# 734 Theory of Probability II, Spring 2018

 Meetings: TR 1-2:15, Sterling 1313 Instructor: Timo Seppäläinen Office: 425 Van Vleck. Office Hours: MW 11-12 and any other time by appointment Office Phone: 263-3624 E-mail: seppalai at the department's address which is math and dot and wisc and dot and edu

This is the second semester of the 2-semester graduate level introduction to probability theory 733-734. The textbook is

Richard Durrett: Probability: Theory and Examples. Version 5 is available from Rick Durrett's homepage at Duke University. The official version for the course is the one dated January 11, 2019.

Material covered. We begin with a quick review of martingales and then take up Sections 4.6 (Uniform integrability) and 4.7 (Backward martingales). The section on backward martingales gives us the opportunity to discuss exchangeable processes and de Finetti's theorem. After that we go on to Markov chains, stationary processes and the ergodic theorem, the subadditive ergodic theorem and some of its applications, and finally Brownian motion and some of its basic properties.

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 and possibly one exam.

### Piazza

We will use Piazza for online class discussion. On Piazza you can post your math questions and answer other students' questions. You can get to Piazza through Canvas. Our class Piazza page is at piazza.com/wisc/spring2019/734/home

### Spring 2019 Schedule

(Section numbers refer to version 5, 01-11-2019, of Durrett's book.)

Week Tuesday Thursday
1 1/21-1/25 4.6 Uniform integability, L1 convergence. 4.6 L1 convergence of martingales, Lévy's 0-1 law, tail σ-algebra, Kolmogorov's 0-1 law.
4.7 Backward martingales.
2 1/28-2/1 4.7 Exchangeable σ-algebra, exchangeable probability measures. 4.7 Hewitt-Savage 0-1 law, de Finetti's theorem, exchangeable measures as mixtures of IID product measures.
3 2/4-2/8 4.7 de Finetti's theorem as an instance of Choquet's theorem. An application of the Hewitt-Savage 0-1 law to random walk.
5.1-5.2 Beginning Markov chains. Markov property of random walk.
Homework 1 due.
5.1-5.2 Transition probabilities. Construction of Markov chains.
4 2/11-2/15 Introduction to the corner growth model.
5.2 Markov property extended to the infinite future. Chapman-Kolmogorov equations. Stopping times and strong Markov property.
5.3 Recurrence, transience and canonical decomposition for discrete state Markov chains.
5 2/18-2/22 5.3-5.4 Recurrence and transience. 5.5 Invariant and reversible measures and distributions. 5.5 Existence and uniqueness of invariant measures and distributions for countable state space. Stationary processes and shift-invariant probability measures. 5.6 Periodicity.
Homework 2 due.
6 2/25-3/1 5.6 Proof of the Markov chain convergence theorem. Dissection principle. Other asymptotic results for Markov chains. 6.1 Stationary processes, measure-preserving dynamical systems, ergodicity.
7 3/4-3/8 6.1 Ergodicity of Markov chains in general state space. Mappings preserve stationarity and ergodicity.
Homework 3 due.
S. Roch lectured on discrete probability.
8 3/11-3/15 6.2 Maximal ergodic lemma and Birkhoff's ergodic theorem. D. Anderson lectured on Markov Chain Monte Carlo.
Homework 4 due.
3/18-22 SPRING BREAK SPRING BREAK
9 3/25-3/29 6.4 Subadditive ergodic theorem. 6.4 Conclusion of the proof of the subadditive ergodic theorem.
Brief discussion about mixing and trivial tail field.
10 4/1-5 Multivariate Gaussian distributions. Lévy's construction of Brownian motion. (B. Valkó lectured.) 7.1 Finite-dimensional distributions of Brownian motion and their consistency. Continuous paths do not form a measurable subset of the product space R[0, ∞).
11 4/8-4/12 7.1 Construction of Brownian motion and Hölder continuity of paths. Kolmogorov-Centsov criterion for path continuity. 7.1 No Brownian path is Hölder continuous with exponent γ>1/2. 7.2 Markov property.
12 4/15-4/19 7.2 Blumenthal's 0-1 law, trivial tail field.
Homework 5 due.
7.2 Recurrence of 1-dimensional Brownian motion. Brownian filtration. 7.3 Stopping times.
13 4/22-4/26 7.3 Sketch of the proof of the strong Markov property. 7.4 Reflection principle and distribution of the hitting time Ta. 7.5 Martingales. 8.1 Discussion of Donsker's theorem. Contrasting weak convergence on C[0,1] and on a countable product space.
14 4/29-5/3 8.1 Skorohod representation.
Homework 6 due.
8.1 Donsker's theorem as a corollary of Skorohod representation. Applications: maximum of random walk, Kolmogorov-Smirnov statistics for the empirical distribution function. 8.2 Brief discussion about the martingale central limit theorem.

### Instructions for Homework

• Homework is handed in on the due date in class or by 3 PM to the instructor's office (VV425) or mailbox (2nd floor of Van Vleck).
• The graders may not be able to grade every problem, in which case they will make some random choice of problems to grade.
• Neatness and clarity are essential. Write one problem per page except in cases of very short problems. Typesetting your work in Latex is highly recommended, especially if your handwriting is not great.
• It is not trivial to learn to write solutions. You have to write enough to show that you understand the flow of ideas and that you are not jumping to unjustified conclusions, but not too much to get lost in details. If you are unsure of the appropriate level of detail to include, you can fill in technical details in separate lemmas at the end. A good rule of thumb is if the grader needs to pick up a pencil to check your assertion, you should have proved it. Points can be deducted in such cases.
• Observe rules of academic integrity: while it is very valuable to discuss ideas for homework problems with your fellow students, it is NOT acceptable to write solutions together or to copy another person's solution. In the end you have to hand in your own personal work. Plagiarism will lead to sanctions.
• IMPORTANT: You can use basic facts from analysis and measure theory in your homework. But do NOT use a special theorem of probability unless it has been covered in class. The reason for this rule is that the literature contains a theorem or a solution to every problem (of course). If you find an idea in the literature or online, use it to write your own solution. In such cases cite your sources properly.

Timo Seppalainen