Theory of Probability Including Measure Theory : Math 8652

Spring 2019

Welcome to the course webpage!

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 online. A fifth edition of the textbook is also avaialble 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%.

Canvas: Homework assignments, lecture notes and grades will be posted on Canvas.


Weekly schedule:
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