Supplementary Materials:
(1) Gartner-Ellis Theorem - read it from the book of Dembo and Zeitouni
(2) Slides of a Talk on Large Deviations for Heavy Tails
(3) ICTS Lectures on Large Deviations for Heavy Tails: Lecture 1 Lecture 2
(4) Practice Problems on Sanov's Theorem for Finitely Supported Random Variables
Lecture 20 (01/05/2021): Notes Video
Content: Sanov's theorem for finitely supported random variables and its proof [please read the proof of the lemma from the book of Dembo and Zeitouni]
Lecture 19 (28/04/2021): Notes Video
Content: Guest lecture by Sayan Das on Large Deviations of Integrable Models
An informal introduction based on joint work of Sayan Das with Li-Cheng Tsai [1], Evgeni Dimitrov [2], and Weitao Zhu [3]. Simulations of TASEP were presented using [4].
[1]: https://arxiv.org/abs/1910.09271
[2]: https://arxiv.org/abs/2103.15227
[3]: https://arxiv.org/abs/2104.00661
[4]: https://wt.iam.uni-bonn.de/ferrari/research/jsanimationtasep
Lecture 18 (24/04/2021): Notes Video
Content: Sanov's theorem for Bernoulli random variables
Lecture 17 (17/04/2021): Notes Video
Content: Contraction principle, preliminaries on Sanov's theorem for finitely supported random variables
Lecture 16 (10/04/2021): Notes Video
Content: Slutsky type theorems for exponential tightness and good LDP
Lecture 15 (07/04/2021): Notes Video
Content: Exponential tightness on a countable product of Polish spaces, comparison between weak convergence and LDP on Polish spaces, analogue of Prokhorov's theorem for exponential tightness
Lecture 14 (03/04/2021): Notes Video
Content: Weak LDP, LDP, good LDP, exponential tightness and their relations [see Theorem 2.49 (Pg 44) of these notes (by Peter Orbanz) for a proof of the fact that any probability measure on a Polish space is tight]
Lecture 13 (31/03/2021): Notes Video
Content: Weak large deviation principle and its connection to existence of limits along an open base
Lecture 12 (24/03/2021): Notes Video
Content: Uniqueness of rate function, Varadhan's integral lemma
Lecture 11 (06/03/2021): Notes Video
Content: The general definition of large deviation principle for a sequence of measures on a Polish space, speed and rate function
Lecture 10 (03/03/2021): Notes Video
Content: Last part of the proof of lower bound in Cramer's theorem for random vectors, Laplace principle
Lecture 09 (27/02/2021): Notes Video
Content: Last part of the proof of upper bound in Cramer's theorem for random vectors, first part of the proof of lower bound in Cramer's theorem for random vectors
Lecture 08 (24/02/2021): Notes Video
Content: First part of the proof of upper bound in Cramer's theorem for random vectors
Lecture 07 (20/02/2021): Notes Video
Content: Cramer's theorem for random vectors, properties of multivariate log-MGF and its Fenchel-Legendre transform
Lecture 06 (17/02/2021): Notes Video
Content: Last part of the proof of large deviation lower bound in Cramer's theorem
Lecture 05 (13/02/2021): Notes Video
Content: First part of the proof of large deviation lower bound in Cramer's theorem
Lecture 04 (10/02/2021): Notes Video
Content: More properties of log-MGF and its Fenchel-Legendre transform, proof of large deviation upper bound in Cramer's theorem
Lecture 03 (06/02/2021): Notes Video
Content: General statement of Cramer's theorem, properties of log-MGF and its Fenchel-Legendre transform
Lecture 02 (03/02/2021): Notes Video
Content: Cramer's theorem for standard normal distribution, upper bound in the general case (the video for the standard normal case is missing - please go through the notes and ask the instructor if you have questions)
Lecture 01 (30/01/2021): Notes Video
Content: Introduction and historical motivation