Math 831 - Theory of Probability
Fall 2012


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

Required Text:
  Probability: Theory and Examples, by Rick Durrett.  We will use the fourth edition.  For background on measure theory, I recommend Gerald Folland's Real Analysis.

Course content: 
This is a graduate level introductory course on mathematical probability theory. Here is the Mathematics Department's information page on the course.

We cover selected portions of Chapters 1-5 of Durrett. These are the main 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).


The course continues in the Spring Semester on topics such as Markov chains, stationary processes and ergodic theory, and Brownian motion.


Prerequisites:
Probability theory operates in a measure-theoretic framework, so it is important to know basic measure theory. A suitable background can be obtained from Math 629 or Math 721. Comfort with rigorous analysis and some elementary probability are also necessary.

Evaluation
: Course grades will be based on home work assignments, a take-home midterm exam and an in class final exam at the end of the semester.
Exam date:  Exam given November 1st.  Collected November 8th.  Only class text and notes are allowed as references.

Final exam information.  Topics covered:  entire course. 
Date: 12/19/12
Time: 5:05PM - 7:05PM
Place: Social Sciences 6102 (NOTE THAT THIS IS NOT OUR NORMAL ROOM!!!)


Check out the
Probability Seminar for talks on topics that might interest you.

Homework Assignments:
  1. Here is assignment 1.  Due date is Thursday, September 13th.  The raw latex is here.
  2. Here is assignment 2.  Due date is Thursday, September 27th.
  3. HW 3:  Exercises 2.2.2 and 2.2.3 from text.  Due Thursday, October 4th.
  4. HW 4.  Due Tuesday, October 16th.
  5. HW 5.  Due Thursday, November 1st.
  6. HW 6.  Due Thursday, December 6th.
  7. HW 7, suggested problems.  5.1.9, 5.2.1, 5.2.4.  Let S_n be a random symmetric random walk (i.e. there are \xi_i, taking values plus and minus one with probability 1/2 and S_n = \sum_{i=1}^{n} \xi_i).  Find a cubic polynomial g so that g(S_n) is a martingale.

Fall 2012 Schedule: This schedule is tentative and is subject to change.  Section numbers refer to Durrett's book.

Week
Tuesday
Thursday
1
Sept. 4th
Topic: Measure Theory/Probability.  Distributions.
Readings: 1.1-1.2.
Sept. 6th
Topic: More on Random variables.  Integration and expectations.
Readings: 1.2-1.4.
2
Sept. 11th
Topic: Integration and expectations.
Readings: 1.4-1.7.
Sept. 13th  HW 1 Due
1.4-1.7.
3
Sept. 18th
2.1, 2.1.1.
Sept. 20th
2.1.2 - 2.1.4, 2.2
4
Sept. 25th
2.2
Sept. 27th HW 2 Due
2.2, 2.3
5
Oct. 2nd
2.3.
Oct. 4th  HW 3 Due
2.3, 2.4.  Includes renewal theory.
6
Oct. 9th
2.4 (Renewal Theory and Glivenko-Cantelli theorem).
Oct. 11th
3.1.
7
Oct. 16th
3.2
Matlab code from lecture is here.
Oct. 18th HW 4 Due
3.2
8
Oct. 23rd
3.2, 3.3
Oct. 25th
3.3.  Characteristic functions and the inversion formula.
9
Oct. 30th
No class.

Nov. 1st
Midterm Exam Handed out.
 
HW 5 Due.

3.3, 3.4.
10
Nov. 6th
3.4: Lindeberg-Feler and applications.
Nov. 8th Midterm Exam Collected.
Finish 3.4, Berry-Esseen bound, start section 3.6.
11
Nov. 13th
Section 3.6.
Nov. 15th
Chapter 3.6.
12
Nov. 20th
Poisson processes.  Slides.
Nov. 22nd
NO CLASS: THANKSGIVING

13
Nov. 27th
Section 3.6 and 3.9.
Nov. 29th
Topic
: Conditional expectation.
Readings
: Section 5.1.
14
Dec. 4th
Topic
: Martingales and their convergence.
Readings
: Section 5.2.
Dec. 6th HW 6 Due.
Section 5.3.
15
Dec. 11th
Chapter 5.
Dec. 13th
Chapter 5.

Instructions for Homework