**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*

## Download

Link | Size | Description | Hits |
---|---|---|---|

gaussian.pdf | 1.62 MB | Gaussian Random Functions (PDF, 55 pages) | 2049 |

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

- Definition of Gaussian Objects
- One Dimensional Case
- Finite Dimensional Case
- General Case

- Examples of Gaussian Objects
- Main Examples
- Special Examples of Gaussian Processes
- Gaussian White Noise and Integration

- Kernels of Gaussian Measures
- Definition and Basic Properties
- Examples
- Factorization Theorem
- Kernel in Literature
- Linear Transformations

- Cameron-Martin Theorem
- Absolute Continuity of Shifted Measures
- Borell Inequality for Shifted Sets

- Isoperimetric Inequalities
- Introduction to Isoperimetric Inequalities
- Gaussian Isoperimetric Inequality
- Concentration Principle

- Large Deviations
- Introduction
- Gaussian Large Deviations

- Convexity and Other Inequalities
- Concavity of Measures
- Correlation Conjecture
- Shift-isoperimetric Inequalities and S-conjecture

- Metric Entropy and Sample Paths
- General Metric Entropy
- Metric Entropy of Gaussian Processes
- Metric Entropy of an Operator

- Expansions
- General Series of Independent Vectors
- Linear Operators and Gaussian Vectors

- Strassen's Law
- Scalar Laws of Iterated Logarithm
- Functional Law
- Proof of Strassen's Law
- Extensions of Strassen's Law