Math/Stat 733 - Theory of Probability I.

Fall 2016

Meetings: TR 1PM-2:15PM, Van Vleck B115  Social Sciences 6203
Instructor: Benedek Valkó
Office: 409 Van Vleck
Phone: 263-2782
Email: valko at math dot wisc dot edu
Office hours: W 2:30PM-3:30PM 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 cover at least the following topics: foundations (probability spaces and existence of processes), independence, zero-one laws, laws of large numbers, weak convergence and the central limit theorem, conditional expectations and their properties, and martingales (convergence theorem and basic properties).

Textbook

Richard Durrett: Probability: Theory and Examples, 4th edition, 2010

There are several good textbooks on probability and it might help to have a look around. Here is a list of textbooks that could be used for extra reading:

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 1-5 of Durrett. These are the main topics:
Foundations, properties of probability spaces
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.

Evaluation

Course grades will be based on biweekly home work assignments (30%), a midterm exam (30%) and the final exam (40%). We will have an evening midterm exam on Wednesday, November 2, 7:15pm-8:45pm in room B239 Van Vleck.  In exchange, the class on Thursday, November 3 will be canceled.

Final exam: Thursday, December 15, 5:30pm-7:30pm, B239 Van Vleck.


Homework assignments

Homework will be posted on Learn@UW.


Piazza

We will be using Piazza for class discussion.  The system is catered to getting you help fast and efficiently from classmates and myself.  Rather than emailing all questions to me, I encourage you to post your questions on Piazza. If you have any problems or feedback for the developers, email team@piazza.com.

Find our class Piazza page at: https://piazza.com/wisc/fall2016/math733_001_fa16/home.

Instructions for homework assignments


Weekly schedule

The following schedule will be updated during the semester.


Covered topics
Suggested reading for next week
Week 1.
9/6, 9/8
Definition of probability space, random variables,  distribution
Sections 1.1-1.5
Sections 1.5-1.7, 2.1
Week 2.
9/13, 9/15
Expectation (inequalities, limits), product measures, independence
Sections 1.5-1.7,  2.1
Sections 2.1-2.3
Week 3.
9/20, 9/22
Kolmogorov extension theorem, various types of convergences,
Weak law of large numbers
(Sections 2.1-2.4)
Sections 2.3-2.5
Week 4.
9/27, 9/29
Borel-Cantelli Lemmas, Strong Law of Large Numbers,
Sections 2.5, 3.1-3.2
Week 5.
10/4, 10/6
Glivenko-Cantelli Theorem, Kolmogorov 0-1 Law
Central Limit Theorem for coin flips, Convergence in distribution
Sections 3.2-3.3
Week 6.
10/11, 10/13
Convergence in distribution, characteristic functions
Sections 3.2-3.4
Week 7.
10/18, 10/20
Characteristic functions, general CLT,
examples
Section 3.6-3.8
Week 8.
10/25, 10/27
Lindeberg-Feller theorem, Moment problem, Local CLT,

Week 9.
11/1
Poisson approximation, coupling
midterm exam


Week 10.
11/8, 11/10
Poisson processm Infinitely divisible and stable distributions,
convergence in distribution in R^d, multivariate CLT,
quick intro to Brownian motion,
Conditional expectation
Section 5.1-5.2
Week 11.
11/15, 11/17
Properties of conditional expectation, martingales

Week 12.
11/22


Week 13.
11/29, 12/1


Week 14.
12/6, 12/8


Week 15.
12/13, 12/15





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