Home Page: An Introduction to Large Deviations, Block 1, 2018

Course Details:

Lecturer: Jeffrey Collamore; ph.: 3532 0782; e-mail: collamore-at-math.ku.dk.

Lectures: Monday 9-12 in Aud. 10; Tuesday 15-17 in Aud. 10.

Evaluation: There will be a 30-minute oral exam and exercises. Your grade will be based primarily on the oral exam, although the exercises will also count for a small part of the grade. In particular, it is required that all exercises are attempted and the exercise set is "passed" in order to participate in the oral exam.

Prerequisites: A basic measure-theoretic course in probability theory.

Course material: Lecture notes.

Course description: This will be an introductory course in the theory of large deviations and its applications. Topics will include: Cramer's theorem for sample means and for level crossings, moderate deviations, heavy tails, and a brief introduction to the multidimensional theory. Applications will include insurance mathematics and finance, statistics, and Monte Carlo methods (importance sampling).

Schedule for the lectures:

03.09.18: Introduction (JC pp. 4-6); Cramer's theorem in R (JC, pp. 7-10).
04.09.18: Cramer's theorem in R (JC, pp. 7-10, cont.); proof of lower bound (JC, pp. 10-12). Note: The truncation and smoothing techniques are described in the notes starting on p. 12; this is needed for a complete proof of the lower bound.
10.09.18: Proof of upper bound (JC, pp. 14-16); sharp asymptotics (first handout, pp. 16-20).
11.09.18: The one-dimensional ruin problem via large deviation methods (preparation for the multidimensional ruin problem, see below).
17.09.18: Cramer's theorem in d-dimensional Euclidean space (JC, pp. 27-36).
18.09.18: Cramer's theorem in higher dimensions, cont. (JC, pp. 27-36).
24.09.18: The multidimensional ruin problem (JC, pp. 51-59).
25.09.18: Multidimensional ruin, cont.; large deviations for paths (JC, pp. 44-50).
01.10.18: LDs for paths cont.; Sanov's theorem (second handout).
02.10.18: Gibbs conditioning; importance sampling (JC, pp. 63-73).
08.10.18: Importance sampling cont.; moderate deviations and heavy tails.
09.10.18: Concentration inequalities (third handout). Last lecture.

"JC" refers to my lecture notes posted on Absalon, while the "handouts" refer to the book excerpts given out in the lectures.


One mandatory homework set has been posted on Absalon (due October 2). A second homework set will be posted around the fourth week of the course (approximately September 28), while a short third homework set will be posted around the sixth week of the course (approximately October 9).