Theory of Stochastic Canonical Equations

1. Front Matter

   Pages i-xlix

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2. Canonical Equation K ₁

   • Vyacheslav L. Girko

   Pages 1-24

3. Canonical Equation K ₂. Necessary and Sufficient Modified Lindeberg’s Condition. The Wigner and Cubic Laws

   • Vyacheslav L. Girko

   Pages 25-50

4. Regularized Stochastic Canonical Equation K ₃ for Symmetric Random Matrices with Infinitely Small Entries

   • Vyacheslav L. Girko

   Pages 51-74

5. Stochastic Canonical Equation K ₄ for Symmetric Random Matrices with Infinitely Small Entries. Necessary and Sufficient Conditions for the Convergence of Normalized Spectral Functions

   • Vyacheslav L. Girko

   Pages 75-86

6. Canonical Equation K ₅ for Symmetric Random Matrices with Infinitely Small Entries

   • Vyacheslav L. Girko

   Pages 87-92

7. Canonical Equation K ₆ for Symmetric Random Matrices with Identically Distributed Entries

   • Vyacheslav L. Girko

   Pages 93-96

8. Canonical Equation K ₇ for Gram Random Matrices

   • Vyacheslav L. Girko

   Pages 97-118

9. Canonical Equation K ₈

   • Vyacheslav L. Girko

   Pages 119-124

10. Canonical Equation K ₉ for Random Matrices whose Entries have Identical Variances

    • Vyacheslav L. Girko

    Pages 125-128

11. Canonical Equation K ₁₀. Necessary and Sufficient Modified Lindeberg Condition

    • Vyacheslav L. Girko

    Pages 129-140

12. Canonical Equation K ₁₁. Limit Theorem for Normalized Spectral Functions of Empirical Covariance Matrices under the Modified Lindeberg Condition

    • Vyacheslav L. Girko

    Pages 141-152

13. Canonical Equation K ₁₂ for Random Gram Matrices with Infinitely Small Entries

    • Vyacheslav L. Girko

    Pages 153-160

14. Canonical Equation K ₁₃ for Random Gram Matrices with Infinitely Small Entries

    • Vyacheslav L. Girko

    Pages 161-164

15. The Method of Random Determinants for Estimating the Permanents of Matrices and the Canonical Equation K ₁₄ for Random Gram Matrices

    • Vyacheslav L. Girko

    Pages 165-180

16. Canonical Equation K ₁₅ for Random Gram Matrices with Identically Distributed Entries

    • Vyacheslav L. Girko

    Pages 181-184

17. Canonical Equation K ₁₆ for Sample Covariance Matrices

    • Vyacheslav L. Girko

    Pages 185-202

18. Canonical Equation K ₁₇ for Identically Distributed Independent Vector Observations and the G ₂-Estimators of the Real Stieltjes Transforms of the Normalized Spectral Functions of the Covariance Matrices

    • Vyacheslav L. Girko

    Pages 203-206

19. Canonical Equation K ₁₈ for the Special Structure of Vector Observations

    • Vyacheslav L. Girko

    Pages 207-224

20. Canonical Equation K ₁₉

    • Vyacheslav L. Girko

    Pages 225-230

Theory of Stochastic Canonical Equations collects the major results of thirty years of the author's work in the creation of the theory of stochastic canonical equations. It is the first book to completely explore this theory and to provide the necessary tools for dealing with these equations. Included are limit phenomena of sequences of random matrices and the asymptotic properties of the eigenvalues of such matrices. The book is especially interesting since it gives readers a chance to study proofs written by the mathematician who discovered them.
All fifty-nine canonical equations are derived and explored along with their applications in such diverse fields as probability and statistics, economics and finance, statistical physics, quantum mechanics, control theory, cryptography, and communications networks. Some of these equations were first published in Russian in 1988 in the book Spectral Theory of Random Matrices, published by Nauka Science, Moscow.
An understanding of the structure of random eigenvalues and eigenvectors is central to random matrices and their applications. Random matrix analysis uses a broad spectrum of other parts of mathematics, linear algebra, geometry, analysis, statistical physics, combinatories, and so forth. In return, random matrix theory is one of the chief tools of modern statistics, to the extent that at times the interface between matrix analysis and statistics is notably blurred.
Volume I of Theory of Stochastic Canonical Equations discusses the key canonical equations in advanced random matrix analysis. Volume II turns its attention to a broad discussion of some concrete examples of matrices. It contains in-depth discussion of modern, highly-specialized topics in matrix analysis, such as unitary random matrices and Jacoby random matrices.
The book is intended for a variety of readers: students, engineers, statisticians, economists and others.

• Estimator
• Variance
• linear algebra
• mathematics
• statistics

• Michigan State University, East Lansing, USA

  Vyacheslav L. Girko

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.

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