Theory of Large Deviations
M. Math and M. Stat 2nd Year
Academic Year 2020-2021, Second Semester
The phrase "large deviations" refers to analytically computing or estimating probabilities of events that correspond to deviating significantly from the typical behaviour. In other words, it is the study of rare events. The goal is to understand how rare events occur so that their probabilities can be estimated. In this course, we will systematically learn the basics of large deviation theory along with its applications.
Class Timing:
Wednesdays 4:30 pm - 6:30 pm
Saturdays 11:00 am - 1:00 pm
Prerequisites:
Measure Theoretic Probability
Radon-Nikodym Theorem and Conditional Expectation
Discrete Parameter Martingales (recommended but not required)
References:
Large Deviations Techniques and Applications by Dembo and Zeitouni (the main reference)
Large Deviations by Deuschel and Stroock
Large Deviations by Hollander
Large Deviations and Applications by Varadhan
A Weak Convergence Approach to the Theory of Large Deviations by Dupuis and Ellis
Course Outline:
Introduction to large deviations. Motivations from insurance and statistics. Classical large deviation for partial sums in the Gaussian case: exact computation using Mills ratio.
Fenchel-Legendre transform: definition, properties and computation.
Cramer’s theorem for general random variables and vectors.
General notion of large deviation principle on Polish spaces: Laplace principle, Varadhan’s lemma, weak large deviation principle, exponential tightness, goodness of rate function, contraction principle. Applications.
Sanov’s theorem for finitely supported random variables and recovering Cramer's theorem in this setup using contraction principle.
Statement of Gartner-Ellis Theorem.
Very brief outline of the notion of large deviation for heavy tailed random variables (if time permits).