Large Deviations

Large Deviations (MATH 234/STAT 374, Spring 2019)

Intended for students somewhat familiar with advanced probability theory, this course is about large deviations probabilities and their applications (for example, in statistics, information theory, queuing theory, statistical mechanics, DNA analysis, communications and control).

The theory starts with combinatorial estimates and centers around Cramer's and Sanov's theorems and their counterparts for Markov chains, first in finite-dimensional vector spaces and then in an abstract set-up, touching upon exact asymptotics, moderate deviations, martingale differences and concentration inequalities.

General properties of the large deviations principle such as existence, uniqueness, contractions and metrizability, are covered, as well as the role of Fenchel-Legendre transform and Varadhan's integration lemma in identifying the rate function.

The large deviations behavior of stochastic processes is explored, starting with random walks and progressing to Brownian motion and diffusion processes, culminating with exit from domain problems.

Meeting: Tue, Thu 1:30-2:50, Herrin T185; On 5/30, meeting - 1:05-2:50 .

Instructor: Amir Dembo, Sequoia 129, office hour Thu 4:30-5:30, or e-mail amir@math.stanford.edu

We use the second edition of ``Large Deviations Techniques and Applications'' by Dembo and Zeitouni as text, from which we post Chapter 2 , Chapter 3 , Chapter 4 , Chapter 5 , and Sections 6.1,6.2.

Other recommended texts (on reserve at math. library) are:

Varadhan, Large Deviations and Applications (Ch. 1-5).
den Hallander, Large Deviations (Ch. I-VI).
Shwartz and Weiss, Large Deviations for Performance Analysis. (Ch. 1-8).
Dupuis and Ellis, Weak Convergence approach to the Theory of Large Deviations.
Deuschel and Stroock, Large Deviations (Ch. 1-3).
Freidlin and Wentzell, Random Perturbation of Dynamical Systems.

Prerequisite: Statistics 310a or its equivalent.

Grading: Registered students present material relevant large deviations and attend all other student presentations (entailing one to an A letter grade). Each student is to give a 25 minute presentation. We shall have 3 teams of 2-3 students, each focusing on one topic of interest. Among possible topics are:

Large deviations for random graphs.
Large deviations for random matrices.
The weak convergence approach to large deviations.
Large deviations for interacting particles.
Large deviations in statistical physics.
Large deviations in polymers, LPP/FPP.

Recommended to solve correctly at least 6 of the following (17) problems (non-mandatory homework):

HW1: Ex. 2.1.21, 2.1.28, 2.2.23, 2.2.26, 2.2.37, 2.2.39.
HW2: Ex. 2.3.17, 2.3.25, 2.3.27, 3.7.11(a), 4.1.31, 4.1.32.
HW3: Ex. 4.3.12, 6.1.16, 6.2.17, 6.2.27, 5.2.12.

Preliminary Syllabus (out of text):

     4/1    Tu (1.1;1.2;2.1.1)         Th (2.2.1;2.2.2)
     4/8    Tu (2.2.2;2.3;3.1.1)       Th (3.7;2.4.1)   
     4/15   Tu (6.1)                   Th (6.1;4.1) 
     4/22   Tu (4.3;4.5.1;4.5.2;6.1)   Th (6.2)
     4/29   Tu (6.2;4.6)               Th (5.1;4.6;4.2.1)       
     5/6    Tu (5.1;4.2.1;5.2)         Th (5.6;4.2.2)
     5/13   Tu (5.6)                   Th (5.7)
     5/20   Tu (5.7;4.4;D.3)           Th (TBD) 
     5/27   Tu (St1:TBD)               Th (St2,St3:AGZ2.6.1;6.6)   
     6/3    Tu (---)

See also seminar for current activity in related areas.