# Coursebooks

## Gaussian processes

Aru Juhan

English

#### Summary

This is an introductory course on Gaussian processes. By discussing both the general theory and concrete examples, we will try to understand where and how Gaussian processes appear, and how to study them.

#### Content

This course serves as an introduction to the world of Gaussian processes. Gaussian processes are omnipresent in modelling random phenomena. There are at least two reasons for it:

1) Gaussian processes appear naturally through the Central Limit Theorem and its relatives;

2) Gaussian models have many special properties that make their mathematical study interesting...and possible.

The aim of this course is to better understand these two reasons by both looking into general properties of Gaussian measures, and by studying in detail some concrete Gaussian models.

Here is a tentative list of topics:

- Different characterisations of standard Gaussians and revisiting the Central Limit Theorem;
- Basic properties of finite-dimensional Gaussian measures (including marginal laws, conditional laws, Cameron-Martin shifts);
- High-dimensional Gaussian processes and the concentration of measure phenomena (some version of Borell-TIS inequality and Dudley entropy bound);
- Existence and constructions of infinite-dimensional Gaussian processes and their basic properties;
- Some of the models potentially discussed: Gaussian random matrices, the Random Energy Model, the Discrete Gaussian free field; Gaussian process regression; Brownian motion/bridge.

#### Learning Prerequisites

##### Required courses

Mathematics Bachelor's level knowledge of analysis, linear algebra and probability (for example, the Bloc "Science de Base" in EPFL Mathematics Bachelor's program).

##### Recommended courses

From the Bachelor's program: Martingales and applications; Stochastic processes;
From the Master's program: Probability theory, Theory of Stochastic calculus.

#### Learning Outcomes

By the end of the course, the student must be able to:
• Recognize Gaussian processes
• Characterize Gaussian processes
• Analyze Gaussian processes

#### Teaching methods

Lectures and exercise classes.

#### Assessment methods

Oral exam

Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.

#### Supervision

 Office hours No Assistants Yes Forum No

#### Resources

##### Bibliography

Will be discussed in class.

### Reference week

MoTuWeThFr
8-9
9-10
10-11
11-12
12-13  MAA331
13-14
14-15
15-16CM1105
16-17
17-18
18-19
19-20
20-21
21-22

Lecture
Exercise, TP
Project, other

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• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German