Math/Stat 733 Theory of Probability I

Fall 2017

This is the course homepage for Math/Stat 733 Theory of Probability I, a graduate level introductory course on mathematical probability theory. This homepage serves also as the syllabus for the course. Below you find basic information about the course and future updates to our course schedule.

Meetings: TR 1-2:15 Ingraham 120
Instructor: Timo Seppäläinen
Office: 425 Van Vleck. Office hours MW 11-12 or any time by appointment.
Phone: 263-3624
E-mail: seppalai at math dot wisc dot edu

Course material will be based on the book

Richard Durrett: Probability: Theory and Examples. (The fourth edition is the newest published one but any edition should work. You can get the book from Rick Durrett's homepage. List of corrections.)

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). Chapter 1 in Durrett covers the measure theory needed. If desired some measure theory can be reviewed at the start. Prior exposure to elementary probability theory is also necessary.

Course Content

We cover selected portions of Chapters 2-5 of Durrett 4th Ed. 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
Random walk
Conditional expectations
Martingales
The course continues in the Spring Semester on topics such as Markov chains, stationary processes and ergodic theory, and Brownian motion.

Course Grades, Exams and Homework

Course grades will be based on take-home work (50%) and one in-class exam (50%) where you can bring 3 sheets of notes. Homework assignments will be posted on Canvas at Learn@UW. Instructions for homework appear at the bottom of this homepage.

Piazza

Piazza is an online platform for class discussion. Post your questions on Piazza and answer other students' questions. Our class Piazza page is at https://piazza.com/wisc/fall2017/mathstat733/home

Related Seminars

Check out the Probability Seminar and the Statistics Seminar for talks that might interest you.

Fall 2017 Schedule

Here we record topics covered during each class period as we progress. Section numbers refer to the 4th edition of Durrett's book.

Week Tuesday Thursday
1 9/5-8 1.1-1.7 Probability spaces, random variables.
2 9/11-15 1.1-1.7 Expectations, inequalities, types of convergence. 2.1 Independence, proof of the π-λ theorem, application to independence.
Homework 1 due.
3 9/18-22 2.1 Independence and product measures, independence and convolution, Kolmogorov's extension theorem. 2.3 Borel-Cantelli lemmas. Applications to convergence in probability, the necessity of finite mean for the strong law of large numbers.
4 9/25-29 2.4 Strong law of large numbers.
Homework 2 due.
2.4 Glivenko-Cantelli theorem. 2.5 Tail σ-algebra, Kolmogorov 0-1 law, Kolmogorov's inequality.
5 10/2-6 2.5 Variance criterion for convergence of random series. The corner growth model, its queueing interpretation, superadditivity, the exactly solvable case with exponentially distributed weights. 3.2 Weak convergence, portmanteau theorem.
6 10/9-13 3.2 Continuous mapping theorem, Scheffé's theorem, Helly's selection theorem, tightness of sets of probability measures.
Homework 3 due.
Midwest Probability Colloquium at Northwestern University. Class rescheduled.
7 10/16-20 3.2 Completion of tightness discussion. 3.3 Characteristic functions. Continuity theorem. 3.3 Completion of the proof of the continuity theorem. An error bound for the Taylor estimate of eit. 3.4 Central limit theorem for IID sequences with finite variance.
Homework 4 due.
8 10/23-27 3.4 Lindeberg-Feller theorem.
9 10/30-11/3
10 11/6-10
11 11/13-17
12 11/20-22 Thanksgiving break.
13 11/27-12/1
14 12/4-8
15 12/11-13

Old Fall 2014 Schedule

Section numbers refer to 4th edition of Durrett's book.

Week Tuesday Thursday
1 9/2-9/5 1.1-1.7 Probability spaces, random variables. 1.1-1.7 Expectations, inequalities, types of convergence.
2 9/8-9/12 2.1 Independence, proof of the π-λ theorem, application to independence. 2.1 Independence and product measures, independence and convolution, Kolmogorov's extension theorem.
3 9/15-19 2.2 Coupon collector's problem. Weak law of large numbers. St. Petersburg paradox (details left for reading). 2.3 Borel-Cantelli lemmas. Applications to convergence in probability, the necessity of finite mean for the strong law of large numbers, SLLN under a finite fourth moment.
4 9/22-26 2.4 Strong law of large numbers. 2.4 Glivenko-Cantelli theorem. 2.5 Kolmogorov 0-1 law, Kolmogorov's inequality, variance criterion for convergence of random series.
5 9/29-10/3 3.2 Weak convergence, portmanteau theorem. 3.2 Continuous mapping theorem, Scheffé's theorem, Helly's selection theorem, tightness of sets of probability measures.
6 10/6-10/10 3.3 Characteristic function, continuity theorem, an error bound for the Taylor estimate of eit. Midwest Probability Colloquium, Northwestern University, Evanston, IL. Class rescheduled.
7 10/13-17 3.4 Central limit theorem, Lindeberg-Feller theorem. 3.5 Discussion of the local limit theorem for lattice distributions. 3.6 Poisson limit. Poisson process.
8 10/20-24 3.7-3.8 Brief discussion about stable and infinitely divisible laws. 3.9 Multivariate normal distribution. CLT in Rd. Brief discussion about Brownian motion and Donsker's invariance principle. 3.3 Moment problem.
9 10/27-31 4.1 Random walk. Exchangeable sets. Hewitt-Savage 0-1 law. 4.1 Stopping times. Strong Markov property for random walk. Wald's identity.
10 11/3-7 4.2 Recurrence and transience of simple random walk on Zd. 5.1 Definition of conditional expectation. 5.1 Properties of conditional expectation.
11 11/10-14 5.1 Conditional probability distributions. Generalizing Fubini's theorem with stochastic kernels. 5.2 Martingales.
12 11/17-21 5.2 Martingale convergence theorem. Class rescheduled.
13 11/24-26 5.3 Examples of martingales. Thanksgiving
14 12/1-5 5.3 Branching process. Review for exam. Exam on Sunday.
15 12/8-12 Discussion about exam problems. Return exams. Last passage percolation and KPZ universality.

Instructions for Homework