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Item Detail

Theory of Stochastic Canonical Equations

Author(s): Girko, Vyacheslav L.

ISBN10: 1402000758

ISBN13: 9781402000751

Cover: Hardcover

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Theory of Stochastic Canonical Equations collects the major results of thirty years of the author's work in the creation of the theory of stochastic canonical equations. It is the first book to completely explore this theory and to provide the necessary tools for dealing with these equations. Included are limit phenomena of sequences of random matrices and the asymptotic properties of the eigenvalues of such matrices. The book is especially interesting since it gives readers a chance to study proofs written by the mathematician who discovered them. All fifty-nine canonical equations are derived and explored along with their applications in such diverse fields as probability and statistics, economics and finance, statistical physics, quantum mechanics, control theory, cryptography, and communications networks. Some of these equations were first published in Russian in 1988 in the book Spectral Theory of Random Matrices, published by Nauka Science, Moscow. An understanding of the structure of random eigenvalues and eigenvectors is central to random matrices and their applications. Random matrix analysis uses a broad spectrum of other parts of mathematics, linear algebra, geometry, analysis, statistical physics, combinatories, and so forth. In return, random matrix theory is one of the chief tools of modern statistics, to the extent that at times the interface between matrix analysis and statistics is notably blurred. Volume I of Theory of Stochastic Canonical Equations discusses the key canonical equations in advanced random matrix analysis. Volume II turns its attention to a broad discussion of some concrete examples of matrices. It contains in-depth discussion of modern, highly-specialized topics in matrix analysis, such as unitary random matrices and Jacoby random matrices. The book is intended for a variety of readers: students, engineers, statisticians, economists and others.

List of basic notations and assumptions

xix

Preface to the second volume

xxiii

Canonical equation K31 for normalized spectral functions of the sum of random Gram block matrix and nonrandom matrix

(8)

Nonsymmetric matrices with independent random blocks

(2)

Canonical equation K31 in the case where random blocks have zero expectations and are identically distributed

(1)

Canonical equation K31. Limit theorems for normalized spectral functions of random matrices with asymptotically independent blocks

(3)

Canonical equation K31 in the case where random entries have zero expectations

(2)

Canonical equation K32 for normalized spectral functions of random Gram matrices with identically distributed independent blocks. Block matrix density

(6)

Block Gram random matrices whose blocks have nonzero expectations and are identically distributed

(2)

Canonical equation K32 for normalized spectral functions of a nonrandom matrix and Gram random matrices whose blocks have nonzero expectations and are identically distributed

(1)

Block Gram random matrices that have zero expectations and are identically distributed

(1)

Block density for block Gram random matrices which have zero expectations

(1)

Convergency of normalized spectral functions of block Gram random matrices to the distribution with block ``One Quarter Law'' density

(2)

Canonical equation K33 for the Fourier transform of the resolvent of a Gram block random matrix

(10)

Block Gram matrices with stationary (in wide sense) random entries

(1)

The boundedness of the norms of the row vectors of the matrix solution of the equation K32

(1)

The asymptotic stationary state of the entries of the matrix solution of the equation K32

(1)

The asymptotics of the normalized traces of the matrix solution of the equation K32

(3)

Description of the limit normalized spectral functions of random matrices with stationary (in wide sense) entries with the help of the canonical equation K33

(2)

Description of limit normalized spectral functions of random matrices with stationary (in wide sense) entries that have zero expectations

(2)

Canonical equation K34 for normalized spectral functions of empirical covariance matrix with asymptotically independent blocks

(8)

A sample of dependent observations of a random vector

(1)

Method of thinning empirical covariance matrices: block empirical covariance matrices

(1)

Condition of asymptotic independence of observations

(1)

Canonical equation K34 for the resolvent of the block empirical covariance matrix

(4)

Canonical equation K34 for the normalized spectral function of an empirical covariance matrix with identically distributed blocks

(2)

Canonical equation K35 for normalized spectral functions of a pencil of random matrices

(12)

Normalized spectral function of nonsingular covariance matrices

(1)

Normalized spectral function of a pencil of empirical covariance matrices

(1)

Main assertion

(2)

Regularized Stieltjes transform

(1)

Elimination of the empirical means from the regularized Stieltjes transform

(1)

Limit in mean for regularized Stieltjes transform

(1)

Invariance principle for a pencil of random matrices

(1)

Limit theorem for regularized Stieltjes transform

(3)

Existence and uniqueness of the solution of the canonical equation K35

(1)

Elimination of the regularization parameter

(1)

Integral representation of the Stieltjes transform on the basis of the normalized trace of the resolvent with regularization parameter

(1)

Convergency of the derivative of the normalized trace of the resolvent with regularization parameter to the derivative of the solution of the canonical equation

(3)

Canonical equation K36 for normalized spectral functions of a pencil of random matrices

(8)

Sample of observations of random vectors with identity covariance matrix

(1)

Main assertion

(1)

Asymptotic density of eigenvalues of a pencil of random matrices

(1)

Limit theorem for normalized spectral functions of a pencil of random matrices

(1)

Calculations of an integral of a nonlinear function

(5)

Canonical equation K37 for normalized spectral functions of a pencil of empirical random matrices

(4)

Sample of observations of a certain random vector

(2)

Large number law for the normalized spectral functions of a pencil of random matrices

(1)

Matrix canonical equation for a pencil of random matrices

(1)

Main assertion

(1)

Canonical equation K38 for normalized spectral functions of a pencil of random nonsymmetric matrices. G-law

(12)

Main assertion

(2)

The setting of the problem for random determinants

(1)

The method of normal random regularization

(4)

Proof of the Logarithmic law

(3)

G-Law

(2)

Twenty five years of stochastic canonical equation K39 for normalized spectral functions of ACE-symmetric matrices

(34)

General formulation of the problem of describing all possible distributions of normalized spectral functions of ACE-symmetric matrices

(1)

The case where the variances of random entries exist but the Lindeberg condition is not satisfied. Some auxiliary formulas

(2)

Limit theorems for random quadratic forms

(4)

Accompanying system of stochastic equations where the variances of random entries are bounded

(1)

A weak convergence of a sum of random variables to a random linear functional

(1)

The replacement of the sum of random variables in the accompanying system of stochastic equations by random functionals

(3)

The replacement of the sum of random variables by random functionals under general conditions

(1)

Proof of the existence of a solution of the canonical system of stochastic equation K39

(2)

The problem of choosing of random linear functional

(1)

The convergence of the solution of the accompanying system of canonical stochastic equations to the canonical system of stochastic equations K39

(1)

The choice of normalization constants for the entries of random matrices. Formulation of the problem

(1)

The replacement of the entries of random ACE-symmetric matrices by infinitely divisible random variables

(1)

General limit theorem for normalized spectral functions of ACE-symmetric random matrices

(3)

Limit theorem for random non negative defined quadratic forms

(1)

Limit theorem for perturbed diagonal entries of the resolvent of random matrix

(1)

Martingale differences method for the proof of limit theorem for the random quadratic forms

(2)

Method of regularization of resolvents of random matrices

(2)

Proof of the existence of the solution of the canonical system of stochastic equations K38

(1)

Proof of the uniqueness of a solution of the canonical system of stochastic equations K39

(1)

The problem of choosing of a random linear functional

(1)

The convergence of the solution of the accompanying system of canonical stochastic equations to the solution of the canonical system of stochastic equations K39

(2)

System of canonical stochastic equations for degenerate random functionals

100

(1)

System of canonical stochastic equations with stable random functionals

101

(2)

Twenty five years of stochastic canonical equation K40 for normalized spectral functions of ACE-Gram matrices

103

(56)

General formulation for the problem of describing all possible distributions of normalized spectral functions of random Gram matrices with asymptotically negligible entries

103

(1)

The case when the variances of random entries exist but Lindeberg condition is not fulfilled. Main assertion

104

(2)

The main auxiliary perturbation formulas for symmetric and Gram matrices. Limit theorems for the entries of the resolvent of random matrices

106

(9)

Asymptotic behavior of random quadratic forms

115

(1)

Perturbation formulas for the resolvent of random matrices

116

(1)

Inequalities for the entries of the resolvent of random matrices

116

(4)

Analytic continuation of the entries of the resolvents of random matrices

120

(1)

Derivation of the accompanying system of canonical equations for the entries of the resolvents of random matrices when the variances of random entries are bounded

121

(1)

Accompanying random linear functionals

122

(1)

A weak convergence of the sum of random variables to a random linear functionals

123

(1)

The replacement of the sum of random variables in the accompanying system of stochastic equations by random functionals

124

(4)

The replacement of the sum of random variables under general conditions

128

(1)

The problem of choosing of random linear functional

129

(2)

Proof of the existence of the solution of the canonical system of stochastic equations

131

(2)

The convergence of the solution of the accompanying system of canonical stochastic equations to the canonical system of stochastic equations

133

(1)

The general formulation of the problem. The choice of normalized constant for the entries of random matrices

134

(1)

The replacement of the entries of random Gram matrices by infinitely divisible random variables

135

(1)

General theorem of normalized spectral functions of ACE-Gram matrices

136

(3)

Limit theorem for random nonnegative definite quadratic forms

139

(1)

Limit Theorem for perturbed diagonal entries of resolvents

140

(1)

Limit theorem for the sum of random entries multiplied by diagonal entries of resolvents

141

(1)

Accompanying random infinitely divisible law for the sum of random entries

142

(1)

Method of martingale differences in the proof of the limit theorem for random quadratic forms

142

(3)

A weak convergence of the sum of random variables to random linear functionals

145

(1)

Limit theorem for perturbed diagonal entries of the resolvent of random matrix

146

(1)

The method of the regularization of the resolvents of random matrices

146

(2)

The problem of choosing of random linear functionals

148

(2)

Proof of the existence of the solution of the canonical system of stochastic equations

150

(1)

Proof of the uniqueness of the solution of the canonical system of stochastic equation

150

(2)

Method of successive approximations for the solution of the accompanying regularized stochastic canonical equation

152

(1)

The convergence of the solution of the accompanying system of canonical stochastic equations to the canonical system of stochastic equations

153

(1)

The system of canonical stochastic equations for the stable random functionals

154

(2)

Canonical Equation for random Gram matrices with identically distributed entries. Stable canonical equation

156

(1)

Stable stochastic canonical equation K15

157

(1)

Limit theorem for individual spectral functions

158

(1)

Stochastic canonical equation K41 for normalized spectral functions of empirical covariance matrices

159

(4)

A sample of independent observations of a random vector for which the Lindeberg condition is not satisfied for their components

159

(1)

Stieltjes transforms of individual spectral functions of empirical covariance matrices

160

(1)

Assumptions on a sample of observations

160

(1)

Accompanying system of canonical equations

161

(1)

System of canonical stochastic equations

161

(2)

Stochastic canonical equation K42 for normalized spectral functions of random symmetric matrices with block structure

163

(6)

Statement of the problem

163

(1)

Block matrices

163

(1)

Method of thinning matrices: block matrices

164

(1)

Discussion of conditions on the random blocks of a matrix

165

(1)

Main assertion. Canonical equation K42

165

(1)

Main assertion for random block matrices in the case where the expectations of random blocks do not exist

166

(3)

Stochastic canonical equation K43 for normalized spectral functions of random Gram block matrices

169

(4)

Nonsymmetric matrices with independent random blocks

169

(1)

Accompanying random block diagonal matrices

170

(1)

Main assertion

171

(2)

Stochastic canonical equation K44 for normalized spectral functions of empirical covariance matrices with block structure

173

(4)

Block empirical covariance matrices

173

(1)

Block empirical covariance matrices with identically distributed random blocks in every series of observations

174

(1)

Canonical equation for distribution functions

175

(2)

Stochastic canonical equation K45 for normalized spectral functions of random matrices pencil

177

(10)

Normalized spectral function of nonsingular covariance matrices

177

(1)

Main assertion

177

(3)

Integral representation of the Stieltjes transform on the basis of the normalized trace of the resolvent with regularization parameter

180

(1)

The inequality for the regularized Stieltjes transform

180

(1)

Elimination of empirical means in the regularized Stieltjes transform

181

(1)

Limit theorem for the regularized Stieltjes transform

181

(1)

Elimination of the regularization parameter

182

(2)

Convergence of the derivative of the normalized trace of the resolvent with regularization parameter to the derivative of the solution of the canonical equation

184

(1)

Stable stochastic canonical equation K45

185

(2)

Canonical equation K46 for the Stieltjes transform of normalized spectral functions of tridiagonal and Jacobi random matrices

187

(16)

The main assertion for normalized spectral functions for tridiagonal random matrices with identically distributed vectors of their entries

187

(1)

Replacement tridiagonal random matrix by sysmmetric one

188

(1)

Self-averaging of normalized spectral functions of tridiagonal random matrices

189

(1)

Method of shortening of the entries of tridiagonal random matrices

189

(2)

A useful inequality

191

(1)

A convergence of the entries of resolvents of tridiagonal random matrices

192

(1)

Proof of the existence of the solution of canonical equation K46

193

(1)

Proof of the uniqueness of the solution of the canonical equation K46

194

(3)

Stochastic Sturm-Liouville problem

197

(1)

The integral representation for the normalized logarithm of the determinant of tridiagonal matrix

198

(1)

Integral representation for the Stieltjes transform of spectral function of stochastic Sturm-Liouville problem

199

(2)

Integral representation for solutions of differential equations of the second order

201

(2)

Class of direct canonical equation K47 for spectral functions of random symmetric banded matrices and Jacobi matrices

203

(12)

The main assertion for normalized spectral functions for symmetric banded random matrices with identically distributed vectors of their entries

203

(2)

Sturm oscillation theorem

205

(1)

Canonical equation K47 for limit spectral functions of banded random matrices

205

(5)

Canonical equation K47 for limit normalized spectral functions of symmetric random tridi agonal matrices

210

(1)

Dyson canonical equation K47 for limit normalized spectral functions of symmetric random Jacobi matrices

211

(1)

One example of a solution of equation K47. Arcsine distribution for limit normalized spectral functions of a nonrandom Jacobi matrices

212

(1)

One example of a solution of equation K47 for symmetrical matrices perturbed by diagonal matrices with random diagonal entries distributed by Cauchy law

212

(3)

Canonical equation K48 for normalized spectral functions of product of random matrices

215

(4)

Hermitian Matrizant

215

(1)

Main Assertion

215

(1)

Strong Law for the Hermitian matrizant

216

(1)

Invariance principle for the Hermitian matrizant

217

(1)

Derivation of the canonical equation K48 for Hermitian matrizant

217

(2)

Canonical equation K49 for normalized spectral functions of a product of random unitary matrices

219

(6)

Unitary matrizant and its normalized spectral function

219

(1)

Modified Stieltjes transform for the normalized spectral function of a unitary matrizant

219

(2)

Strong law for the unitary matrizant

221

(1)

Invariance principle for the unitary matrizant

222

(1)

Derivation of the canonical equation K49 for unitary matrizant

223

(2)

Class of canonical equation K50 for the entries of random S-matrices

225

(12)

Stochastic scattering matrix

225

(1)

Canonical equation K50 for the stochastic S-matrix

226

(5)

Asymptotic behavior of the nondiagonal entries of the random S-matrix

231

(1)

The integral representation for the S-matrix

232

(1)

The regularized integral representation for the S-matrix

233

(1)

Canonical equation K50 for the stochastic S-matrix with different variances of their entries

233

(4)

Canonical equation K51 for normalized spectral functions of a product of random independent matrices

237

(14)

G-stochastic matrizant of increasing dimension

237

(1)

Modified V-transform for the normalized spectral function of the stochastic matrizant

238

(1)

Strong law for normalized spectral functions of the product of two independent matrices with independent entries

238

(2)

Existence of the expected logarithm of the determinant of the G-stochastic matrizant

240

(1)

Regularized logarithm of the determinant of the G-stochastic matrizant

240

(1)

Regularized V-transform

241

(1)

Strong law for the G-stochastic matrizant

242

(1)

Invariance principle for the G-stochastic matrizant

243

(1)

Derivation of the canonical equation K51 for the stochastic matrizant

243

(5)

An example of a stochastic matrizant

248

(3)

Canonical equation K52 for Hankel and Toeplitz random matrices

251

(14)

Limit theorem of the type of the law of large numbers

251

(3)

Method of integral representation for the determinants of Hankel random matrices

254

(4)

Stochastic analog of the Szego theorem

258

(3)

Method of perturbation for determinants of some Hankel and Toeplitz random matrices

261

(4)

The class of canonical equation K53 for the solutions of the system of linear algebraic equations with random coefficients. Inverse tangent and canonical laws

265

(46)

Formulation of the problem. Large dimensional SLAERC around us

266

(1)

The classical least squares method

267

(1)

The stochastic least squares method

268

(1)

The inverse tangent law

269

(1)

Canonical law

270

(4)

Stochastic Leontief model

274

(1)

The first Victory (V-transform or the method of Hermitization) based on the integral representation for determinants

274

(1)

Limit theorem for random determinants

275

(2)

Victory-transform (integral representation method or the method of Hermitization) for Solutions of SLAERC

277

(1)

Simulation in linear algebra. The G-formula for calculation of a determinant without the Gauss algorithum

278

(1)

Simulation in linear algebra. The G-formula for calculation of a solution of linear algebraic equations without the Gauss algorithm

279

(1)

Formulation of the problem

279

(1)

Canonical equation K53 for the solutions of a system of linear algebraic equations with independent random coefficients

280

(2)

G-conditions of the theory of stochastic canonical equations

282

(1)

V1-transform for solutions of SLAE

282

(1)

V2-transform for solutions of SLAE

283

(1)

V3-finite increment transform for solutions of SLAE

283

(1)

REFORM Method

283

(1)

Limit theorems for entries of the resolvent of random matrices

284

(8)

Analytic continuation of entries of resolvents

292

(1)

Calculation of the derivative of a resolvent of a random matrix

293

(1)

The main assertion

294

(1)

The canonical equation K7

295

(3)

The SLAERC with special structure of a matrix of coefficients

298

(1)

Canonical equation K53 for the Solution of SLAERC whose coefficients have identity variances

299

(1)

Stochastic canonical equation K53 for the solution of SLAERC with symmetric matrix of coefficients

300

(1)

Stochastic canonical equation K53

301

(2)

Canonical equation K53 for the solution of SLAERC with independent symmetric block structure

303

(1)

Canonical equation K53 for the solution of SLAERC with block structure

304

(1)

Canonical equation K53 for the solution of SLAERC with asymptotically independent symmetric blocks structure

305

(2)

Canonical equation K53 for the solution of SLAERC with an asymptotically independent random blocks

307

(1)

Class of G8-estimators of the solutions of systems of linear algebraic equations (SLAE)

307

(1)

Modified G8-estimator of the solution of SLAE

308

(1)

G8-estimator of the solutions of SLAE with block structure

309

(1)

G8-estimator of the solutions of SLAE with symmetric block structure

310

(1)

Canonical equation K54 for normalized spectral functions of nonself-adjoint random Jacobi matrices

311

(28)

Random nonsymmetric Jacobi matrices and their normalized spectral functions

311

(1)

V-transform of nonsymmetric Jacobi matrices

312

(1)

Strong law for normalized spectral functions of nonselfadjoint random Jacobi matrices with independent row vectors

313

(10)

Canonical equation K54 for nonselfadjoint random Jacobi matrices with independent entries

323

(1)

Non-Hermitian method for the proof of a limit theorem for normalized spectral functions of nonselfadjoint random Jacobi matrices with independent entries

323

(6)

Equation K54 for the densities

329

(1)

Limit normalized spectral functions of non Hermitian matrices arisen in certain non-Hermitian Anderson models

329

(2)

V-transform approach

331

(1)

Truncated and regularized V1-transform approach

332

(1)

Calculation of a limit of the determinant of Jacobi matrix

333

(2)

Triply regularized V-transform

335

(3)

Main assertion for limit normalized spectral functions of nonrandom matrices

338

(1)

Canonical equation K55 for normalized spectral functions of a product of two independent nonsymmetric random matrices

339

(12)

A product of two independent matrices with independent entries and their normalized spectral functions

339

(1)

V-transform for the product of two matrices

339

(1)

Strong law for normalized spectral functions of the product of two independent matrices with independent entries

340

(2)

Existence of the expected logarithm of the determinant of G- matrices with independent entries

342

(1)

Regularized logarithm of the determinant of G-matrices with independent entries

342

(1)

Inequality for the minimal eigenvalue of the Gram matrix with independent entries

343

(1)

The regularized V-transform

344

(1)

Limit theorem for the G-matrix with independent entries

344

(1)

Invariance principle for the G-matrix with independent entries

345

(4)

Main assertion for the normalized spectral functions of the product of two independent matrices with independent entries

349

(2)

Canonical equation K56 for the solution of the system of linear differential equations with random coefficients

351

(14)

V1-transform of the solution of the system of linear differential equations with random coefficients

351

(2)

V2-transform of the solution of the system of linear differential equations

353

(1)

V3-transform of the solution of the system of linear differential equations

353

(2)

Limit theorem for singular values of random complex matrices

355

(1)

Limit theorem for V-transforms of the solution of the system of linear differential equations

356

(1)

Vanishing of random coefficients of a system of differential equations

357

(2)

The V-transform of individual spectral functions based on the general V-transform (Girko 1982)

359

(1)

The inverse formula for the modified V-transform of individual spectral functions

360

(1)

Stochastic canonical equation K56 for the solution of SLDERC with symmetric matrix of coefficients

361

(4)

Canonical equation K57, the cubic law, the invariance principle and related topics in the theory of analytic functions of random matrices

365

(28)

Strong self-averaging law for analytic functions of random matrices

365

(3)

Invariance principle for analytic functions of random matrices

368

(2)

The Cauchy integral representation for analytic function of matrix

370

(1)

Limit theorems for random quadratic forms

370

(5)

Canonical equation K57

375

(2)

Canonical equation K57 for matrices Ξ 2nxnAnxn(Ξ 2nxn)*

377

(1)

Cubic Law

378

(4)

Law of independency for analytic functions of random matrices ΞΞ and Ξ*Ξ

382

(2)

The First Law for the eigenvalues and eigenvectors of random symmetric matrices

384

(2)

The second law for the singular values of random matrices

386

(2)

The third law for the eigenvalues and eigenvectors of empirical covariance matrices

388

(3)

Limit value for the norm of squared random nonsymmetric matrix

391

(2)

Canonical equation K58. Universality and arcsine laws for random matrices A + Um B (U*)m

393

(22)

Eleven classes of distributions of random unitary matrices

393

(6)

The main formula of REFORM method

399

(1)

The integral representations for the square root of a matrix

400

(1)

The main perturbation formula for the integral representation of square root of a matrix

401

(1)

Method of reqularization based on the expending of unitary matrices

402

(1)

Method of regularization of unitary matrices

402

(2)

Geometrical progression for the resolvents of unitary matrices

404

(2)

Canonical equation K58

406

(1)

Limit theorems for random quadratic forms

407

(2)

Analytic continuation of entries of resolvent of random matrix

409

(1)

The completion of deduction of the system of equation K58

409

(2)

One example of solution of the system of equation K58

411

(2)

Arcsine law for matrices A + UBU*

413

(1)

Limit theorem for individual spectral functions of matrices An + UnBnUn*

413

(1)

Universality law for random matrices An + UnmBn (Un*)m

414

(1)

Canonical equation K59 and universality law for random matrices (A + UB) (A + UB)*. Arcsine law

415

(18)

Class C11 of distributions of random unitary matrices

415

(1)

The first auxiliary formula of REFORM method

415

(2)

The second auxiliary formula

417

(1)

The third class of auxiliary formula

418

(3)

Canonical equation K59

421

(4)

Limit theorems for random quadratic forms

425

(1)

Analytic continuation of entries of resolvent of random matrix

426

(1)

The completion of deduction of the system of equations K59

426

(2)

One example of the system of equation K59

428

(2)

One simple example of the system of equations K59

430

(3)

References

433

(26)

Index

459

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