Course Instructor: Arnab Sen
Office: 238 Vincent Hall
Class time: MW 4:00 - 5:15 pm
Location: Vincent Hall 211
Office hours: M 1:00-2:00 pm, W 11:30 - 12:30 pm, and by appointment.
Course Description: This is the second half of a yearly
sequence of graduate probability theory at the measure-theoretic
level. There will be an emphasis on rigorous proofs.
In this course, we aim to cover the following main topics:
Prerequisite: Math 8651.
Textbook (for both 8651 and 8652):
Probability: Theory and Examples by Rick Durrett, Cambridge Series in
Statistical and Probabilistic Mathematics, 4th Edition. Available
fifth edition of the textbook is also avaialble
Other recommended books:
1. Probability and Measure (3rd Edition) by Patrick Billingsley.
2. A Modern Approach to Probability theory by Bert Fristedt and Lawrence Gray.
3. Probability with Martingales by David Williams.
4. Brownian motion by Peter Morters and Yuval Peres.
Homework: There will be 6 homework assignments. The lowest score will be dropped in calculating the final score.
Final Exam: There will be a take-home final exam.
Grading: Homework 50%, Final 50%.
Canvas: Homework assignments, lecture notes and grades will be posted on Canvas.
Announcements: The take home final exam will be emailed to your UMN account before 12:00pm on Wednesday, May 8. The deadline is 11:59 am, Friday May 10.
|Jan 23||martingales, examples.|
|Jan 28, 30||Doob's decomposition, stopping times. No class on Jan 30.|
|Feb 4, 6||Doob's upcrossing inequality, martingale convergence theorem, maximal inequalities, L^p-martingale convergence theorem|
|Feb 11, 13||Galton-Watson Process, uniform integrability, uniform integrable martingales, Optional Stopping theorem, Gambler's ruin problem|
|Feb 18, 20||proof of OST, backward martingale, convergence, SLLN using backward martingale, Hewitt-Savage 0-1 law, exchangeable sequence|
|Feb 25, 27||de Finetti's theorem, polya's urn. Markov chains: defintion, construction and example, Markov and strong Markov property.|
|Mar 4, 6||Markov and strong Markov property (contd.), recurrent and transient states, characterization of recurrent and transient states for finite state Markov chain|
|Mar 11, 13||recurrence/transience of random walks on Z^d, existence and uniqueness of stationary measure and stationary distribution, positive and null recurrence|
|Mar 18, 20||spring break|
|Mar 25, 27||reverisble Markov chains, asymptotic density of states, periodicity, convergence to equilibrium, an introduction to mixing time of Markov chains: RW on cycle, RW on hypercube [You can find much more about the mixing time of Markov chains in this book by Levin, Peres and Wilmer].|
|Apr 1, 3||Measure preserving systems, stationary sequence, ergodicity, examples: i.i.d. sequence, function of stationary sequence, stationary Markov chain, rotation on circle, Birkhoff's ergodic theorem|
|Apr 8, 10||asymptotic frequency of a fixed pattern in Markov chain, range of random walk, Kingman's subadditive ergodic theorem, first passage percolation, longest common subsequence|
|Apr 15, 17||Introduction to Brownian Motion. Construction of BM. Basic properties of BM. Holder exponent of Brownian paths.|
|Apr 22, 24|
|Apr 29, May 1|