Math 832 - Theory of Probability, II
Spring
2013


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, but earlier editions should be fine.  Just make sure you are completing the correct homework assignments.

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 will cover chapters 5 through 8 of Durrett's book.


Prerequisites:
Some familiarity with key parts from the first semester, such as measure-theoretic foundations of probability, laws of large numbers, central limit theorem.

Evaluation
: Course grades will be based on homework.


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

Homework Assignments:

  1. Due Tuesday, January 29th.  HW1.
  2. Due Thursday, February 7th.  HW2.
  3. Due Tuesday, February 19th.  HW3.
  4. Due Tuesday, March 5th.  HW4.
  5. Due Thursday, March 21st.  HW 5.
  6. Due Tuesday, April 23rd.  HW 6.
  7. Due Thursday, May 16th at 12pm in my mailbox in Van Vleck.  HW 7.
Fall 2012 Schedule: Section numbers refer to Durrett's book.

Week
Tuesday
Thursday
1
Jan. 22nd
Topic. Review of 831: Chapters 1 - 3.  WLLN, SLLN, CLTs
.
Readings: Chapters 1 - 3.
Slides.
Jan. 24th
Topic. Begin review/lectures on Sections 5.1 and 5.2. 
Readings:
Sections 5.1, 5.2.
2
Jan. 29th HW 1 Due.
Topic. Martingales: definition, basic properties, convergence.
Readings: Sections 5.1, 5.2.
Jan. 31st
Topics.  Martingales: Doob's decomposition theorem, examples, optional stopping theorem (not general).
Readings:  Sections 5.2, 5.3, and 5.4.
3
Feb. 5th
Topic.  Martingales: Maximal inequalities and convergence in Lp, p>1.  Begin considering uniform integrability.
Readings: Section 5.4.  Begin 5.5.
Feb. 7th HW 2 Due (changed).
Topic.  Uniform integrability, convergence in L1.
Readings:  Sections 5.5.
4
Feb. 12th
Topic.  Finish U.I., convergence in L1, Levy and Kolmogorov 0-1 laws.
Readings.  Section 5.5.
Feb. 14th
Topics.  Finish Chapter 5:  Optional Stopping theorem, backwards martingales.
Readings:  Sections 5.6 and 5.7.
5
Feb. 19th HW 3 Due.
Topic: Markov chains: construction, uniqueness, examples.
Sections: 6.1
Feb. 21st
Topics: Strong Markov property. Beginning recurrence and transience of countable state space chains.
Readings: 6.2, 6.3.
6
Feb. 26th
No class.
Feb. 28th
Readings: 6.3 - 6.4.
7
March 5th  HW 4 Due.
Topic:
Recurrence and transience of Markov chains.
Readings: 6.4.
March 7th
Topic: Stationary distributions.
Readings: 6.5.
8
March 12th
Readings: 6.5 and 6.6
March 14th
Readings: 6.6.
9
March 19th
Topic: Coupling and strong law for additive functionals.  Begin chapter 7: stationary sequences.

Readings: 6.6.  7.1.
March 21st HW 5 Due.
Topic:  Stationary sequences. 
Readings: 7.1.
10
March 26th
No class due to spring break.
March 28th
No class due to spring break.
11
April 2nd
Topic: Birkhoff's Egodic Theorem.  Benford's law.  Link to radio lab show.
Readings: 7.2.
April 4th
No Class.
12
April 9th
Topic: Definition and first properties of Brownian Motion.  Maybe start Levy's construction.
Readings: 8.1
April 11th
Topic:  Levy's construction of Brownian motion.
Readings: 8.1
13
April 16th
Topic: Path properties of Brownian motion.
Readings: 8.1.
April 18th
Topic: Path properties of Brownian motion.  Markov property.
Readings: 8.1, 8.2.
14
April 23rd HW 6 Due.
Topic: Markov property. Blumenthal 0-1 law and consequences.
 Readings 8.2.
April 25th
Topic:  Stopping times and the strong Markov property.
Readings: 8.3.
15
April 30th
Topic: Martingales and Brownian motion.
Readings: 8.4.
May 2nd
Topic: Integration with respect to Brownian motion.
16
May 7th
Topic: Integration with respect to Brownian motion.
May 9th
Topic: Integration with respect to Brownian motion.
17
May 14th
No Class: Finals week.
May 16th HW 7 Due.
No Class: Finals week.

Instructions for Homework