Theory of Stochastic Canonical Equations: Volumes I and II

1009 Pages·2001·28.507 MB·English

by  Vyacheslav L. Girko (auth.)

Theory of Stochastic Canonical Equations
Volume I
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 535
Theory of Stochastic
Canonical Equations
Volume 1
by
Vyacheslav L. Girko
Michigan State University,
East Lansing, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-3882-9 ISBN 978-94-010-0989-8 (eBook)
DOI 10.1007/978-94-010-0989-8
AII Rights Reserved
© 2001 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2001
Softcover reprint ofthe hardcover lst edition 2001
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
CONTENTS
List of basic notations and assumptions xv
How the stochastic canonical equation was found XIX
Chapter 1. Canonical equation K I 1
1.1. Main assertion 1
1.2. Perturbation formulas for the entries of resolvent of a matrix 3
1.3. Strong Law for normalized spectral functions of random matrix.
The method of martingale differences 4
1.4. Limit theorem for random quadratic forms 8
1.5. Inequalities for the entries of the resolvents of random matrices 10
1.6. Limit theorem for a sum of random entries multiplied by diagonal
entries of the resolvents of random matrices 11
1.7. Proof of the limit theorem for the sum of diagonal entries of the
resolvents of random matrices by the method of martingale differences 13
1.8. Main inequality. Accompanying system of canonical equations Kl 13
1.9. Existence of solution of the system of canonical equations Kl 14
1.10. Uniqueness of the solution of the system of canonical
equations KI 16
1.11. Existence of the densities of accompanying normalized spectral
functions. The completion of the proof of Theorem 1.1 17
1.12. Limit theorem for individual spectral functions 18
1.13. Strong Law for individual spectral functions of random
symmetric matrices 19
1.14. Weak Law for random matrices 22
1.15. Canonical equation Kl for sparse random symmetric matrices 23
Chapter 2. Canonical equation K2. Necessary and sufficient modified
Lindeberg's condition. The Wigner and Cubic laws 25
2.1. Formulation of the main assertion 25
2.2. Invariance principle for the entries of the resolvents of random matrices 26
2.3. Equation Ml for the trace of the resolvent of a random
symmetric matrix 34
2.4. Solvability of the accompanying equation Ll 36
2.5. Proof of the existence of the density of the accompanying normalized
spectral function based on the unique solvability of the
spectral equation Ll 37
2.6. Uniform inequality for normalized spectral functions of random
VI Contents
symmetric matrices. Completion of the proof of the main assertion 40
2.7. Canonical equation K2 for individual spectral functions 40
2.8. Canonical equation K2. Modified Lindeberg condition
for the Wigner Semicircle Law 41
2.9. Canonical equation K2. Necessary and sufficient modified Lindeberg
condition for the Wigner Semicircle Law 42
2.10. Canonical equation K2. Sufficient condition for the
Cubic Law. Limit cubic density for two different eigenvalues
of a nonrandom matrix 48
Chapter 3. Regularized stochastic canonical equation K3 for symmetric
random matrices with infinitely small entries 51
3.1. Main theorem for ACE-matrices 51
3.2. Limit theorem for random nonnegative definite quadratic forms 53
3.3. Accompanying random infinitely divisible law for random 57
quadratic forms
3.4. Self-averaging of accompanying random infinitely divisible law 60
3.5. Limit Theorem for perturbed diagonal entries of resolvents 63
3.6. Limit theorem for the sum of random entries multiplied by diagonal
entries of a resolvents 64
3.7. Accompanying random infinitely divisible law for the sum
of random entries 65
3.8. Method of martingale differences in the proof of the limit theorem
for random quadratic forms 65
3.9. Method of the regularization of the resolvents of ACE-matrices 68
3.10. Vanishing of the imaginary parts of the entries of the resolvents
of ACE-matrices 69
3.11. Accompanying regularized stochastic canonical equation K 3 71
3.12. Uniqueness of the solution of the accompanying regularized
stochastic canonical equation K3 72
3.13. Method of successive approximations for the solution of the
accompanying regularized stochastic canonical equation 74
Chapter 4. Stochastic canonical equation K4 for symmetric
random matrices with infinitely small entries. Necessary and sufficient
conditions for the convergence of normalized spectral functions 75
4.1. Stochastic equation K4 with a random functional of a special form 75
4.2. Limit theorems for random spectral functions. The case
of weak convergence of spectral functions to a random process 77
4.3. Stochastic canonical equation K4 85
4.4. Limit theorem for the individual spectral functions 86
Chapter 5. Canonical equation K5 for symmetric random matrices with
infinitely small entries 87
5.1. Degenerate random linear functional 87
Contents vii
5.2. Limit theorem for individual spectral functions 88
5.3. Canonical equation K4. Necessary and sufficient conditions
for the Wigner semicircle law 89
Chapter 6. Canonical equation K6 for symmetric random matrices with
identically distributed entries 93
6.1. Random symmetric matrices whose entries belong to the region
of attraction of a stable law 93
6.2. Stable stochastic canonical equation K6 95
6.3. The case where the random entries belong to the domain
of attraction of the stable law with parameter a = 1/2 96
6.4. Stable stochastic canonical equation K6 for individual
spectral functions of random symmetric matrices 96
Chapter 7. Canonical equation K7 for Gram random matrices 97
7.1. Canonical equation K7 for Gram random matrices, whose entries
have bounded variances 97
7.2. Limit theorems for the entries of the resolvent of random matrices 99
7.3. Limit theorems for random quadratic forms 106
7.4. Asymptotics of randomly normalized resolvent of random matrices 107
7.5. Perturbation formulas for the resolvent of random matrices 107
7.6. Inequalities for the entries of the resolvent of random matrices 109
7.7. Analytic continuation of the entries of the resolvents of
random matrices 112
7.8. Derivation of the system of canonical equations for the
entries of the resolvents of random matrices 113
7.9. Proof of the unique solvability of the system of canonical equations K7 114
7.10. Convergence of the solution of the accompanying system of canonical
equations to the solution of the system of canonical equations K7 116
7.11. Canonical equation K7 for the Gram random matrix whose entries
have variances satisfying the double stochastic condition 117
Chapter 8. Canonical equation Kg 119
8.1. Limit theorem in the case where Lindeberg's condition is satisfied 119
8.2. Canonical equation Kg for random Gram matrix, whose entries have
variances satisfying the double stochastic condition 122
8.3. Canonical equation Kg for random symmetric matrices some entries
of which have equal variances 124
Chapter 9. Canonical equation Kg for random matrices whose
entries have identical variances 125
9.1. The case where normalized spectral functions of nonrandom
matrices converge to certain distribution functions 125
9.2. The case where the entries of nonrandom matrices are equal to zero 126
9.3. Rate of convergence of expected spectral functions of the sample
Vlll Contents
covariance matrix Rmn (n) is equal to O(n-l/2) under the condition
that mnn-1 s: c < 1 127
Chapter 10. Canonical equation KlO. Necessary and sufficient
modified Lindeberg condition 129
10.1. Limit theorem for normalized spectral functions of random
matrices with expectation equal to zero 129
10.2. Cubic Law for random Gram matrices 130
10.3. Monte-Carlo simulations 131
10.4. Necessary and sufficient condition for the convergence of normalized
spectral functins to Bronk-Marchenko-Pastur (BMP) density
in the case where the double stochastic condition is satisfied 131
10.5. Necessity of the modified Lindeberg condition for the
convergence of normalized spectral functions of random matrices
to the BMP distribution 134
10.6. Accompanied infinitely divisible distributions for the sum
of independent random variables 135
Chapter 11. Canonical equation Kl1. Limit theorem for normalized
spectral functions of empirical covariance matrices under the
modified Lindeberg condition 141
11.1. Accompanying equations for the densities of normalized
spectral functions 141
11.2. Canonical equation Kl1. Bronk-Marchenko-Pastur density 150
11.3. Canonical equation Kl1. Cubic Density 150
11.4. Canonical Equation Kl1. Simulation technique 152
Chapter 12. Canonical Equation K12 for random Gram matrices
with infinitely small entries 153
12.1. Fundamental Result 153
12.2. Limit Theorem for random nonnegative definite quadratic forms 155
12.3. The method of martingale differences in the proof of the limit
theorem for random quadratic forms 158
12.4. The Method of regularization of the resolvents of random matrices 159
Chapter 13. Canonical Equation K13 for random Gram matrices
with infinitely small entries 161
13.1. Stochastic canonical equation with random functional of
different form 161
13.2. Limit theorem for individual spectral functions of random
Gram matrices 163
Chapter 14. The method of random determinants for estimating the
permanents of matrices and the canonical equation K14 for random
Gram matrices 165
14.1. Main assertion 165
Contents IX
14.2. Limit theorem for individual spectral functions of random
Gram matrices 166
14.3. The method of random determinants for estimating the
permanents of matrices 167
14.4. The method of random determinants 168
14.5. The invariance principle for nonrandom permanents 179
Chapter 15. Canonical Equation K for random Gram matrices
15
with identically distributed entries 181
15.1. Stable canonical equation 181
15.2. Stable stochastic canonical equation K15 182
15.3. Limit theorem for individual spectral functions 182
15.4. Limit theorem for eigenvalues of random Gram matrices when
Lindeberg's condition is not fulfilled. Stochastic power method 183
Chapter 16. Canonical Equation K16 for sample covariance matrices 185
16.1. Canonical equation K16 185
16.2. Conditional canonical equation K16 187
16.3. Canonical equation K16 for sample covariance matrices 187
16.4. Canonical equation K16 for random matrices with special structure 189
16.5. Proof of Theorem 16.3 190
16.6. Substitution of the mean vector for an empirical mean vector 190
16.7. Self-averaging of normalized spectral functions 190
16.8. Method of shortening of entries of empirical covariance matrices 192
16.9. Self-averaging of random quadratic forms 193
16.10. Proof of the uniqueness of the solution of the
conditional canonical equation 196
16.11. Proof of the existence of a solution of the conditional
canonical equation 199
16.12. Substitution of a solution of the conditional canonical equation
for an empirical covariance matrix 200
16.13. Proof of the main assertion 202
Chapter 17. Canonical Equation K17 for identically distributed
independent vector observations and the G2-estimators of the real
Stieltjes transforms of the normalized spectral functions of the
covariance matrices 203
17.1. Identically distributed independent observations 203
17.2. Limit theorem for individual spectral functions of empirical
covariance matrices 204
17.3. G -estimator of the real Stieltjes transforms of the normalized
2
spectral functions of covariance matrices 204
17.4. G -estimators of the complex Stieltjes transforms of the normalized
2
spectral functions of covariance matrices 205
17.5. Modified G -estimator 206
2
x Contents
Chapter 18. Canonical equation K for the special structure
18
of vector observations 207
18.1. Canonical equations for observations with special structure 207
18.2. Simpler canonical equation for observations with special structure 208
18.3. The case of the identity covariance matrix 210
18.4. Canonical equation K for the special structure of observations 211
18
18.5. Accompanying Canonical equation K for the densities of the
18
normalized spectral functions of empirical covariance matrices 212
18.6. Invariance principle for normalized spectral functions
of empirical covariance matrices 213
18.7. Existence and uniqueness of the solution of the canonical
Equation KI8 221
Chapter 19. Canonical equation K 225
I9
19.1. G-equations for estimators of differentiable functions of unknown
parameters 225
19.2. G-equation of higher orders 227
19.3. G-equation for functions of the empirical vector of expectations
and the covariance matrix 228
19.4. G-equation for functions of empirical expectations 229
19.5. Estimator G of regularized function of unknown parameters 230
I9
Chapter 20. Canonical equation K20. Strong law for normalized spectral
functions of nonselfadjoint random matrices with independent row
vectors. Simple rigorous proof of the strong Circular law 231
20.1. Modified V-transform of spectral functions 232
20.2. Inverse formula for the modified V-transform 233
20.3. Strong law for normalized spectral functions of nonselfadjoint
random matrices with independent row vectors 233
20.4. Method of perpendiculars for proving the strong circular law 242
20.5. Substitution of the determinant of a Gram matrix for the
determinant of a random matrix 243
20.6. Regularized modified V-transform for a spectral function 248
20.7. Canonical equation K20. Estimate of the rate of convergence
of the Stieltjes transformation of spectral functions 252
20.8. Rigorous proof of the strong circular law 259
Chapter 21. Canonical equation K2I for random matrices
with independent pairs of entries with zero expectations.
Circular and Elliptic laws 261
21.1. Basic Equation 261
21.2. Elliptic Law 262
21.3. Spectral functions and G functions 263
21.4. Modified V-transform of spectral and G-functions 263
21.5. Truncated conditional V I-transform and V 2-transform 264