Course "Large Deviations", TU Berlin, Summer Semester 2020

Basic info

• Lecturer: Michiel Renger
• Weekly hourse (SWS): 2.0
• Material:
  • Wolfgang König - Große Abweichungen, Techniken und Anwendungen (lecture notes) 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). The TU Berlin has made this book digitally available for you through this link (should work from Campusnetz or VPN).
  • Frank den Hollander - Large Deviations (AMS). The TU Berlin has made this book digitally available for you through this link.
  In the program below I indicate where the discussed material of each week can be found.
• Language: English
• Prerequisites: Probability Theory (WTI and WTII) and basic knowledge of topology

Less information can be found on the corresponding TU Berlin website (LSF).

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 WTI and WTII and basic knowledge of 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, if time allows, sequences of Markov processes.

Format

Please contact me if you are interested in the course.

Due to the corona outbreak, the course will be fully online. More precisely:

• Every Tuesday (starting 21st April) I will upload new lecture videos/pencasts and exercises on this website.
• The exercises are an essential part of the course. Please email me your solution of the exercises (or attempts thereof) by friday. A simple scan or photo of hand-written notes is fine, as long as it's readible!
  
• Every Tuesday 8:15--9:45 (starting 28th April) I will post a (Jitsi or Zoom) link, where anyone can join a question-and-answer session.

Of course, I'm also available for questions by email, and open for feedback on the material and format of the course.

Exams

The exam will be held either in person (TU or WIAS), or online. I'm quite flexible (although I advice not to wait too long), but we'll need a second examinator, so it will be best to organise multiple exams on the same day. If you want to take the exam, please email me: 1) whether you want to do the exam in person or online, and 2) your preferred week.

Weekly program

21st April
Recall and/or look up the definition of weak convergence and the corresponding Portemanteau Theorem (in your favorite book or website). Pencast #1a Pencast #1b Exercises (please mail me your solutions by Friday 24th April)	König, Th. 1.4.3 den Hollander, Th. I.4
28st April
Q&A session over Jitsi Pencast #2 Exercises (please mail me your solutions by Friday 1st May)	König, Th. 1.4.3 den Hollander, Th. I.4
5th May
Q&A session over Jitsi Pencast #3 Exercises (please mail me your solutions by Friday 8th May)	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
12th May
Q&A session over Jitsi Pencast #4 Exercises (please mail me your solutions by Friday 15th May)	König, Lem. 1.4.1.2(iii) & Satz 2.2.1
19th May
Q&A session over Jitsi Pencast #5a (last weeks exercise) Pencast #5b Exercises (please mail me your solutions by Friday 22th May) Remark: this weeks videos are pretty long. There's not much hurry to watch them though, and there's only one very short exercise. I will make a shorter video next week. In general, please don't spend tooo much time on the exercises, you can always contact me in you're stuck... Remark on literature: both books by Dembo & Zeitouni and den Hollander are made digitally available (at least from the TU) by the links above. The German lecture notes by König will be officially published soon. This means that it will be available as a book (Wolfgang says it will be cheap), but he will need to remove the link from his website soon...!	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))
25th May
Q&A session over Jitsi 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 (please mail me your solutions by Friday 29th May)	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)
2nd June
Q&A session over Jitsi Pencast #7 Exercises (please mail me your solutions by Friday 5th June)	König, Satz 3.3.1 & Satz 3.3.3 Dembo & Zeitouni Th. 4.3.1 & Th. 4.4.2 (den Hollander, Th. III.13)
8th June
here are the solution to last weeks exercises. Q&A session over Jitsi 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 (please mail me your solutions by Friday 12th June)	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 ℝd).
16th June
Q&A session over Jitsi Pencast #9A Pencast #9B Only one simple exercise this week (please mail me your solution by Friday 19th June)	König, Satz 3.5.7 & Satz 3.6.1 Dembo & Zeitouni Lem. 4.1.5(b)
23rd June
Q&A session over Jitsi Pencast #10A Pencast #10B Pencast #10C Exercises (please mail me your solution by Friday 26th June)	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)
30th June
Remark: updated info on exams Remark 2: there will be one more lecture Remark 3: in case you like this course, there are definitely possibilities to do a bachelor or master project with me and/or Wolfgang König. If you are interested, please email me when you plan to start your bachelor or master. Q&A session over https://meet.jit.si/TULargeDeviations Pencast #11 Exercises (please mail me your solution by Friday 3rd July)	Dembo & Zeitouni Section 5.1 & Th. 5.2.3 ( & Section 5.6) König Satz 2.3.1 & Satz 3.5.6
7th July
Q&A session over https://meet.jit.si/TULargeDeviations 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)
14th July
Q&A session Feel free to contact me any time for more questions!