832 Theory of Probability II, Spring 2012

Meetings: TR 11-12:15, B129 Van Vleck
Instructor: Timo Seppäläinen
Office: 419 Van Vleck. Office Hours: Tuesdays after class and any other time by appointment
Phone: 263-2812
E-mail: seppalai at math dot wisc dot edu

This is the second semester of the 2-semester graduate level introduction to probability theory 831-832. The textbook is

Richard Durrett: Probability: Theory and Examples. 4th Edition.

We begin with a quick review of conditional expectations from Section 5.1 and martingales from Section 5.2. New material starts with Section 5.4 on martingales. First we finish Chapter 5 on martingales. Next follow Markov chains (Chapter 6), stationary processes and the ergodic theorem, including the subadditive ergodic theorem and some of its applications (Chapter 7), and finally Brownian motion and some of its basic properties (Chapter 8). If it looks like a topic you are interested in is not touched then let me know and we can try to work it in.

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

Course grades will be based on take-home work.


  1. Homework 1 due February 9.
  2. Homework 2 due February 28.
  3. Homework 3 due March 6.
  4. Homework 4 due March 13.
  5. Homework 5 due April 10.
  6. Homework 6 due April 24.
  7. Homework 7 due May 3.

Spring 2012 Schedule

(Section numbers refer to 4th edition of Durrett's book.)

Week Tuesday Thursday
1 5.1-5.2 Review of conditional expectation and martingales. 5.4 Doob's inequality. 5.4 Lp convergence of martingales. 5.5 Uniform integrability.
2 5.5 L1 convergence of martingales. Applications to conditional expectations and Kolmogorov's 0-1 law. 5.7 Optional stopping. Application to random walk.
3 5.7 Wald's identity from optional stopping. 5.6 Backward martingales. (Homework 1 due.) 5.6 Backward martingales applied to SLLN. Exchangeable σ-algebra.
4 5.6 Hewitt-Savage 0-1 law, de Finetti's theorem. Discussion about de Finetti's theorem, IID measures as extreme points of the set of exchangeable measures.
Extra classes: de Finetti's theorem as a special case of Choquet's theorem. Martingale central limit theorem. 6. Markov chains: first definitions.
5 6.1 Markov chains: construction, uniqueness, examples. 6.2 Strong Markov property. 6.3 Beginning recurrence and transience of countable state space chains.
6 Canonical decomposition. Recurrence of symmetric simple random walk. Martingale technique for recurrence of a birth and death chain.
7 6.5 Invariant and reversible distributions, stationarity and shift-invariance, examples. Existence and uniqueness of invariant measures and distributions for countable state space Markov chains.
8 Markov chain convergence theorem. Final remarks on Markov chains. 7. Measure preserving transformations, ergodicity.
9 Class rescheduled. Class rescheduled.
10 7.2 The ergodic theorem. 7.2 The ergodic theorem.
11 Application of the ergodic theorem to a Markov chain central limit theorem. 7.4 Longest common subsequences and longest increasing subsequences. Subadditive ergodic theorem and beginning of its proof.
12 7.4. Proof of the subadditive ergodic theorem. 8. Definition of Brownian motion. The Gaussian (heat) kernel. 8. Existence of Brownian motion. Kolmogorov's criterion for path continuity.
13 8. Brownian motion: conclusion of existence discussion, nondifferentiability of paths, Markov property. Blumenthal's 0-1 law, 0-1 law for tail events, recurrence in 1 dimension.
14 Class rescheduled. Strong Markov property, zero set of Brownian motion, reflection principle.
15 Hitting times as an application of strong Markov property and reflection principle. Martingales. Donsker's invariance principle (functional central limit theorem). Application to empirical distribution functions.

Timo Seppalainen