1 Wireless Communications 433

12.1.1 Channel Models 435

12.1.2 random matrix channelRandom Matrix Channels 436

12.1.3 Linearly Precoded Systems 438

12.1.4 Channel Capacity for MIMO Antenna Systems 442

12.1.5 Limiting Capacity of Random MIMO Channels 450

12.1.6 A General DS-CDMA Model 452

12.2 Application to Finance 454

12.2.1 A Review of Portfolio and Risk Management 455

12.2.2 Enhancement to a Plug-in Portfolio 460

A SomeResults in Linear Algebra 469

A.1 Inverse Matrices and Resolvent 469

A.1.1 Inverse Matrix Formula 469

A.1.2 Holing a Matrix 470

A.1.3 Trace of an Inverse Matrix 470

A.1.4 Difference of Traces of a Matrix A and its Major Sub-matrices 471

A.1.5 Inverse Matrix of Complex Matrices 472

A.2 Inequalities Involving Spectral Distributions 473

A.2.1 Singular-Value Inequalities 473

A.3 Hadamard Product and Odot Product 480

A.4 Extensions of Singular-Value Inequalities 483

A.4.1 Definitions and Properties 484

A.4.2 Graph-Associated Multiple Matrices 485

A.4.3 Fundamental Theorem on Graph-Associated MMs 488

A.5 Perturbation Inequalities 496

A.6 Rank Inequalities 503

A.7 A Norm Inequality 505

B Miscellanies 507

B.1 Moment Convergence Theorem 507

B.2 Stieltjes Transform 514

B.2.1 Preliminary Properties 514

B.2.2 Inequalities of Distance between Distributions in Terms of Their Stieltjes Transforms 517

B.2.3 Lemmas Concerning Levy Distance 521

B.3 Some Lemmas about Integrals of Stieltjes Transforms 523

B.4 A Lemma on the Strong Law of Large Numbers 526

B.5 A Lemma on Quadratic Forms 530

Relevant Literature 533

Index 547