Math/Stat 831 Theory of Probability

Fall 2009

Meetings: TR 11-12:15 B135 Van Vleck

Office: 419 Van Vleck, Office hours after class on Tuesdays, or any time by appointment

Phone: 263-2812

E-mail: seppalai at math dot wisc dot edu

This is a graduate-level introductory course on mathematical probability theory. The material will be based on the book

Richard Durrett: Probability: Theory and Examples. 3rd edition.

There are numerous good books on probability and it may be helpful to look at other books besides Durrett. For example, these authors have written graduate texts: Patrick Billingsley, Leo Breiman, Kai Lai Chung, Richard M. Dudley, Bert Fristedt and Lawrence Gray, Olav Kallenberg, Sidney Resnick, Albert Shiryaev, Daniel Stroock.

Prerequisites

Measure theory is a basic tool for this course. A suitable background can be obtained from Math 629 or Math 721 (possibly concurrently). An appendix in Durrett covers the measure theory needed. Upon request some aspects of measure theory can be reviewed at the start. Prior exposure to elementary probability theory is also necessary.

Course Content

We cover selected portions of the first four chapters of Durrett. These are the main topics:

Foundations, existence of stochastic processes

Independence, 0-1 laws, strong law of large numbers

Characteristic functions, weak convergence and the central limit theorem

Conditional expectations

Martingales

The course continues in the Spring Semester on topics such as Markov chains, stationary processes and ergodic theory, and Brownian motion. Requests for particular topics for the spring term are welcome.

Exams and Grades

Course grades will be based on take-home work and one in-class exam where you can bring 3 sheets of notes. Homework assignments, updates on rescheduling classes, and other matters will appear on the course homepage.