Cyril Marzouk
École polytechnique
Centre de Mathématiques Appliquées - UMR 7641 CNRS
Route de Saclay
91128 Palaiseau Cedex
Email : cyril (dot) marzouk (at) polytechnique (dot) edu
Office : 00 30 23 (wing 0, 2nd floor)
Phone : +33 (0) 1 69 33 45 78
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Limit theorems and applications
(course from the M2 Mathématiques de l'aléatoire)
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 four parts:
- Convergence of real-valued random variables.
Our aim is to generalise the central limit theorem when the variance is infinite.
Key words are:
characteristic functions,
infinitely divisible distributions,
Lévy-Khintchine formula,
stable distributions,
domains of attraction.
- Weak convergence in metric spaces.
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.
Key words are:
Portmanteau theorem,
Polish spaces,
Lévy-Prohorov metric,
Prohorov’s theorem.
- 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.
Key words are:
continuous-paths stochastic processes,
tightness criteria,
Donsker theorem.
- Poisson random measures & Lévy processes.
The theory of Poisson random measures goes way beyond the scope of this course and we only offer an introduction here to these measure-valued random variables.
We then apply it to construct the family of stochastic processes that generalise the Brownian motion (which are thus eventually the limits of random walks with infinite variance); again, they deserve an entire course and we only introduce them here.
Key words are:
Poisson measures,
compensated integrals,
Poisson point processes,
Lévy processes,
Lévy-Itō construction.
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 more familiarity with stochastic processes or advanced probability theory is essential.
Although we will take another path, Part 2 and Part 3 of the course can be studied with analysis and function analysis methods.
The course is selfcontained, but some of the topics relate to other courses:
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Stable distributions and their domain of attraction appear whenever you want to consider limits involving laws 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.
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Part 2 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.
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Part 3 will share similarities with the course on stochastic calculus, and offer different perspective on the same objects.
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Poisson random measures are a standard tool in probability and are used both in the modelisation of many phenomena (in random graphs, queues, Boolean percolation etc.) and in theory (for example in excursion theory of processes).
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Finally, Lévy processes, in addition to describing the limits of random walks, are deeply connected to branching processes and also relate to random graphs.
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 final 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 Feller and Sato for Part 1 and those of Billingsley for Part 2 and 3, with some incursion in that of Ethier & Kurtz for Part 2.
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.
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Some books cover all the topics discussed here (and way more):
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Fristedt & Gray - A Modern Approach to Probability Theory
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Kallenberg - Foundations of modern probability
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Stroock - Probability theory an analytic view
In addition, some books cover only some parts of the course:
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For infinitely divisible laws and stable laws:
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Bingham, Goldie, & Teugels - Regular variation
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Feller - An Introduction to Probability Theory and Its Applications, Volume 2
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Sato - Lévy Processes and Infinitely Divisible Distributions
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Seneta - Regularly Varying Functions
For generalities on weak convergence and applications to stochastic processes:
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Billingsley - Convergence of Probability Measures
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Dudley - Real Analysis and Probability
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Ethier & Kurtz - Markov processes characterization and convergence
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Klenke - Probability Theory. A Comprehensive Course
For Poisson random measures:
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Kingman - Poisson Processes
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Last & Penrose - Lectures on the Poisson Process
For Lévy processes:
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Kyprianou - Fluctuations of Lévy Processes with Applications
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Sato - Lévy Processes and Infinitely Divisible Distributions