Math/Stat 733 - Theory of Probability I.

Fall 2018

Meetings: TR 1PM-2:15PM, Van Hise 115
Instructor: David Anderson
Office: 617 Van Vleck
Email: anderson@math.wisc.edu
Office hours: W 1:00PM-2:00PM or by appointment

I will use the class email list to send out corrections, announcements, please check your wisc.edu email from time to time.

If you try to register, you may find the class full. In that case sign up for the wait-list. Wait-listed students will typically find a place in the course eventually, and you may also attend lectures in the meantime (assuming there is space in the classroom).

Course description

This is the first semester of a two-semester graduate-level introduction to probability theory and it also serves as a stand-alone introduction to the subject. The course will focus on the basics of probability and will roughly cover the first four chapters of the text.  This includes foundations (probability spaces, random variables, expectations, etc.), laws of large numbers, central limit theorems, and martingales.  Time permitting, we will cover a bit on Markov chains.

Textbook

Richard Durrett: Probability: Theory and Examples, 5th edition, 2017.  The link for the text is here.

Prerequisites

Measure theory is a basic tool for this course. A suitable background can be obtained from Math 629 or Math 721. Chapter 1 in Durrett covers the measure theory needed. We will very briefly review some measure theory at the beginning of the semester. Prior exposure to elementary probability theory could be useful.

Course content

We cover selected portions of Chapters 1-4 of Durrett. These are the main topics:
Foundations, properties of probability spaces
laws of large numbers
Characteristic functions, weak convergence and the central limit theorem
Conditional expectations
Martingales
Markov chains (time permitting)
The course continues in the Spring Semester on topics such as Markov chains, stationary processes and ergodic theory, and Brownian motion.

Evaluation

Course grades will be based on biweekly (once per two weeks) homework assignments (30%), a midterm exam (30%) and the final exam (40%). We will have an evening midterm exam on Monday, October 29th, 7:15pm-8:45pm.  In exchange, the class on Tuesday, October 30th will be canceled.   The room for the first exam is B239 in Van Vleck Hall.

Final exam: Saturday, December 15, 10:05am - 12:05pm, room is Psychology 107.


Homework assignments

Homework will be posted on our Canvas page.

Instructions for homework assignments


Gradescope

We will use the software Gradescope, which is already being succesfully tested in other courses here at UW-Madison. The advantages for you are the following:

Here is a guide on how you have to submit your homework. You can also watch the video “For students: submitting homework” here.

Weekly schedule

The following is a (very) tentative schedule for the semester.  It will be continuously updated during the semester.


Covered topics
Suggested reading for next week
Week 1.
9/6
Introduction and Probability space.
Sections 1.1.
Sections 1.1 - 1.7
Week 2.
9/11, 9/13
Finish chapter 1.
Sections 1.1-1.7.
Sections 2.1
Week 3.
9/18, 9/20
Finish chapter 1.  Begin Independence
Tuesday: Finish chapter 1.  Thurs: Independence (Section 2.1)
Section 2.2
Week 4.
9/25, 9/27
Independence and the weak law of large numbers
Sections 2.1 and 2.2
Sections 2.3, 2.4, and 2.5
Week 5.
10/2, 10/4
Weak law and Borel-Cantelli, strong law if time
Sections 2.2, 2.3, and 2.4
Sections 2.4 - 2.7
Week 6.
10/9, 10/11
Strong law of large numbers, renewal theory or large deviations(?)
Sections 2.4 - 2.7
Sections 3.1 - 3.2
Week 7.
10/16, 10/18
Weak convergence
Sections 3.1 and 3.2
Section 3.3 and 3.4
Week 8.
10/23, 10/25
Characteristic functions and CLTs
Sections 3.3 and 3.4
Sections 3.3 and 3.4
Week 9.
10/29, 11/1
midterm (Monday night), 10/30 class canceled,
Central limit theorem, Lindeberg-Feller
Sections 3.3 and 3.4
Section 3.4 and 3.6 and 3.7
Week 10.
11/6, 11/8
Finish Lindeberg-Feller, Poisson convergence
Section 3.4, 3.6
Section 3.7
Week 11.
11/13, 11/15
Poisson processes
Sections 3.7 + extra material
Section 4.1
Week 12.
11/20
Conditional expectation.  Thanksgiving (no class Thursday)
Section 4.1
Sections 4.1 - 4.3
Week 13.
11/27, 11/29
Conditional expectation, Martingales, and convergence
Sections 4.1, 4.2, and 4.3
Section 4.4 and 4.5
Week 14.
12/4, 12/6
Doob's inequality, square integrable Martingales
Sections 4.4 and 4.5
Sections 4.6, 4.7, and 4.8
Week 15.
12/11
UI, Backwards Martingales, Optional Stopping
Sections 4.6, 4.7, and 4.8
N.A.



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