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Gaussian random functions

Update: As of 2012, Mikhail Lifshits has now published a Springer booklet on Lectures on Gaussian Processes.

Abstract

This document is based on my notes taken at lectures about Gaussian Random Functions which were held by visiting Prof. Dr. M. Lifshits (University of St. Petersburg) at Technische Universität Darmstadt from July 11 to 22, 2005. These pages only reflect what I picked up from this fine lecture but not necessarily what is mathematically true. I therefore appreciate any hint about things I messed up. — ''August 23, 2005''

For this second version I have fixed a lot of typos and corrected some minor errors. Nonetheless, there should be lots of them still to be found. — ''March 7, 2006''

Many thanks to Prof. Dr. M. Lifshits for reading these notes. His corrections have been incorporated in this third version. — ''April 2, 2006''

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For those who wonder: all graphics (except the two formal diagrams) were made using PyX and Simply Draw. You can also find some examples extracted from this article.

Table of Contents

  1. Definition of Gaussian Objects
    1. One Dimensional Case
    2. Finite Dimensional Case
    3. General Case
  2. Examples of Gaussian Objects
    1. Main Examples
    2. Special Examples of Gaussian Processes
    3. Gaussian White Noise and Integration
  3. Kernels of Gaussian Measures
    1. Definition and Basic Properties
    2. Examples
    3. Factorization Theorem
    4. Kernel in Literature
    5. Linear Transformations
  4. Cameron-Martin Theorem
    1. Absolute Continuity of Shifted Measures
    2. Borell Inequality for Shifted Sets
  5. Isoperimetric Inequalities
    1. Introduction to Isoperimetric Inequalities
    2. Gaussian Isoperimetric Inequality
    3. Concentration Principle
  6. Large Deviations
    1. Introduction
    2. Gaussian Large Deviations
  7. Convexity and Other Inequalities
    1. Concavity of Measures
    2. Correlation Conjecture
    3. Shift-isoperimetric Inequalities and S-conjecture
  8. Metric Entropy and Sample Paths
    1. General Metric Entropy
    2. Metric Entropy of Gaussian Processes
    3. Metric Entropy of an Operator
  9. Expansions
    1. General Series of Independent Vectors
    2. Linear Operators and Gaussian Vectors
  10. Strassen's Law
    1. Scalar Laws of Iterated Logarithm
    2. Functional Law
    3. Proof of Strassen's Law
    4. Extensions of Strassen's Law