Theory of Random Determinants

ISBN-13: 9789401073288 / Angielski / Miękka / 2013 / 678 str.

V. L. Girko

Opis

'Et mm. ..., si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point all': '' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf IIClI.t to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series."

Spis treści

1. Generalized Wishart Density and Integral Representation for Determinants.- §1. The Haar Measure.- §2. The Haar Measure On the Group of Orthogonal Matrices.- §3. The Generalized Wishart Density.- §4. Integral Representations for Determinants.- §5. Integration on Grassmann and Clifford Algebras.- 2. Moments of Random Matrix Determinants.- §1. Moments of Random Gram Matrix Determinants.- §2. Moments of Random Vandermond Determinants. Hypothesis by Mehta and Dyson.- §3. Methods of Calculating the Moments of Random Determinants.- §4. Moments of Random Permanents.- §5. Formulas of Random Determinant Perturbation.- 3. Distribution of Eigenvalues and Eigenvectors of Random Matrices.- §1. Distribution of Eigenvalues and Eigenvectors of Hermitian Random Matrices.- §2. Distribution of the Eigenvalues and Eigenvectors of Antisymmetric Random Matrices.- §3. Distribution of Eigenvalues and Eigenvectors of Nonsymmetric Random Matrices.- §4. Distribution of the Eigenvalues and Eigenvectors of Complex Random Matrices.- §5. Distribution of Eigenvalues of Gaussian Real Random Matrices.- §6. Distribution of Eigenvalues and Eigenvectors of Unitary Random Matrices.- §7. Distribution of Eigenvalues and Eigenvectors of Orthogonal Random Matrices.- §8. Distribution of Roots of Algebraic Equations with Random Coefficients.- 4. Inequalities for Random Determinants.- §1. The Stochastic Hadamard Inequality.- §2. Inequalities for Random Determinants.- §3. The Frechet Hypothesis.- §4.On Inequalities for Sums of the Martingale Difference and Random Quadratic Forms.- 5. Limit Theorems for the Borel Functions of Independent Random Variables.- §1. Limit Theorems with the Lindeberg Condition.- §2. Limit Theorems for Polynomial Functions of Independent Random Variables.- §3. Accompanying Infinitely Divisible Distributions for Borel Functions of Independent Random Variables.- §4. Limit Theorems for Sums of Martingale Differences.- §5. Limit Theorems for Sums of Martingale Differences in Nonclassical Situations.- §6. Limit Theorems for Generalized U Statistics.- §7. Central Limit Theorem for Some Functional of Random Walk.- §8. Limit Theorems for Sums of Random Variables Connected in a Markov Chain.- 6. Limit Theorems of the Law of Large Numbers and Central Limit Theorem Types for Random Determinants.- §1. Limit Theorems of the Law of Large Numbers Type for Random Determinants.- §2. The Perturbation Method.- §3. The Orthogonalization Method.- §4. Logarithmic Law.- §5. The Central Limit Theorem for the Determinants of Random Matrices of Finite Order.- 7. Accompanying Infinitely Divisible Laws for Random Determinants.- §1. Perturbation Method and Accompanying Infinitely Divisible Laws for Random Determinants.- §2. The Method of Orthogonalization and Accompanying Infinitely Divisible Laws.- §3. Central Limit Theorem for Random Permanents.- 8. Integral Representation Method.- §1. Limit Theorem for the Random Analytical Functions.- §2. Limit Theorems for Random Determinants.- §3. Method of Integral Representations and Accompanying Infinitely Divisible Laws.- §4. Limit Theorems of the General Form for Random Determinants.- §5. Limit Theorems for the Determinants of Random Matrices with Dependent Random Elements.- 9. The Connection between the Convergence of Random Determinants and the Convergence of Functionals of Random Functions.- §1. The Method of Integral Representations and Limit Theorems for Functionals of Random Functions.- §2. The Spectral Functions Method of Proving Limit Theorems for Random Determinants.- §3. The Canonical Spectral Equation.- §4. The Wigner Semicircle Law.- §5. The General Form of Limit Spectral Functions.- §6. Normalized Spectral Functions of Symmetric Random Matrices with Dependent Random Entries.- 10. Limit Theorems for Random Gram Determinants.- §1. Spectral Equation for Gram Matrices.- §2. Limit Theorems for Random Gram Determinants with Identically Distributed Elements.- §3. Limit Spectral Functions.- 11. The Determinants of Toeplitz and Hankel Random Matrices.- §1. Limit Theorem of the Law of Large Numbers Type.- §2. The Method of Integral Representations for Determinants of Toeplitz and Hankel Random Matrices.- §3. The Stochastic Analogue of the Szegö Theorem.- §4. The Method of Perturbation for Determinants of some Toeplitz Random Matrices.- 12. Limit Theorems for Determinants of Random Jacobi Matrices.- §1. Limit Theorems of the Law of Large Numbers Type.- §2. The Dyson Equation.- §3. The Stochastic Sturm-Liouville Problem.- §4. The Sturm Oscillation Theorem.- §5. The Central Limit Theorem for Determinants of Random Jacobi Matrices.- §6.The Central Limit Theorem for Normalized Spectral Functions of Random Jacobi Matrices.- 13. The Fredholm Random Determinants.- §1. Fredholm Determinants of Symmetric Random Matrices.- §2. Limit Theorems for Eigenvalues of Symmetric Random Matrices.- §3. Fredholm Determinants of Nonsymmetric Random Matrices and Limit Theorems for Eigenvalues.- §4. Fredholm Determinants of Random Linear Operators in the Hilbert Space.- 14. The Systems of Linear Algebraic Equations with Random Coefficients.- §1. The Systems of Normal Linear Algebraic Equations.- §2. The Stochastic Method of Least Squares.- §3. Spectral Method for the Calculation of Moments of Inverse Random Matrices.- 15. Limit Theorems for the Solution of the Systems of Linear Algebraic Equations with Random Coefficients.- §1. The Arctangent Law.- §2. Method of Integral Representations of the Solution of Systems of Linear Random Algebraic Equations.- §3. The Resolvent Method of Solutions of the Systems of Linear Random Algebraic Equations.- §4. Limit Theorems for Solutions of Difference Equations.- 16. Integral Equations with Random Degenerate Kernels.- §1. Fredholm Integral Equations with Degenerate Random Kernels.- §2. Limit Theorem for Normalized Spectral Functions.- §3. Limit Theorems for Spectral Functions of Integral Equations with Random Kernels.- 17. Random Determinants in the Spectral Theory of Non-Self-Adjoint Random Matrices.- §1. Limit Theorems for the Normalized Spectral Functions of Complex Gaussian Matrices.- §2. The V-Transform of Spectral Functions.- §3. Limit Theorems like the Law of Large Numbers for Normalized Spectral Functions of Non-Self-Adjoint Random Matrices with Independent Entries.- §4. The Regularized V-Transform for Spectral Functions.- §5. An Estimate of the Rate of Convergence of the Stieltjes Transforms of Spectral Functions to the Limit Function.- §6. The Estimates of the Deviations of Spectral Functions from the Limit Functions.- §7. The Circle Law.- §8. The Elliptic Law.- §9. Limit Theorems for the Spectral Functions of Non-Self-Adjoint Random Jacobi Matrices.- §10. The Unimodal Law.- 18. The Distribution of Eigenvalues and Eigenvectors of Additive Random Matrix-Valued Processes.- §1. Distribution of Eigenvalues and Eigenvectors of Random Symmetric Matrix-Valued Processes.- §2. Perturbation Formulas.- §3. Continuity and Nondegeneration of Eigenvalues of Random Matrix-Valued Processes with Independent Increments.- §4. Straight and Back Spectral Kolmogorov Equations for Distribution Densities of Eigenvalues of Random Matrix Processes with Independent Increments.- §5. Spectral Stochastic Differential Equations for Random Symmetric Matrix Processes with Independent Increments.- §6. Spectral Stochastic Differential Equations for Random Matrix-Valued Processes with Multiplicative Independent Increments.- §7. Stochastic Differential Equations for Differences of Eigenvalues of Random Matrix-Valued Processes.- §8. Resolvent Stochastic Differential Equation for Self-Adjoint Random Matrix-Valued Processes.- §9. Resolvent Stochastic Differential Equation for Non-Self-Adjoint Random Matrix-Valued Processes.- 19. The Stochastic Ljapunov Problem for Systems of Stationary Linear Differential Equations.- §1. The Stochastic Ljapunov Problem for Systems of Linear Differential Equations with the Symmetric Matrix of Coefficients.- §2. Hyperdeterminants.- §3. The Stochastic Ljapunov Problem for Systems of Linear Differential Equations with a Nonsymmetric Matrix of Coefficients.- §4. The Spectral Method of Calculating a Probability of Stationary Stochastic Systems Stability.- §5. The Resolvent Method of Proving the Stability of the Solutions of Stochastic Systems.- §6. The Spectral Method of Calculating Mathematical Expectations of Exponents of Random Matrices.- §7. Method of Stochastic Diffusion Equations.- 20. Random Determinants in the Theory of Estimation of Parameters of Some Systems.- §1. The Estimation of Solutions of Equation Systems with Multiplicative Errors in the Series of Observation.- §2. Spectral Equations for Minimax Estimations of Parameters of Linear Systems.- §3. The Estimation of the Parameters of Stable Discrete Control Systems.- §4. The Parameter Estimation of Nonlinear Control Systems.- §5. Limit Theorems of the General Form for the Parameter Estimation of Discrete Control Systems.- §6. Limit Theorem for Estimating Parameters of Discrete Control Systems with Multiplicative Noises.- §7. Estimating Spectra of Stochastic Linear Control Systems and Spectral Equations in the Theory of the Parameters Estimation.- 21. Random Determinants in Some Problems of Control Theory of Stochastic Systems.- §1. The Kaiman Stochastic Condition.- §2. Spectrum Control in Systems Described by Linear Equations in Hilbert Spaces.- §3. Adaptive Approach to the Control of Manipulator Motion.- §4. The Perturbation Method of Linear Operators in the Theory of Optimal Control of Stochastic Systems.- 22. Random Determinants in Some Linear Stochastic Programming Problems.- §1. Formulation of the Linear Stochastic Programming Problem.- §2. Systems of Inequalities with Random Coefficients in Linear Stochastic Programming.- §3. Integral Representation Method for Solving Linear Stochastic Programming Problems.- 23. Random Determinants in General Statistical Analysis.- §1. The Equation for Estimation of Parameters of Fixed Functions.- §2. The Equations for Estimation of Twice-Differentiable Functions of Unknown Parameters.- §3. The Quasiinversion Method for Solving G1-Equations.- §4. The Fourier Transformation Method.- §5. Equations for Estimations of Functions of Unknown Parameters.- §6. G-Equations of Higher Orders.- §7. G-Equation for the Resolvent of Empirical Covariance Matrices if the Lindeberg Condition Holds.- §8. G-Equation for the Stieltjes Transformation of Normal Spectral Functions of the Empirical Covariance Matrices Beam.- §9. G1-Estimate of Generalized Variance.- §10. G2-Estimate of the Stieltjes Transform of the Normalized Spectral Function of Covariance Matrices.- §11. G3-Estimation of the Inverse Covariance Matrix.- §12. G4Estimates of Traces of Covariance Matrix Powers.- §13. G5-Estimates of Smoothed Normalized Spectral Functions of Empirical Covariance Matrices.- §14. Parameter Estimation of Stable Discrete Control Systems under G-Conditions.- 24. Estimate of the Solution of the Kolmogorov-Wiener Filter.- §1. The G9-Estimate of the Solution of the Kolmogorov-Wiener Filter.- §2. Asymptotic Normality of the G9-Estimate of the Solution of the Kolmogorov-Wiener Equation.- §3. The G10-Estimate of the Solution of the Regularized Kolmogorov-Wiener Filter.- 25. Random Determinants in Pattern Recognition.- §1. The Bayes Method for Classification of Two Populations.- §2. Observation Classifications in the Case of Two Populations Having Known Multivariate Normal Distributions with Identical Covariance Matrices.- §3. The G11-Estimate of the Mahalanobis Distance.- §4. Asymptotic Normality of Estimate G11.- §5. The G12-Estimate of the Regularized Mahalanobis Distance.- §6. The G13-Anderson-Fisher Statistics Estimate.- §7. The G15-Estimate of the Nonlinear Discriminant Function, Obtained by Observations over Random Vectors with Different Covariance Matrices.- 26. Random Determinants in the Experiment Design.- §1. The Resolvent Method in the Theory of Experiment Design.- §2. The G16-Estimate of the Estimation Errors in the Theory of the Design of Experiments.- 27. Random Determinants in Physics.- §1. The Wigner Hypothesis.- §2. Some Properties of the Stochastic Scattering Matrix.- §3. Application of Random Determinants in Some Mathematical Models of Solid-State Physics.- 28. Random Determinants in Numerical Analysis.- §1. Consistent Estimations of the Solutions of Systems of Linear Algebraic Equations, Obtained during Observations of Independent Random Coefficients with Identical Variances.- §2. Consistent Estimations of the Solutions of a System of Linear Algebraic Equations with a Symmetric Matrix of Coefficients.- References.

O autorze

Vyacheslav L. Girko is Professor of Mathematics in the Department of Applied Statistics at the National University of Kiev and the University of Kiev Mohyla Academy. He is also affiliated with the Institute of Mathematics, Ukrainian Academy of Sciences. His research interests include multivariate statistical analysis, discriminant analysis, experiment planning, identification and control of complex systems, statistical methods in physics, noise filtration, matrix analysis, and stochastic optimization. He has published widely in the areas of multidimensional statistical analysis and theory of random matrices.