Course Instructor: Arnab Sen
Office: 238 Vincent Hall
Class time: MW 4:00 - 5:15 pm
Location: Vincent Hall 211
Office hours: MW 11:40-12:40pm or 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. Also available
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%.
Announcements: 1. The deadline of Homework 6 has been extended to May 4.
2. Your take home final exam will be emailed to your UMN account on Sunday 9 am. The deadline is 12 noon, Wednesday May 11.
|Jan 20||martingales, stopping times.|
|Jan 25, 27||optional stopping theorem, Doob's decomposition, Doob's upcrossing inequality, martingale convergence theorem|
|Feb 1, 3||Polya's urn, recurrence of random walk on Z, more on optional stopping theorem, maximal inequalities, L^p-martingale convergence theorem, uniform integrability, martinagle convergence in L^1, Lévy's forward law|
|Feb 8, 10||Reverse martingale, proof of SLLN using reverse martingale, gambler's ruin problem using optional stopping theorem, Ballot theorem|
|Feb 15, 17||Discrete Dirichlet problem, Galton-Watson process. [You can find more about random walks on graphs and electrical networks here] , Markov chains : construction and examples, Kolmogorov's extension theorem|
|Feb 22, 24||proof of existence of Markov chains, Markov and strong Markov property, recurrent and transient states|
|Feb 29, Mar 2||characterization of recurrent and transient states for finite state Markov chain, recurrence/transience of random walks on Z^d, birth death chains|
|Mar 7, 9||existence and uniqueness of stationary measure and stationary distribution, asymptotic density of states|
|Mar 14, 16||spring break|
|Mar 21, 23||periodicity, convergence to equilibrium. An introduction to mixing time of Markov chains: coupling and strong stationary time RW on cycle, RW on hypercube and Top-to-random shuffle [You can find much more about this in this wonderful book by Levin, Peres and Wilmer].|
|Mar 28, 30||Measure preserving systems, stationary sequence, ergodicity, examples: i.i.d. sequence, function of stationary sequence, Markov chain, rotation on circle, doubling map on circle|
|April 4, 6||examples continued, Birkhoff's ergodic theory, asymptotic frequency of a fixed pattern in Markov chain, range of random walk|
|April 11, 13||Kingman's subadditive ergodic theory, applications: product of i.i.d. random matrices and first passage percolation. Introduction to Brownian Motion. An attempt to construct BM on the product space R^[0, \infty).|
|April 18, 20||Construction of BM. Properties of BM, non-differentiability, Holder exponent of Brownian paths|
|April 25, 27||Markov and strong property of BM, Blumenthal's 0-1 law, reflection principle, distribution of hitting times and maxima of BM, arsine law for last zero of BM.|
|May 2, 4||Properties of zero set of BM, Brownian motion as a martingale, Donsker's theorem (without proof)|