Seminar in Large Deviations – Spring 2010
Announcements
Dmitry Kramkov will be giving the seminar on Thursday, February 4. He will be giving a review on weak convergence of measures and dicussing the connections between weak convergence and LDP.
Basic Class Information
Time : Thursdays from 4:30 to 6:30 PM
Room : Wean Hall 6423
Lectures
Lecture 1 – January 21, 2010 – Introduction/Cramer's Theorem
Assigned Exercises – Due February 4, 2010
Example of a weak but not full LDP
It was asked if every exponentially tight sequence of measures on a Polish space contains a subsequence which satisfies a LDP with good rate function I. This is in fact true and is proved as Theorem 1.3.7 (page 26) in the Dupuis and Ellis book.
Lecture 2 – January 28, 2010 – Cramer's Theorem
Assigned Exercises – Due February 11, 2010
Answers to some questions raised in Office Hours
Lecture 3 – February 4, 2010 – Review of Weak Convergence
No Assigned Exercises
Lecture 4 – February 11, 2010 – Applications of Cramer's Theorem and the Gartner Ellis Theorem
Assigned Exercises- Due February 25, 2010
Lecture 5 – February 18, 2010 – The Gartner Ellis Theorem
Assigned Exercies – Due March 4, 2010
Lecture 6 – February 25, 2010 – Varadhan's Integral Lemma
Assigned Exercises – Due March 18, 2010
Lecture 7 – March 4, 2010 – Existence Results, Contraction and Approximate Contraction Principles
Assigned Exercices - Due March 18, 2010
Lecture 8 – March 18, 2010 - Sample path LDP : Mogulskii's and Schilder's Theorems
Assigned Exercises - Due April 1, 2010
Lecture 9 – March 25, 2010 – Hypothesis Testing with an application to Wireless Sensor Networks
No Assigned Exercises
Lecture 10 – April 1, 2010 – Mogulskii's Theorem, Freidlin-Wentzell Theory.
Assigned Exercises - Due April 15, 2010
Lecture 11 – April 8, 2010 – Occupancy Time LDP for ergodic Markov Processes I – Finite State Space Markov Chains
Assigned Exercises - Due April 22, 2010
Lecture 12 – April 15, 2010 – Occupancy Time LDP for ergodic Markov Processes II – Markov Chains with General State Spaces
Assigned Exercises - Due April 29, 2010
Lecture 13 – April 22, 2010 – Asymptotically Optimal Importance Sampling with Explicit Formulas in Continuous Time
No Assigned Exercises
Lecture 14 – April 29, 2010 – Occupancy Time LDP for ergodic Markov Processes III – Continuous time processes with a focus on one dimensional diffusions.
No Assigned Exercises