Theory of Random Determinants

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.