Instructor: Igor
Kortchemski. The course is taught is English.

Random phenomena are modelled using modern probability theory, defined
in the 1930s by Kolmogorov using measure theory as a cornerstrone. The
theoretical understanding of these foundations is an asset to forge
intuition, to understand the objects of interest and to mobilize them in
an applied or theoretical framework. Without any necessary
prerequisites, the aim of this EA will be to take the time to
consolidate some of the foundations of probability related to measure
theory in an accessible way, emphasizing ideas and giving several
applications on important models in probability theory (such as
percolation, random walks in random media, random matrices,
Brownian motion, random fractals, random graphs, etc.).

This course is designed for an audience with a variety of interests: on the one hand, it may be of interest to students willing to study probability in greater depth, and on the other hand, it may be of interest to students interested in business applications (a good understanding of probability theory is essential to know how to navigate and innovate in the world of applications).

Evaluation is done through an oral presentation of a research paper or a significant application.Moodle webpage of the course.

The teaching takes place on Tuesday, from 13h30 to 15h.

The course contains a "lecture" part and an "exercise" part. Given the
context, the "lecture" part will be organised in a "flipped classroom"
way: at week *n*, lecture notes are given (covering the content
of the theme of week *n+1*), together with an exercise. You are
asked to read these lecture notes for week *n+1* try to solve
the exercise and upload your work on Moodle. At week *n+1*, the
key elements of the lecture notes which were given at week *n*
will be discussed, the exercise will be discussed, and then a tutorial
session with new exercises will take place.

Provisional programme:

- Week 0 (September 22): Introduction, description of the course.
- Week 1 (September 29): Mesures. (σ-fields, generated σ-field, mesures, monotone class theorem)
- X-Forum
- Week 2 (October 13): Measurable functions. (measurable functions, product σ-field, approximation by simple functions, Doob-Dynkin lemma, contructing measures and Stieltjes measures)
- Week 3 (October 20): Integration. (integral with respect to a measure, monotone convergence, Radon-Nikodym theorem, Fubini theorem, convergence theorems)
*Holidays*- Week 4 (November 3): Independence (Independence of σ-fields and of random variables, transfer theorem, independence and integration, dummy function method, independence of infinite families, Kolmogorov 0-1 law, Borel-Cantelli lemmas) and recap on different notions of convergence for random variables.
- Week 5 (November 10): Conditioning. (conditional expectation, conditional law)
- Week 6 (November 17): Brownian motion (1/2). (construction)
- Week 7 (November 24): Brownian motion (2/2).
(properties)

- Week 8 (December 1): Random Poisson measures (construction and properties)
- Oral presentations December 15th.

Evaluation is done through an oral presentation of a research paper or a significant application.