Department of Mathematics and Statistics
Concordia University

Winter 2019

MAST 679I/MAST 881I: Rare Events/Large Deviations and Applications

Thu 13:00-15:20, LB-759-6 (SGW)

 
Instructor: Lea Popovic
Office: LB 921-07 (SGW)
Phone: (514) 848-2424 Ext 5854
email: lea.popovic@concordia.ca 

Office Hours: after class and by appointment

Course Outline

The theory of large deviations studies probabilities of events which, in a large sample, are exponentially rare in the number of samples. We will cover standard methods including large deviations for i.i.d. sequences, occupation measures, Markov chains, and diffusions with small noise. This theory has found many applications in finance and risk management, simulation and sampling, as well as operations research and statistical mechanics.
The main part of the course will cover:
  • Introduction to large deviations
  • The large deviations principle
  • Cramer's theorem for sample mean
  • Sanov's theorem for empirical measure
  • Gartner-Ellis theorem
  • Concentration inequalities
  • Large deviations for Markov chains
  • Contraction principle
  • Varadhan's and Bryc's lemmas
  • Sample path large deviations
  • Importance sampling methods

Prerequisites

A course in probability theory and stochastic processes, including: LLNs, CLTs, Markov chains, (martingales, Brownian motion).

Text

"Large Deviation Techniques" by F. den Hollander (AMS, Fields Institute Monographs).
"Large Deviation Techniques and Applications" by A. Dembo and O. Zeitouni (Springer).

Grading

Homework will be assigned approximately once every two weeks, during lecture and posted on the Homework Schedule. Students are encouraged to work together on homework problems, however each student must write up the homework set on their own. There will be a take-home test or project handed out in the last week of class and due a week later. The final grade will be evaluated using: Homework assignments 50%, Final exam 50%.

Important Dates

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Lecture Schedule     Homework Schedule