Theory of Probability Including Measure Theory : Math 8652

Spring 2016

Welcome to the course webpage!

Course Instructor: Arnab Sen
Office: 238 Vincent Hall
Email: arnab@math.umn.edu

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 online.

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.

Weekly schedule:
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)