Gaussian Random Functions

1. Front Matter

   Pages i-xi

   PDF

2. Gaussian Distributions and Random Variables

   • M. A. Lifshits

   Pages 1-7

3. Multi-Dimensional Gaussian Distributions

   • M. A. Lifshits

   Pages 8-15

4. Covariances

   • M. A. Lifshits

   Pages 16-21

5. Random Functions

   • M. A. Lifshits

   Pages 22-29

6. Examples of Gaussian Random Functions

   • M. A. Lifshits

   Pages 30-40

7. Modelling the Covariances

   • M. A. Lifshits

   Pages 41-52

8. Oscillations

   • M. A. Lifshits

   Pages 53-67

9. Infinite-Dimensional Gaussian Distributions

   • M. A. Lifshits

   Pages 68-83

10. Linear Functionals, Admissible Shifts, and the Kernel

    • M. A. Lifshits

    Pages 84-100

11. The Most Important Gaussian Distributions

    • M. A. Lifshits

    Pages 101-107

12. Convexity and the Isoperimetric Property

    • M. A. Lifshits

    Pages 108-138

13. The Large Deviations Principle

    • M. A. Lifshits

    Pages 139-155

14. Exact Asymptotics of Large Deviations

    • M. A. Lifshits

    Pages 156-176

15. Metric Entropy and the Comparison Principle

    • M. A. Lifshits

    Pages 177-210

16. Continuity and Boundedness

    • M. A. Lifshits

    Pages 211-229

17. Majorizing Measures

    • M. A. Lifshits

    Pages 230-245

18. The Functional Law of the Iterated Logarithm

    • M. A. Lifshits

    Pages 246-257

19. Small Deviations

    • M. A. Lifshits

    Pages 258-275

20. Several Open Problems

    • M. A. Lifshits

    Pages 276-281

It is well known that the normal distribution is the most pleasant, one can even say, an exemplary object in the probability theory. It combines almost all conceivable nice properties that a distribution may ever have: symmetry, stability, indecomposability, a regular tail behavior, etc. Gaussian measures (the distributions of Gaussian random functions), as infinite-dimensional analogues of tht< classical normal distribution, go to work as such exemplary objects in the theory of Gaussian random functions. When one switches to the infinite dimension, some "one-dimensional" properties are extended almost literally, while some others should be profoundly justified, or even must be reconsidered. What is more, the infinite-dimensional situation reveals important links and structures, which either have looked trivial or have not played an independent role in the classical case. The complex of concepts and problems emerging here has become a subject of the theory of Gaussian random functions and their distributions, one of the most advanced fields of the probability science. Although the basic elements in this field were formed in the sixties-seventies, it has been still until recently when a substantial part of the corresponding material has either existed in the form of odd articles in various journals, or has served only as a background for considering some special issues in monographs.

• Gaussian distribution
• Gaussian measure
• Probability theory
• Variance
• distribution
• law of the iterated logarithm
• random function

• St. Petersburg University of Finance and Economics, St. Petersburg, Russia

  M. A. Lifshits

Policies and ethics