Theory of Random Determinants

__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. Front Matter....Pages i-xxv Generalized Wishart Density and Integral Representation for Determinants....Pages 1-21 Moments of Random Matrix Determinants....Pages 22-53 Distribution of Eigenvalues and Eigenvectors of Random Matrices....Pages 54-101 Inequalities for Random Determinants....Pages 102-113 Limit Theorems for the Borel Functions of Independent Random Variables....Pages 114-169 Limit Theorems of the Law of Large Numbers and Central Limit Theorem Types for Random Determinants....Pages 170-203 Accompanying Infinitely Divisible Laws for Random Determinants....Pages 204-217 Integral Representation Method....Pages 218-254 The Connection Between the Convergence of Random Determinants and the Convergence of Functionals of Random Functions....Pages 255-296 Limit Theorems for Random Gram Determinants....Pages 297-308 The Determinants of Toeplitz and Hankel Random Matrices....Pages 309-320 Limit Theorems for Determinants of Random Jacobi Matrices....Pages 321-346 The Fredholm Random Determinants....Pages 347-365 The Systems of Linear Algebraic Equations with Random Coefficients....Pages 366-376 Limit Theorems for the Solution of the Systems of Linear Algebraic Equations with Random Coefficients....Pages 377-390 Integral Equations with Random Degenerate Kernels....Pages 391-400 Random Determinants in the Spectral Theory of Non-Self-Adjoint Random Matrices....Pages 401-441 The Distribution of Eigenvalues and Eigenvectors of Additive Random Matrix-Valued Processes....Pages 442-467 The Stochastic Ljapunov Problem for Systems of Stationary Linear Differential Equations....Pages 468-480 Random Determinants in the Theory of Estimation of Parameters of Some Systems....Pages 481-502 Random Determinants in Some Problems of Control Theory of Stochastic Systems....Pages 503-526 Random Determinants in Some Linear Stochastic Programming Problems....Pages 527-538 Random Determinants in General Statistical Analysis....Pages 539-587 Estimate of the Solution of the Kolmogorov-Wiener Filter....Pages 588-594 Random Determinants in Pattern Recognition....Pages 595-615 Random Determinants in the Experiment Design....Pages 616-626 Random Determinants in Physics....Pages 627-643 Random Determinants in Numerical Analysis....Pages 644-656 Back Matter....Pages 657-677 '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