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.

Final Exam: Monday 12/18/2017, 2:45PM - 4:45PM, SOC SCI 6203.

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.
Separate lecture notes: 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 e^it. 3.4 Central limit theorem for IID sequences with finite variance.
Homework 4 due.

8 10/23-27 3.3, 3.5 Discussion of the Berry-Esseen theorem and the local limit theorem. 3.4 Lindeberg-Feller theorem. 3.6 Poisson limit. Poisson process. 3.7-3.8 Brief discussion about stable and infinitely divisible laws. 3.9 Weak convergence in R^d.

9 10/30-11/3 3.9 Multivariate normal distribution. CLT in R^d.
Separate lecture notes: Brief discussion of the Tracy-Widom distribution and the fluctuations of the corner growth model. 4.1 Random walk. Exchangeable sets. Hewitt-Savage 0-1 law.
Homework 5 due.

10 11/6-10 4.1 Stopping times. Strong Markov property for random walk. Wald's identity. Gambler's ruin. 4.2 Recurrence and transience of simple random walk on Z^d.
5.1 Conditional expectation.

11 11/13-17 5.1 Properties of conditional expactation. 5.1 Conditional probability distributions.

12 11/20-22 5.1 Generalizing Fubini's theorem with stochastic kernels. 5.2 Martingales.
Homework 6 due. Thanksgiving break.

13 11/27-12/1 5.2 You cannot beat an unfavorable game of chance. Upcrossing lemma. B. Valkó lectures on random matrices.

14 12/4-8 5.2 Martingale convergence theorem. Random walk example of failure of L¹ convergence. 5.3 Pólya's urn. Galton-Watson branching process: extinction when μ<1. 5.4 Doob's inequality. L^p convergence of martingales for p>1. 5.5 Definition of uniform integrability.
Homework 7 due.

15 12/11-13 Last class of the semester. S. Roch lectures.

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. Separate lecture notes: 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 e^it. 3.4 Central limit theorem for IID sequences with finite variance. Homework 4 due.
8 10/23-27	3.3, 3.5 Discussion of the Berry-Esseen theorem and the local limit theorem. 3.4 Lindeberg-Feller theorem.	3.6 Poisson limit. Poisson process. 3.7-3.8 Brief discussion about stable and infinitely divisible laws. 3.9 Weak convergence in R^d.
9 10/30-11/3	3.9 Multivariate normal distribution. CLT in R^d. Separate lecture notes: Brief discussion of the Tracy-Widom distribution and the fluctuations of the corner growth model.	4.1 Random walk. Exchangeable sets. Hewitt-Savage 0-1 law. Homework 5 due.
10 11/6-10	4.1 Stopping times. Strong Markov property for random walk. Wald's identity. Gambler's ruin.	4.2 Recurrence and transience of simple random walk on *Z^d*. 5.1 Conditional expectation.
11 11/13-17	5.1 Properties of conditional expactation.	5.1 Conditional probability distributions.
12 11/20-22	5.1 Generalizing Fubini's theorem with stochastic kernels. 5.2 Martingales. Homework 6 due.	Thanksgiving break.
13 11/27-12/1	5.2 You cannot beat an unfavorable game of chance. Upcrossing lemma.	B. Valkó lectures on random matrices.
14 12/4-8	5.2 Martingale convergence theorem. Random walk example of failure of L¹ convergence. 5.3 Pólya's urn. Galton-Watson branching process: extinction when μ<1.	5.4 Doob's inequality. L^p convergence of martingales for p>1. 5.5 Definition of uniform integrability. Homework 7 due.
15 12/11-13	Last class of the semester. S. Roch lectures.

Instructions for Homework

Homework must be handed in by the due date, either in class or by 3 PM either in the instructor's office or mailbox, or as a PDF file uploaded on Canvas. Late submissions cannot be accepted.
Neatness and clarity are essential. Write one problem per page except in cases of very short problems. Staple you sheets together. You are welcome to use LaTeX to typeset your solutions.
It is not trivial to learn to write solutions. You have to write enough to show that you understand the flow of ideas and that you are not jumping to unjustified conclusions, but not too much to get lost in details. If you are unsure of the appropriate level of detail to include, you can separate some of the technical details as "Lemmas" and put them at the end of the solution. A good rule of thumb is if the grader needs to pick up a pencil to check your assertion, you should have proved it. The grader can deduct points in such cases.
You can use basic facts from analysis and measure theory in your homework, and the theorems we cover in class without reproving them. If you find a helpful theorem or passage in another book, do not copy the passage but use the idea to write up your own solution. If you do use other literature for help, cite your sources properly. However, it is better to attack the problems with your own resources instead of searching the literature or the internet. The purpose of the homework is to strengthen your problem solving skills, not literature search skills. (In particular: do not use Lyapunov's theorem as a substitute for Lindeberg-Feller unless we have covered Lyapunov's theorem or it has been proved in a homework.)
It is valuable to discuss ideas for homework problems with other students. But it is not acceptable to write solutions together or to copy another person's solution. In the end you have to hand in your own personal work. Similarly, finding solutions on the internet is tantamount to cheating. It is the same as copying someone else's solution.