- 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.
- Gartner and Ellis theorem.
- Sanov’s theorem : Donsker-Varadhan variational formula (if time permits)
Prerequisites:
- Measure Theoretic Probability
- Advanced Probability
References:
- Large Deviations Techniques and Application by Dembo and Zeitouni
- 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
- Teacher: Parthanil Roy