Theory of random determinants. Transl. from the Russian.

*(English)*Zbl 0704.60003
Mathematics and Its Applications (Soviet Series), 45. Dordrecht etc.: Kluwer Academic Publishers. xxv, 677 p. Dfl. 420.00; $ 245.00; £147.00 (1988).

Matrices and determinants play a fundamental role in mathematics and its applications. Nevertheless the calculation of large size determinants usually turns out to be a very difficult problem. Instead of exact calculations we may try to find limit distributions for \(n\to \infty\) of determinants under the assumption that their elements are independent random variables. Such randomization of matrix elements allows us to find reasonable probabilistic descriptions for hardly solvable problems in a deterministic way.

The book concerns most of the classical theory applied to random determinants and wide extensions to special topics like limit theorems, multivariate statistics, pattern recognition, some questions of nuclear physics and others. It is written mostly for specialists in mathematics and its applications and can be used as a fundamental monograph in the mentioned areas. All theorems are supplied by proofs with all details. But exercises, numerical examples and computer assistance are unfortunately absent; there was no attempt to present the material like a handbook.

In 28 chapters the book contains all main concepts of the theory of random determinants, including:

Haare measures on the group of orthogonal matrices and generalized Wishart densities; Moments of random determinants; Distributions of eigenvalues and eigenvectors of random matrices; Limit theorems for random determinants;

Random Gram determinants; Determinants of Toeplitz and Hankel random matrices; Jacobi random matrices; Fredholm random determinants; Linear equations with random coefficients;

Integral equations with random kernels; Distributions of eigenvalues of additive random matrix-valued processes; The stochastic Lyapunov problem for systems of stationary linear differential equations;

Random determinants in the theory of parameter estimation; Random determinants in control theory; Random determinants in general statistics; Random determinants in pattern recognition, experimental design, numerical analysis and physics.

The book concerns most of the classical theory applied to random determinants and wide extensions to special topics like limit theorems, multivariate statistics, pattern recognition, some questions of nuclear physics and others. It is written mostly for specialists in mathematics and its applications and can be used as a fundamental monograph in the mentioned areas. All theorems are supplied by proofs with all details. But exercises, numerical examples and computer assistance are unfortunately absent; there was no attempt to present the material like a handbook.

In 28 chapters the book contains all main concepts of the theory of random determinants, including:

Haare measures on the group of orthogonal matrices and generalized Wishart densities; Moments of random determinants; Distributions of eigenvalues and eigenvectors of random matrices; Limit theorems for random determinants;

Random Gram determinants; Determinants of Toeplitz and Hankel random matrices; Jacobi random matrices; Fredholm random determinants; Linear equations with random coefficients;

Integral equations with random kernels; Distributions of eigenvalues of additive random matrix-valued processes; The stochastic Lyapunov problem for systems of stationary linear differential equations;

Random determinants in the theory of parameter estimation; Random determinants in control theory; Random determinants in general statistics; Random determinants in pattern recognition, experimental design, numerical analysis and physics.

Reviewer: I.G.Zhurbenko

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60E05 | Probability distributions: general theory |

60F99 | Limit theorems in probability theory |

60H25 | Random operators and equations (aspects of stochastic analysis) |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |