**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: ** 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:

- Discrete-time martingales.
- Markov chains
- Ergodic theory
- Brownian motion.

** 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.

**Announcements:**

** 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 |