Spectral Analysis of Large Dimensional Random Matrices

by Bai, Zhidong; Silverstein, Jack W.

• ISBN13: 9781441906601
• ISBN10: 1441906606
• Edition: 2nd
• Format: Hardcover
• Copyright: 2010-04-01
• Publisher: Springer Verlag

Summary

The aim of the book is to introduce basic concepts, main results, and widely applied mathematical tools in the spectral analysis of large dimensional random matrices. The material selection of the book focuses on results established under moment conditions on random variables using probabilistic methods, and is thus easily applicable to statistics and other areas of science. The book introduces fundamental results, most of them investigated by the authors, such as the semicircular law of Wigner matrices, the Marcenko-Pastur law, the limiting spectral distribution of the multivariate F matrix, limits of extreme eigenvalues, spectrum separation theorems, convergence rates of empirical distributions, central limit theorems of linear spectral statistics, and the partial solution of the famous circular law. While deriving the main results, the book simultaneously emphasizes the ideas and methodologies of the fundamental mathematical tools, among them being: truncation techniques, matrix identities, moment convergence theorems, and the Stieltjes transform. Its treatment is especially fitting to the needs of mathematics and statistics graduate students and beginning researchers, having a basic knowledge of matrix theory and an understanding of probability theory at the graduate level, who desire to learn the concepts and tools in solving problems in this area. It can also serve as a detailed handbook on results of large dimensional random matrices for practical users.This second edition includes two additional chapters, one on the authors' results on the limiting behavior of eigenvectors of sample covariance matrices, another on applications to wireless communications and finance. While attempting to bring this edition up-to-date on recent work, it also provides summaries of other areas which are typically considered part of the general field of random matrix theory.

Author Biography

Zhidong Bat is a professor of the School of Mathematics and Statistics at Northeast Normal University and Department of Statistics and Applied Probability at National University of Singapore. He is a Fellow of the Third World Academy of Science and a Fellow of the Institute of Mathematical Statistics. Jack W. Silverstein is a professor in the Department of Mathematics at North Carolina State University. He is a Fellow of the Institute of Mathematical Statistics.

Table of Contents

Preface to the Second Edition	p. vii
Preface to the First Edition	p. ix
Introduction	p. 1
Large Dimensional Data Analysis	p. 1
Random Matrix Theory	p. 4
Spectral Analysis of Large Dimensional Random Matrices	p. 4
Limits of Extreme Eigenvalues	p. 6
Convergence Rate of the ESD	p. 6
Circular Law	p. 7
CLT of Linear Spectral Statistics	p. 8
Limiting Distributions of Extreme Eigenvalues and Spacings	p. 9
Methodologies	p. 9
Moment Method	p. 9
Stieltjes Transform	p. 10
Orthogonal Polynomial Decomposition	p. 11
Free Probability	p. 13
Wigner Matrices and Semicircular Law	p. 15
Semicircular Law by the Moment Method	p. 16
Moments of the Semicircular Law	p. 16
Some Lemmas in Combinatorics	p. 16
Semicircular Law for the iid Case	p. 20
Generalizations to the Non-iid Case	p. 26
Proof of Theorem 2.9	p. 26
Semicircular Law by the Stieltjes Transform	p. 31
Stieltjes Transform of the Semicircular Law	p. 31
Proof of Theorem 2.9	p. 33
Sample Covariance Matrices and the Marčenko-Pastur Law	p. 39
M-P Law for the iid Case	p. 40
Moments of the M-P Law	p. 40
Some Lemmas on Graph Theory and Combinatorics	p. 41
M-P Law for the iid Case	p. 47
Generalization to the Non-iid Case	p. 51
Proof of Theorem 3.10 by the Stieltjes Transform	p. 52
Stieltjes Transform of the M-P Law	p. 52
Proof of Theorem 3.10	p. 53
Product of two Random Matrices	p. 59
Main Results	p. 60
Some Graph Theory and Combinatorial Results	p. 61
Proof of Theorem 4.1	p. 68
Truncation of the ESD of Tn	p. 68
Truncation, Centralization, and Rescaling of the X-variables	p. 70
Completing the Proof	p. 71
LSD of the F-Matrix	p. 75
Generating Function for the LSD of Sn Tn	p. 75
Completing the Proof of Theorem 4.10	p. 77
Proof of Theorem 4.3	p. 80
Truncation and Centralization	p. 80
Proof by the Stieltjes Transform	p. 82
Limits of Extreme Eigenvalues	p. 91
Limit of Extreme Eigenvalues of the Wigner Matrix	p. 92
Sufficiency of Conditions of Theorem 5.1	p. 93
Necessity of Conditions of Theorem 5.1	p. 101
Limits of Extreme Eigenvalues of the Sample Covariance Matrix	p. 105
Proof of Theorem 5.10	p. 106
Proof of Theorem 5.11	p. 113
Necessity of the Conditions	p. 113
Miscellanies	p. 114
Spectral Radius of a Nonsymmetric Matrix	p. 114
TW Law for the Wigner Matrix	p. 115
TW Law for a Sample Covariance Matrix	p. 117
Spectrum Separation	p. 119
What is Spectrum Separation?	p. 119
Mathematical Tools	p. 126
Proof of (1)	p. 128
Truncation and Some Simple Facts	p. 128
A Preliminary Convergence Rate	p. 129
Convergence of Sn - Esn	p. 139
Convergence of the Expected Value	p. 144
Completing the Proof	p. 148
Proof of (2)	p. 149
Proof of (3)	p. 151
Convergence of a Random Quadratic Form	p. 151
Dependence on y	p. 157
Completing the Proof of (3)	p. 160
Semicircular Law for Hadamard Products	p. 165
Sparse Matrix and Hadamard Product	p. 165
Truncation and Normalization	p. 168
Truncation and Centralization	p. 169
Proof.of Theorem 7.1 by the Moment Approach	p. 172
Convergence Rates of ESD	p. 181
Convergence Rates of the Expected ESD of Wigner Matrices	p. 181
Lemmas on Truncation, Centralization, and Rescaling	p. 182
Proof of Theorem 8.2	p. 185
Some Lemmas on Preliminary Calculation	p. 189
Further Extensions	p. 194
Convergence Rates of the Expected ESD of Sample Covariance Matrices	p. 195
Assumptions and Results	p. 195
Truncation and Centralization	p. 197
Proof of Theorem 8.10	p. 198
Some Elementary Calculus	p. 204
Increment of M-P Density	p. 204
Integral of Tail Probability	p. 206
Bounds of Stieltjes Transforms of the M-P Law	p. 207
Bounds for <$$>	p. 209
Integrals of Squared Absolute Values of Stieltjes Transforms	p. 212
Higher Central Moments of Stieltjes Transforms	p. 213
Integral of ¿	p. 217
Rates of Convergence in Probability and Almost Surely	p. 219
CLT for Linear Spectral Statistics	p. 223
Motivation and Strategy	p. 223
CLT of LSS for the Wigner Matrix	p. 227
Strategy of the Proof	p. 229
Truncation and Renormalization	p. 231
Mean Function of Mn	p. 232
Proof of the Nonrandom Part of (9.2.13) for j = l, r	p. 238
Convergence of the Process Mn - EMn	p. 239
Finite-Dimensional Convergence of Mn - EMn	p. 239
Limit of S1	p. 242
Completion of the Proof of (9.2.13) for j = l, r	p. 250
Tightness of the Process Mn(z) û EMn(z)	p. 251
Computation of the Mean and Covariance Function of G(f)	p. 252
Mean Function	p. 252
Covariance Function	p. 254
Application to Linear Spectral Statistics and Related Results	p. 256
Tchebychev Polynomials	p. 256
Technical Lemmas	p. 257
CLT of the LSS for Sample Covariance Matrices	p. 259
Truncation	p. 261
Convergence of Stieltjes Transforms	p. 263
Convergence of Finite-Dimensional Distributions	p. 269
Tightness of <$$>	p. 280
Convergence of <$$>	p. 286
Some Derivations and Calculations	p. 292
Verification of (9.8.8)	p. 292
Verification of (9.8.9)	p. 295
Derivation of Quantities in Example (1.1)	p. 296
Verification of Quantities in Jonsson's Results	p. 298
Verification of (9.7.8) and (9.7.9)	p. 300
CLT for the F-Matrix	p. 304
CLT for LSS of the F-Matrix	p. 306
Proof of Theorem 9.14	p. 308
Lemmas	p. 308
Proof of Theorem 9.14	p. 318
CLT for the LSS of a Large Dimensional Beta-Matrix	p. 325
Some Examples	p. 326
Eigenvectors of Sample Covariance Matrices	p. 331
Formulation and Conjectures	p. 332
Haar Measure and Haar Matrices	p. 332
Universality	p. 335
A Necessary Condition for Property 5'	p. 336
Moments of <$$>	p. 339
Proof of (10.3.1) → (10.3.2)	p. 340
Proof of (b)	p. 341
Proof of (10.3.2) → (10.3.1)	p. 341
Proof of (c)	p. 349
An Example of Weak Convergence	p. 349
Converting to D[0, ∞)	p. 350
A New Condition for Weak Convergence	p. 357
Completing the Proof	p. 362
Extension of (10.2.6) to <$$>	p. 366
First-Order Limit	p. 366
CLT of Linear Functional of Bp	p. 367
Proof of Theorem 10.16	p. 368
Proof of Theorem 10.21	p. 372
An Intermediate Lemma	p. 372
Convergence of the Finite-Dimensional Distributions	p. 373
Tightness of <$$> and Convergence of <$$>	p. 385
Proof of Theorem 10.23	p. 388
Circular Law	p. 391
The Problem and Difficulty	p. 391
Failure of Techniques Dealing with Hermitian Matrices	p. 392
Revisiting Stieltjes Transformation	p. 393
A Theorem Establishing a Partial Answer to the Circular Law	p. 396
Lemmas on Integral Range Reduction	p. 397
Characterization of the Circular Law	p. 401
A Rough Rate on the Convergence of vn(x, z)	p. 409
Truncation and Centralization	p. 409
A Convergence Rate of the Stieltjes Transform of vn ( , z)	p. 411
Proofs of (11.2.3) and (11.2.4)	p. 420
Proof of Theorem 11.4	p. 424
Comments and Extensions	p. 425
Relaxation of Conditions Assumed in Theorem 11.4	p. 425
Some Elementary Mathematics	p. 428
New Developments	p. 430
Some Applications of RMT	p. 433
Wireless Communications	p. 433
Channel Models	p. 435
random matrix channelRandom Matrix Channels	p. 436
Linearly Precoded Systems	p. 438
Channel Capacity for MIMO Antenna Systems	p. 442
Limiting Capacity of Random MIMO Channels	p. 450
A General DS-CDMA Model	p. 452
Application to Finance	p. 454
A Review of Portfolio and Risk Management	p. 455
Enhancement to a Plug-in Portfolio	p. 460
SomeResults in Linear Algebra	p. 469
Inverse Matrices and Resolvent	p. 469
Inverse Matrix Formula	p. 469
Holing a Matrix	p. 470
Trace of an Inverse Matrix	p. 470
Difference of Traces of a Matrix A and its Major Sub-matrices	p. 471
Inverse Matrix of Complex Matrices	p. 472
Inequalities Involving Spectral Distributions	p. 473
Singular-Value Inequalities	p. 473
Hadamard Product and Odot Product	p. 480
Extensions of Singular-Value Inequalities	p. 483
Definitions and Properties	p. 484
Graph-Associated Multiple Matrices	p. 485
Fundamental Theorem on Graph-Associated MMs	p. 488
Perturbation Inequalities	p. 496
Rank Inequalities	p. 503
A Norm Inequality	p. 505
Miscellanies	p. 507
Moment Convergence Theorem	p. 507
Stieltjes Transform	p. 514
Preliminary Properties	p. 514
Inequalities of Distance between Distributions in Terms of Their Stieltjes Transforms	p. 517
Lemmas Concerning Levy Distance	p. 521
Some Lemmas about Integrals of Stieltjes Transforms	p. 523
A Lemma on the Strong Law of Large Numbers	p. 526
A Lemma on Quadratic Forms	p. 530
Relevant Literature	p. 533
Index	p. 547
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