Theory of Random Determinants

1. Generalized Wishart Density and Integral Representation for Determinants.- x1. The Haar Measure.- x2. The Haar Measure On the Group of Orthogonal Matrices.- x3. The Generalized Wishart Density.- x4. Integral Representations for Determinants.- x5. Integration on Grassmann and Clifford Algebras.- 2. Moments of Random Matrix Determinants.- x1. Moments of Random Gram Matrix Determinants.- x2. Moments of Random Vandermond Determinants. Hypothesis by Mehta and Dyson.- x3. Methods of Calculating the Moments of Random Determinants.- x4. Moments of Random Permanents.- x5. Formulas of Random Determinant Perturbation.- 3. Distribution of Eigenvalues and Eigenvectors of Random Matrices.- x1. Distribution of Eigenvalues and Eigenvectors of Hermitian Random Matrices.- x2. Distribution of the Eigenvalues and Eigenvectors of Antisymmetric Random Matrices.- x3. Distribution of Eigenvalues and Eigenvectors of Nonsymmetric Random Matrices.- x4. Distribution of the Eigenvalues and Eigenvectors of Complex Random Matrices.- x5. Distribution of Eigenvalues of Gaussian Real Random Matrices.- x6. Distribution of Eigenvalues and Eigenvectors of Unitary Random Matrices.- x7. Distribution of Eigenvalues and Eigenvectors of Orthogonal Random Matrices.- x8. Distribution of Roots of Algebraic Equations with Random Coefficients.- 4. Inequalities for Random Determinants.- x1. The Stochastic Hadamard Inequality.- x2. Inequalities for Random Determinants.- x3. The Frechet Hypothesis.- x4.On Inequalities for Sums of the Martingale Difference and Random Quadratic Forms.- 5. Limit Theorems for the Borel Functions of Independent Random Variables.- x1. Limit Theorems with the Lindeberg Condition.- x2. Limit Theorems for Polynomial Functions of Independent Random Variables.- x3. Accompanying Infinitely Divisible Distributions for Borel Functions of Independent Random Variables.- x4. Limit Theorems for Sums of Martingale Differences.- x5. Limit Theorems for Sums of Martingale Differences in Nonclassical Situations.- x6. Limit Theorems for Generalized U Statistics.- x7. Central Limit Theorem for Some Functional of Random Walk.- x8. 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.- x1. Limit Theorems of the Law of Large Numbers Type for Random Determinants.- x2. The Perturbation Method.- x3. The Orthogonalization Method.- x4. Logarithmic Law.- x5. The Central Limit Theorem for the Determinants of Random Matrices of Finite Order.- 7. Accompanying Infinitely Divisible Laws for Random Determinants.- x1. Perturbation Method and Accompanying Infinitely Divisible Laws for Random Determinants.- x2. The Method of Orthogonalization and Accompanying Infinitely Divisible Laws.- x3. Central Limit Theorem for Random Permanents.- 8. Integral Representation Method.- x1. Limit Theorem for the Random Analytical Functions.- x2. Limit Theorems for Random Determinants.- x3. Method of Integral Representations and Accompanying Infinitely Divisible Laws.- x4. Limit Theorems of the General Form for Random Determinants.- x5. 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.- x1. The Method of Integral Representations and Limit Theorems for Functionals of Random Functions.- x2. The Spectral Functions Method of Proving Limit Theorems for Random Determinants.- x3. The Canonical Spectral Equation.- x4. The Wigner Semicircle Law.- x5. The General Form of Limit Spectral Functions.- x6. Normalized Spectral Functions of Symmetric Random Matrices with Dependent Random Entries.- 10. Limit Theorems for Random Gram Determinants.- x1. Spectral Equation for Gram Matrices.- x2. Limit Theorems for Random Gram Determinants with Identically Distributed Elements.- x3. Limit Spectral Functions.- 11. The Determinants of Toeplitz and Hankel Random Matrices.- x1. Limit Theorem of the Law of Large Numbers Type.- x2. The Method of Integral Representations for Determinants of Toeplitz and Hankel Random Matrices.- x3. The Stochastic Analogue of the Szego Theorem.- x4. The Method of Perturbation for Determinants of some Toeplitz Random Matrices.- 12. Limit Theorems for Determinants of Random Jacobi Matrices.- x1. Limit Theorems of the Law of Large Numbers Type.- x2. The Dyson Equation.- x3. The Stochastic Sturm-Liouville Problem.- x4. The Sturm Oscillation Theorem.- x5. The Central Limit Theorem for Determinants of Random Jacobi Matrices.- x6.The Central Limit Theorem for Normalized Spectral Functions of Random Jacobi Matrices.- 13. The Fredholm Random Determinants.- x1. Fredholm Determinants of Symmetric Random Matrices.- x2. Limit Theorems for Eigenvalues of Symmetric Random Matrices.- x3. Fredholm Determinants of Nonsymmetric Random Matrices and Limit Theorems for Eigenvalues.- x4. Fredholm Determinants of Random Linear Operators in the Hilbert Space.- 14. The Systems of Linear Algebraic Equations with Random Coefficients.- x1. The Systems of Normal Linear Algebraic Equations.- x2. The Stochastic Method of Least Squares.- x3. 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.- x1. The Arctangent Law.- x2. Method of Integral Representations of the Solution of Systems of Linear Random Algebraic Equations.- x3. The Resolvent Method of Solutions of the Systems of Linear Random Algebraic Equations.- x4. Limit Theorems for Solutions of Difference Equations.- 16. Integral Equations with Random Degenerate Kernels.- x1. Fredholm Integral Equations with Degenerate Random Kernels.- x2. Limit Theorem for Normalized Spectral Functions.- x3. Limit Theorems for Spectral Functions of Integral Equations with Random Kernels.- 17. Random Determinants in the Spectral Theory of Non-Self-Adjoint Random Matrices.- x1. Limit Theorems for the Normalized Spectral Functions of Complex Gaussian Matrices.- x2. The V-Transform of Spectral Functions.- x3. Limit Theorems like the Law of Large Numbers for Normalized Spectral Functions of Non-Self-Adjoint Random Matrices with Independent Entries.- x4. The Regularized V-Transform for Spectral Functions.- x5. An Estimate of the Rate of Convergence of the Stieltjes Transforms of Spectral Functions to the Limit Function.- x6. The Estimates of the Deviations of Spectral Functions from the Limit Functions.- x7. The Circle Law.- x8. The Elliptic Law.- x9. Limit Theorems for the Spectral Functions of Non-Self-Adjoint Random Jacobi Matrices.- x10. The Unimodal Law.- 18. The Distribution of Eigenvalues and Eigenvectors of Additive Random Matrix-Valued Processes.- x1. Distribution of Eigenvalues and Eigenvectors of Random Symmetric Matrix-Valued Processes.- x2. Perturbation Formulas.- x3. Continuity and Nondegeneration of Eigenvalues of Random Matrix-Valued Processes with Independent Increments.- x4. Straight and Back Spectral Kolmogorov Equations for Distribution Densities of Eigenvalues of Random Matrix Processes with Independent Increments.- x5. Spectral Stochastic Differential Equations for Random Symmetric Matrix Processes with Independent Increments.- x6. Spectral Stochastic Differential Equations for Random Matrix-Valued Processes with Multiplicative Independent Increments.- x7. Stochastic Differential Equations for Differences of Eigenvalues of Random Matrix-Valued Processes.- x8. Resolvent Stochastic Differential Equation for Self-Adjoint Random Matrix-Valued Processes.- x9. Resolvent Stochastic Differential Equation for Non-Self-Adjoint Random Matrix-Valued Processes.- 19. The Stochastic Ljapunov Problem for Systems of Stationary Linear Differential Equations.- x1. The Stochastic Ljapunov Problem for Systems of Linear Differential Equations with the Symmetric Matrix of Coefficients.- x2. Hyperdeterminants.- x3. The Stochastic Ljapunov Problem for Systems of Linear Differential Equations with a Nonsymmetric Matrix of Coefficients.- x4. The Spectral Method of Calculating a Probability of Stationary Stochastic Systems Stability.- x5. The Resolvent Method of Proving the Stability of the Solutions of Stochastic Systems.- x6. The Spectral Method of Calculating Mathematical Expectations of Exponents of Random Matrices.- x7. Method of Stochastic Diffusion Equations.- 20. Random Determinants in the Theory of Estimation of Parameters of Some Systems.- x1. The Estimation of Solutions of Equation Systems with Multiplicative Errors in the Series of Observation.- x2. Spectral Equations for Minimax Estimations of Parameters of Linear Systems.- x3. The Estimation of the Parameters of Stable Discrete Control Systems.- x4. The Parameter Estimation of Nonlinear Control Systems.- x5. Limit Theorems of the General Form for the Parameter Estimation of Discrete Control Systems.- x6. Limit Theorem for Estimating Parameters of Discrete Control Systems with Multiplicative Noises.- x7. 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.- x1. The Kaiman Stochastic Condition.- x2. Spectrum Control in Systems Described by Linear Equations in Hilbert Spaces.- x3. Adaptive Approach to the Control of Manipulator Motion.- x4. The Perturbation Method of Linear Operators in the Theory of Optimal Control of Stochastic Systems.- 22. Random Determinants in Some Linear Stochastic Programming Problems.- x1. Formulation of the Linear Stochastic Programming Problem.- x2. Systems of Inequalities with Random Coefficients in Linear Stochastic Programming.- x3. Integral Representation Method for Solving Linear Stochastic Programming Problems.- 23. Random Determinants in General Statistical Analysis.- x1. The Equation for Estimation of Parameters of Fixed Functions.- x2. The Equations for Estimation of Twice-Differentiable Functions of Unknown Parameters.- x3. The Quasiinversion Method for Solving G1-Equations.- x4. The Fourier Transformation Method.- x5. Equations for Estimations of Functions of Unknown Parameters.- x6. G-Equations of Higher Orders.- x7. G-Equation for the Resolvent of Empirical Covariance Matrices if the Lindeberg Condition Holds.- x8. G-Equation for the Stieltjes Transformation of Normal Spectral Functions of the Empirical Covariance Matrices Beam.- x9. G1-Estimate of Generalized Variance.- x10. G2-Estimate of the Stieltjes Transform of the Normalized Spectral Function of Covariance Matrices.- x11. G3-Estimation of the Inverse Covariance Matrix.- x12. G4Estimates of Traces of Covariance Matrix Powers.- x13. G5-Estimates of Smoothed Normalized Spectral Functions of Empirical Covariance Matrices.- x14. Parameter Estimation of Stable Discrete Control Systems under G-Conditions.- 24. Estimate of the Solution of the Kolmogorov-Wiener Filter.- x1. The G9-Estimate of the Solution of the Kolmogorov-Wiener Filter.- x2. Asymptotic Normality of the G9-Estimate of the Solution of the Kolmogorov-Wiener Equation.- x3. The G10-Estimate of the Solution of the Regularized Kolmogorov-Wiener Filter.- 25. Random Determinants in Pattern Recognition.- x1. The Bayes Method for Classification of Two Populations.- x2. Observation Classifications in the Case of Two Populations Having Known Multivariate Normal Distributions with Identical Covariance Matrices.- x3. The G11-Estimate of the Mahalanobis Distance.- x4. Asymptotic Normality of Estimate G11.- x5. The G12-Estimate of the Regularized Mahalanobis Distance.- x6. The G13-Anderson-Fisher Statistics Estimate.- x7. 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.- x1. The Resolvent Method in the Theory of Experiment Design.- x2. The G16-Estimate of the Estimation Errors in the Theory of the Design of Experiments.- 27. Random Determinants in Physics.- x1. The Wigner Hypothesis.- x2. Some Properties of the Stochastic Scattering Matrix.- x3. Application of Random Determinants in Some Mathematical Models of Solid-State Physics.- 28. Random Determinants in Numerical Analysis.- x1. Consistent Estimations of the Solutions of Systems of Linear Algebraic Equations, Obtained during Observations of Independent Random Coefficients with Identical Variances.- x2. Consistent Estimations of the Solutions of a System of Linear Algebraic Equations with a Symmetric Matrix of Coefficients.- References.