Gaussian processes
MATH-426 / 5 credits
Teacher:
Language: English
Remark: pas donné en 2022-2023
Summary
This is an introductory course on Gaussian fields and processes - or more shortly, on Gaussian magic. 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 (via stable laws, entropy etc) and revisiting the Central Limit Theorem;
- Basic properties of finite-dimensional Gaussian measures (including marginal laws, conditional laws);
- Existence and constructions of infinite-dimensional Gaussian processes, Reproducing Kernel Hilbert Spaces and their basic properties;
- Suprema and continuity of Gaussian processes;
- Some of the models potentially discussed: Gaussian random matrices, the Random Energy Model, the Discrete Gaussian free field; Gaussian process regression; Brownian motion/bridge.
With high motivation, the course can be followed already at BA6 level too.
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.
In the programs
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Gaussian processes
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Gaussian processes
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Gaussian processes
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Gaussian processes
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
Reference week
| Mo | Tu | We | Th | Fr | |
| 8-9 | |||||
| 9-10 | |||||
| 10-11 | |||||
| 11-12 | |||||
| 12-13 | |||||
| 13-14 | |||||
| 14-15 | |||||
| 15-16 | |||||
| 16-17 | |||||
| 17-18 | |||||
| 18-19 | |||||
| 19-20 | |||||
| 20-21 | |||||
| 21-22 |
Légendes:
Lecture
Exercise, TP
Project, other