Theory of random determinants:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Dordrecht [u.a.]
Kluwer Acad. Publ.
1990
|
| Series: | Mathematics and its applications / Soviet series
45 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus d. Russ. übers. |
| Physical Description: | XXV, 677 S. |
| ISBN: | 0792302338 |
Staff View
MARC
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| 240 | 1 | 0 | |a Teorija slučajnych determinantov |
| 245 | 1 | 0 | |a Theory of random determinants |c by V. L. Girko |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer Acad. Publ. |c 1990 | |
| 300 | |a XXV, 677 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Mathematics and its applications / Soviet series |v 45 | |
| 500 | |a Aus d. Russ. übers. | ||
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| 650 | 7 | |a Matrices stochastiques |2 ram | |
| 650 | 4 | |a Stochastic matrices | |
| 650 | 4 | |a Determinants | |
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Record in the Search Index
| _version_ | 1804118453394604032 |
|---|---|
| adam_text | Contents
List of Basic Notations and Assumptions xv
Introduction to the English Edition xvii
Chapter 1. Generalized Wishart Density and Integral
Representation for Determinants 1
§1. The Haar Measure 1
§2. The Haar Measure On the Group of Orthogonal Matrices 4
§3. The Generalized Wishart Density 7
§4. Integral Representations for Determinants 16
§5. Integration on Grassmann and Clifford Algebras 19
Chapter 2. Moments of Random Matrix Determinants 22
§1. Moments of Random Gram Matrix Determinants 22
§2. Moments of Random Vandermond Determinants.
Hypothesis by Mehta and Dyson 29
§3. Methods of Calculating the Moments of Random Determinants 43
§4. Moments of Random Permanents 46
§5. Formulas of Random Determinant Perturbation 48
Chapter 3. Distribution of Eigenvalues and Eigenvectors
of Random Matrices 54
§1. Distribution of Eigenvalues and Eigenvectors of
Hermitian Random Matrices 54
§2. Distribution of the Eigenvalues and Eigenvectors of
Antisymmetric Random Matrices 62
§3. Distribution of Eigenvalues and Eigenvectors of
Nonsymmetric Random Matrices 67
§4. Distribution of the Eigenvalues and Eigenvectors of
Complex Random Matrices 72
§5. Distribution of Eigenvalues of Gaussian Real
Random Matrices 78
§6. Distribution of Eigenvalues and Eigenvectors of
Unitary Random Matrices 84
viii Contents
§7. Distribution of Eigenvalues and Eigenvectors of
Orthogonal Random Matrices 87
§8. Distribution of Roots of Algebraic Equations
with Random Coefficients 93
Chapter 4. Inequalities for Random Determinants 102
§1. The Stochastic Hadamard Inequality 102
§2. Inequalities for Random Determinants 105
§3. The Frechet Hypothesis 110
§4. On Inequalities for Sums of the Martingale Difference and
Random Quadratic Forms 112
Chapter 5. Limit Theorems for the Borel Functions of
Independent Random Variables 114
§1. Limit Theorems with the Lindeberg Condition 114
§2. Limit Theorems for Polynomial Functions of
Independent Random Variables 119
§3. Accompanying Infinitely Divisible Distributions for Borel
Functions of Independent Random Variables 128
§4. Limit Theorems for Sums of Martingale Differences 134
§5. Limit Theorems for Sums of Martingale Differences in
Nonclassical Situations 158
§6. Limit Theorems for Generalized U Statistics 161
§7. Central Limit Theorem for Some Functional of Random Walk 164
§8. Limit Theorems for Sums of Random Variables Connected
in a Markov Chain 167
Chapter 6. Limit Theorems of the Law of Large Numbers and
Central Limit Theorem Types for Random Determinants 170
§1. Limit Theorems of the Law of Large Numbers Type
for Random Determinants 170
§2. The Perturbation Method 174
§3. The Orthogonalization Method 188
§4. Logarithmic Law 199
§5. The Central Limit Theorem for the Determinants of
Random Matrices of Finite Order 202
Chapter 7. Accompanying Infinitely Divisible Laws
for Random Determinants 204
§1. Perturbation Method and Accompanying Infinitely Divisible
Laws for Random Determinants 204
§2. The Method of Orthogonalization and Accompanying
Infinitely Divisible Laws 213
Contents ix
§3. Central Limit Theorem for Random Permanents 215
Chapter 8. Integral Representation Method 218
§1. Limit Theorem for the Random Analytical Functions 218
§2. Limit Theorems for Random Determinants 220
§3. Method of Integral Representations and Accompanying
Infinitely Divisible Laws 235
§4. Limit Theorems of the General Form for Random Determinants 247
§5. Limit Theorems for the Determinants of Random Matrices
with Dependent Random Elements 252
Chapter 9. The Connection between the Convergence of
Random Determinants and the Convergence of
Functional of Random Functions 255
§1. The Method of Integral Representations and Limit Theorems
for Functional of Random Functions 255
§2. The Spectral Functions Method of Proving Limit Theorems
for Random Determinants 263
§3. The Canonical Spectral Equation 269
§4. The Wigner Semicircle Law 282
§5. The General Form of Limit Spectral Functions 289
§6. Normalized Spectral Functions of Symmetric Random Matrices
with Dependent Random Entries 295
Chapter 10. Limit Theorems for Random Gram Determinants 297
§1. Spectral Equation for Gram Matrices 297
§2. Limit Theorems for Random Gram Determinants with
Identically Distributed Elements 302
§3. Limit Spectral Functions 305
Chapter 11. The Determinants of Toeplitz and Hankel
Random Matrices 309
§1. Limit Theorem of the Law of Large Numbers Type 309
§2. The Method of Integral Representations for Determinants
of Toeplitz and Hankel Random Matrices 312
§3. The Stochastic Analogue of the Szego Theorem 315
§4. The Method of Perturbation for Determinants of some
Toeplitz Random Matrices 318
Chapter 12. Limit Theorems for Determinants of Random
Jacobi Matrices 321
§1. Limit Theorems of the Law of Large Numbers Type 321
x Contents
§2. The Dyson Equation 323
§3. The Stochastic Sturm Liouville Problem 333
§4. The Sturm Oscillation Theorem 337
§5. The Central Limit Theorem for Determinants of
Random Jacobi Matrices 339
§6. The Central Limit Theorem for Normalized Spectral Functions
of Random Jacobi Matrices 342
Chapter 13. The Fredholm Random Determinants 347
§1. Fredholm Determinants of Symmetric Random Matrices 349
§2. Limit Theorems for Eigenvalues of Symmetric Random Matrices 351
§3. Fredholm Determinants of Nonsymmetric Random Matrices
and Limit Theorems for Eigenvalues 361
§4. Fredholm Determinants of Random Linear Operators
in the Hilbert Space 364
Chapter 14. The Systems of Linear Algebraic Equations
with Random Coefficients 366
§1. The Systems of Normal Linear Algebraic Equations 366
§2. The Stochastic Method of Least Squares 370
§3. Spectral Method for the Calculation of Moments of Inverse
Random Matrices 372
Chapter 15. Limit Theorems for the Solution of the Systems of
Linear Algebraic Equations with Random Coefficients 377
§1. The Arctangent Law 377
§2. Method of Integral Representations of the Solution of
Systems of Linear Random Algebraic Equations 381
§3. The Resolvent Method of Solutions of the Systems of
Linear Random Algebraic Equations 383
§4. Limit Theorems for Solutions of Difference Equations 388
Chapter 16. Integral Equations with Random Degenerate Kernels 391
§1. Fredholm Integral Equations with Degenerate Random Kernels 391
§2. Limit Theorem for Normalized Spectral Functions 396
§3. Limit Theorems for Spectral Functions of Integral Equations
with Random Kernels 398
Chapter 17. Random Determinants in the Spectral Theory of
Non Self Adjoint Random Matrices 401
§1. Limit Theorems for the Normalized Spectral Functions of
Complex Gaussian Matrices 401
Contents xi
§2. The V Transform of Spectral Functions 404
§3. Limit Theorems like the Law of Large Numbers for Normalized
Spectral Functions of Non Self Adjoint Random Matrices
with Independent Entries 407
§4. The Regularized V Transform for Spectral Functions 411
§5. An Estimate of the Rate of Convergence of the Stieltjes
Transforms of Spectral Functions to the Limit Function 414
§6. The Estimates of the Deviations of Spectral Functions from
the Limit Functions 425
§7. The Circle Law 428
§8. The Elliptic Law 431
§9. Limit Theorems for the Spectral Functions of Non Self Adjoint
Random Jacobi Matrices 433
§10. The Unimodal Law 437
Chapter 18. The Distribution of Eigenvalues and Eigenvectors of
Additive Random Matrix Valued Processes 442
§1. Distribution of Eigenvalues and Eigenvectors of
Random Symmetric Matrix Valued Processes 442
§2. Perturbation Formulas 443
§3. Continuity and Nondegeneration of Eigenvalues of Random
Matrix Valued Processes with Independent Increments 446
§4. Straight and Back Spectral Kolmogorov Equations for
Distribution Densities of Eigenvalues of Random Matrix
Processes with Independent Increments 451
§5. Spectral Stochastic Differential Equations for Random Symmetric
Matrix Processes with Independent Increments 456
§6. Spectral Stochastic Differential Equations for Random
Matrix Valued Processes with Multiplicative
Independent Increments 461
§7. Stochastic Differential Equations for Differences of
Eigenvalues of Random Matrix Valued Processes 463
§8. Resolvent Stochastic Differential Equation for Self Adjoint
Random Matrix Valued Processes 465
§9. Resolvent Stochastic Differential Equation for Non Self Adjoint
Random Matrix Valued Processes 466
Chapter 19. The Stochastic Ljapunov Problem for Systems of
Stationary Linear Differential Equations 468
§1. The Stochastic Ljapunov Problem for Systems of Linear
Differential Equations with the Symmetric Matrix of
Coefficients 468
§2. Hyperdeterminants 470
xii Contents
§3. The Stochastic Ljapunov Problem for Systems of Linear
Differential Equations with a Nonsymmetric Matrix of
Coefficients 471
§4. The Spectral Method of Calculating a Probability of
Stationary Stochastic Systems Stability 472
§5. The Resolvent Method of Proving the Stability of the
Solutions of Stochastic Systems 473
§6. The Spectral Method of Calculating Mathematical
Expectations of Exponents of Random Matrices 476
§7. Method of Stochastic Diffusion Equations 478
Chapter 20. Random Determinants in the Theory of Estimation
of Parameters of Some Systems 481
§1. The Estimation of Solutions of Equation Systems with
Multiplicative Errors in the Series of Observation 481
§2. Spectral Equations for Minimax Estimations of
Parameters of Linear Systems 484
§3. The Estimation of the Parameters of Stable Discrete
Control Systems 486
§4. The Parameter Estimation of Nonlinear Control Systems 491
§5. Limit Theorems of the General Form for the Parameter
Estimation of Discrete Control Systems 495
§6. Limit Theorem for Estimating Parameters of Discrete
Control Systems with Multiplicative Noises 498
§7. Estimating Spectra of Stochastic Linear Control Systems
and Spectral Equations in the Theory of the
Parameters Estimation 499
Chapter 21. Random Determinants in Some Problems of
Control Theory of Stochastic Systems 503
§1. The Kalman Stochastic Condition 503
§2. Spectrum Control in Systems Described by Linear Equations
in Hilbert Spaces 508
§3. Adaptive Approach to the Control of Manipulator Motion 518
§4. The Perturbation Method of Linear Operators in the Theory
of Optimal Control of Stochastic Systems 524
Chapter 22. Random Determinants in Some Linear Stochastic
Programming Problems 527
§1. Formulation of the Linear Stochastic Programming Problem 528
§2. Systems of Inequalities with Random Coefficients in
Linear Stochastic Programming 529
Contents xiii
§3. Integral Representation Method for Solving Linear
Stochastic Programming Problems 535
Chapter 23. Random Determinants in General Statistical Analysis 539
§1. The Equation for Estimation of Parameters of Fixed
Functions 540
§2. The Equations for Estimation of Twice Differentiable
Functions of Unknown Parameters 542
§3. The Quasiinversion Method for Solving Gi Equations 543
§4. The Fourier Transformation Method 545
§5. Equations for Estimations of Functions of Unknown
Parameters 547
§6. G Equations of Higher Orders 549
§7. G Equation for the Resolvent of Empirical Covariance
Matrices if the Lindeberg Condition Holds 550
§8. G Equation for the Stieltjes Transformation of Normal Spectral
Functions of the Empirical Covariance Matrices Beam 557
§9. Gi Estimate of Generalized Variance 565
§10. G2 Estimate of the Stieltjes Transform of the Normalized
Spectral Function of Covariance Matrices 568
§11. G3 Estimation of the Inverse Covariance Matrix 577
§12. G4 Estimates of Traces of Covariance Matrix Powers 580
§13. G5 Estimates of Smoothed Normalized Spectral Functions
of Empirical Covariance Matrices 581
§14. Parameter Estimation of Stable Discrete Control Systems
under G Conditions 583
Chapter 24. Estimate of the Solution of the Kolmogorov Wiener
Filter 588
§1. The Gg Estimate of the Solution of the Kolmogorov Wiener
Filter 589
§2. Asymptotic Normality of the Gg Estimate of the Solution
of the Kolmogorov Wiener Equation 591
§3. The G10 Estimate of the Solution of the Regularized
Kolmogorov Wiener Filter 593
Chapter 25. Random Determinants in Pattern Recognition 595
§1. The Bayes Method for Classification of Two Populations 595
§2. Observation Classifications in the Case of Two Populations
Having Known Multivariate Normal Distributions with
Identical Covariance Matrices 597
§3. The Gn Estimate of the Mahalanobis Distance 598
§4. Asymptotic Normality of Estimate G 600
xiv Contents
§5. The Gi2 Estimate of the Regularized Mahalanobis Distance 601
§6. The Gi3 Anderson Fisher Statistics Estimate 611
§7. The Gi5 Estimate of the Nonlinear Discriminant Function,
Obtained by Observations over Random Vectors with
Different Covariance Matrices 615
Chapter 26. Random Determinants in the Experiment Design 616
§1. The Resolvent Method in the Theory of Experiment Design 617
§2. The C?i6 Estimate of the Estimation Errors in the Theory of
the Design of Experiments 624
Chapter 27. Random Determinants in Physics 627
§1. The Wigner Hypothesis 627
§2. Some Properties of the Stochastic Scattering Matrix 635
§3. Application of Random Determinants in Some Mathematical
Models of Solid State Physics 642
Chapter 28. Random Determinants in Numerical Analysis 644
§1. Consistent Estimations of the Solutions of Systems of Linear
Algebraic Equations, Obtained during Observations of
Independent Random Coefficients with Identical Variances 645
§2. Consistent Estimations of the Solutions of a System of Linear
Algebraic Equations with a Symmetric Matrix of Coefficients 655
References 657
Index 668
|
| any_adam_object | 1 |
| author | Girko, Vʼjačeslav Leonidovyč 1946- |
| author_GND | (DE-588)1055764976 |
| author_facet | Girko, Vʼjačeslav Leonidovyč 1946- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.9/434 512.9/434 20 |
| dewey-search | 512.9/434 512.9/434 20 |
| dewey-sort | 3512.9 3434 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV004257370 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T16:10:32Z |
| institution | BVB |
| isbn | 0792302338 |
| language | English Russian |
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| spelling | Girko, Vʼjačeslav Leonidovyč 1946- Verfasser (DE-588)1055764976 aut Teorija slučajnych determinantov Theory of random determinants by V. L. Girko Dordrecht [u.a.] Kluwer Acad. Publ. 1990 XXV, 677 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications / Soviet series 45 Aus d. Russ. übers. Déterminants (Mathématiques) ram Matrices stochastiques ram Stochastic matrices Determinants Determinante (DE-588)4138983-9 gnd rswk-swf Stochastische Matrix (DE-588)4057624-3 gnd rswk-swf Stochastische Matrix (DE-588)4057624-3 s Determinante (DE-588)4138983-9 s DE-604 Soviet series Mathematics and its applications 45 (DE-604)BV004708148 45 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002648348&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Girko, Vʼjačeslav Leonidovyč 1946- Theory of random determinants Déterminants (Mathématiques) ram Matrices stochastiques ram Stochastic matrices Determinants Determinante (DE-588)4138983-9 gnd Stochastische Matrix (DE-588)4057624-3 gnd |
| subject_GND | (DE-588)4138983-9 (DE-588)4057624-3 |
| title | Theory of random determinants |
| title_alt | Teorija slučajnych determinantov |
| title_auth | Theory of random determinants |
| title_exact_search | Theory of random determinants |
| title_full | Theory of random determinants by V. L. Girko |
| title_fullStr | Theory of random determinants by V. L. Girko |
| title_full_unstemmed | Theory of random determinants by V. L. Girko |
| title_short | Theory of random determinants |
| title_sort | theory of random determinants |
| topic | Déterminants (Mathématiques) ram Matrices stochastiques ram Stochastic matrices Determinants Determinante (DE-588)4138983-9 gnd Stochastische Matrix (DE-588)4057624-3 gnd |
| topic_facet | Déterminants (Mathématiques) Matrices stochastiques Stochastic matrices Determinants Determinante Stochastische Matrix |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002648348&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV004708148 |
| work_keys_str_mv | AT girkovʼjaceslavleonidovyc teorijaslucajnychdeterminantov AT girkovʼjaceslavleonidovyc theoryofrandomdeterminants |