Limit theorems and applications

(course from the M2 Mathématiques de l'aléatoire)

lecture notes and the print friendly version

Brief presentation

In short, this course deals with the convergence in distribution of random variables, or equivalently weak convergence of probability measures. It is split into three parts:

Convergence of real-valued random variables.
Our aim is to develop a general theory of convergence in distribution for random variables which are not real-valued, but live in a general metric space. This part is the most fundamental in that the tools we will develop are used in many contexts. We also review and extend (?) some results on probability on the real line and Euclidean spaces.
Key words are: Portmanteau theorem, Polish spaces, Lévy-Prohorov metric, Prohorov’s theorem, characteristic functions, distribution functions.
Continuous-paths processes.
Here we apply the previous general theory to the case of random continuous functions and prove convergence in distribution in this setting; we culminate with the proof of the convergence of finite-variance random walks to the Brownian motion. We also briefly discuss the case of discontinuous functions.
Key words are: continuous-paths stochastic processes, tightness criteria, Donsker theorem, cadlag functions, Skorokhod topology.
Infinitely divisible laws & Lévy processes.
We present the family of infinitely divisible laws, which are the possible limits of sums of i.i.d. random variables, in an extension of the CLT. We then construct the related family of stochastic processes that generalise the Brownian motion, which are thus eventually the limits of random walks, although we shall not prove any convergence.
Key words are: Lévy-Khintchine formula, Poisson processes, Lévy processes, Lévy-Itō construction, stable distributions, domains of attraction.

Prerequisites and relation with other courses

Most importantly, we will assume familiarity with the basic theory of convergence in distribution for real-valued random variables: definition, link with the distribution function, link with the characteristic function, central limit theorem. No familiarity with stochastic processes or advanced probability theory, such as martingales or Markov chains is essential, but mastery of the basics (such as monotone class arguments) is key.

The course is selfcontained, but some of the topics relate to other courses:

Part 1 is fairly general, and the tools developped there, such as Prohorov's theorem, are used almost in all situations in which ones want to prove convergence in distribution. Here we will apply it to stochastic processes, but the theory applies to random graphs, random matrices, particle systems, etc.
Part 2 on continuous stochastic processes will share similarities with stochastic calculus, and offer different perspective on the same objects. For example, we will prove the existence of Brownian motion by constructing it as limits of random walks.
Part 3 on Lévy processes, in addition to describing the limits of random walks, is deeply connected to branching processes and also relates to random graphs and other objects. These processes are also related to Poisson random measures, another standard tool in probability (studied in the random graph course). Stable distributions and their domain of attraction appear whenever we want to consider limits of rescaled of sums of i.i.d. random variables with infinite variance, would they be the law of the degrees in a random graph, the increments of a random walk, the entries of a random matrix, etc.

Practical informations

The course consists of 10 sessions of 3h. Some time will be dedicated to exercises, quite irregularly, but you will be told in advance (and ask to prepare the exercises for the sessions to be useful).

The validation of the course is via a written exam organised at the end.

References

Here are some books that can be useful in relation with this course. The books closer to this course would be those of Billingsley for Part 1 and Part 2 and Sato for Part 3. However feel free to look at the other ones in the list and also outside this list: the important point is to find one or more that you enjoy reading and find complementary to the lectures.

Some books cover all the topics discussed here (and much more):
- Fristedt & Gray - A Modern Approach to Probability Theory
- Kallenberg - Foundations of modern probability
- Stroock - Probability theory an analytic view

In addition, some books cover only some parts of the course:

For generalities on weak convergence and applications to stochastic processes:
- Billingsley - Convergence of Probability Measures
- Ethier & Kurtz - Markov processes characterization and convergence
- Klenke - Probability Theory. A Comprehensive Course
- Parthasarathy - Probability measures on metric spaces
For infinitely divisible laws, stable laws, and Lévy processes:
- Feller - An Introduction to Probability Theory and Its Applications, Volume 2
- Kyprianou - Fluctuations of Lévy Processes with Applications
- Sato - Lévy Processes and Infinitely Divisible Distributions