Teaching 2020 - Large Deviations - Department of Mathematics

Dr. D.R. Michiel Renger - Course "Large Deviations", TU Berlin, Summer Semester 2020

I gave this online course during the corona pandemic, but will keep the material online.

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Course description

Large deviations theory aims to characterise the exponential rate of convergence for sequences of probability measures. It has important applications to information theory and physics; in particular, large deviations theory explains the origin, meaning and role of entropy. Although a subbranch of probability theory, it can be rather analytic, using for example convexity arguments, topological arguments and some variational calculus. However, only basic knowledge of probability and topology are required to follow the course.

The course will cover both the fundamental theory as well as a number of important applications, like the local times of a random walk, large-time asymptotics of a Markov process and sequences of Markov processes.

Format

The course consists of weekly pencasts and exercises, both posted below. The exercises are an essential part of the course, and contain arguments and heuristics that are not in the pencasts, and sometimes even difficult to find in the literature.

Material

Wolfgang König - Größe Abweichungen, Techniken und Anwendungen (Springer 2020). For those who do not speak German; most material can also be found in the books by Dembo & Zeitouni, or the book by den Hollander.
Amir Dembo and Ofer Zeituni - Large Deviations, Techniques and Applications (Springer 1998).
Frank den Hollander - Large Deviations (AMS 2000).

In the program below I indicate where the discussed material of each week can be found.

Prerequisites: probability theory and basic knowledge of topology.

Weekly program

Week 1
Pencast #1a Pencast #1b Exercises	König, Th. 1.4.3 den Hollander, Th. I.4
Week 2
Pencast #2 Exercises	König, Th. 1.4.3 den Hollander, Th. I.4
Week 3
Pencast #3 Exercises	den Hollander, Th. III.8 (Dembo & Zeitouni Lem. 4.1.4) den Hollander, Th. III.3 König, Lem. 2.1.5 & Bem. 2.1.6 (Dembo & Zeitouni Lem. 1.2.18) König, Lem. 2.1.3 (Dembo & Zeitouni Sect. 1.2) Dembo & Zeitouni, Lem. 4.1.23
Week 4
Pencast #4 Exercises	König, Lem. 1.4.1.2(iii) & Satz 2.2.1
Week 5
Pencast #5a (last weeks exercise) Pencast #5b Exercises	den Hollander, Th. II.1 & Lem. II.4 (combinatoric proof of Sanov on finite state space) König, Satz 2.4.1 & Lem. 2.4.3 (general proof of Sanov) (Dembo & Zeitouni Th. 6.2.10 (ridiculously general and advanced proof of Sanov))
Week 6
For those who are curious how to get the Cramér rate functional from Sanov; I couldn't find it in the books (except as an exercise in Den Hollander), so I wrote it down here. Pencast #6 Exercises	König, Lem. 2.5.1 & Satz 2.5.2 den Hollander, Th. II.8 (Dembo & Zeitouni Th. 3.1.13 finite space & Cor. 6.5.10 on a Polish space)
Week 7
Pencast #7 Exercises	König, Satz 3.3.1 & Satz 3.3.3 Dembo & Zeitouni Th. 4.3.1 & Th. 4.4.2 (den Hollander, Th. III.13)
Week 8
here are the solution to last weeks exercises. Pencast #8A (sorry there's typo in the definition of the subdifferential, it should be ∂f(x):={ x^* ∈ X^: f(y) ≥ f(x) + 〈 x^,y-x 〉} ) Pencast #8B Pencast #8C Exercises	König, Lem. 3.4.1 & 3.4.3, Satz 3.4.4 Dembo & Zeitouni Th. 4.5.3 & Th. 4.5.20 (under relaxed assumptions) & Th. 4.5.27 den Hollander, Th. V.6 (in ℝ^d).
Week 9
Pencast #9A Pencast #9B Only one simple exercise this week	König, Satz 3.5.7 & Satz 3.6.1 Dembo & Zeitouni Lem. 4.1.5(b)
Week 10
Pencast #10A Pencast #10B Pencast #10C Exercises	Dembo & Zeitouni Cor. 4.2.6 & Th. 4.6.1 (See also Feng & Kurtz - Large deviations for stochastic processes, Th. 4.28 for a version in Skorohod space)
Week 11
Pencast #11 Exercises	Dembo & Zeitouni Section 5.1 & Th. 5.2.3 ( & Section 5.6) König Satz 2.3.1 & Satz 3.5.6
Week 12
Pencast #12 (last) (no exercises)	This is unfortunately not in the literature we've used so far Girsanov Theorem: Kipnis and Landim - Scaling Limits of Interacting Particle Systems (Prop. A.7.1) Empirical Process: Shwartz and Weiss - Large deviations for performance analysis (Th. 4.1)