# Math/Stat 831 -- Theory of Probability

## Math/Stat 831 - Theory of Probability, Fall 2010

**Meetings:** TR 11-12:15, 5231 SOCIAL SCIENCES

**Instructor:** Benedek Valkó

**Office:** 409 Van Vleck

**Phone:** 263-2782

**Email:** valko at math dot wisc dot edu

**Office hours:** Tuesdays 1-2PM or by appointment

**Grader:** Diane Holcomb

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

**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 discrete-time stochastic processes 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

The fourth edition of the book will be published at the end of August, 2010. If you are not able to get the new edition, the third will also suffice.

There are several good textbooks on probability and it might help to have a look around. An excellent reference book (which was actually used as the textbook for 831 recently) is: Olav Kallenberg: Foundations of Modern Probability.

**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. The appendix of Durrett covers the measure theory we need. If needed, some aspects of measure theory can be reviewed at the beginning.
Comfort with rigorous analysis and some elementary probability are also necessary.

**Course Content:** we will cover (at least) the following topics (mostly contained in the first four chapters of Durrett):

- 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
- martingales (convergence theorems and basic properties)

**Evaluation:** Course grades will be based on home work assignments, a take-home midterm exam and a written final exam at the end of the semester.

- Homework: 30%
- Midterm: 20%
- Final Exam: 50%

**Midterm exam:** November 2

**Homework:**

- Homework 1 due September 16
- Homework 2 due September 30
- Homework 3 due October 14

**Instructions for the homework assignments:**
Homework must be handed in by the due date, either in class or by 12PM in the instructor's mailbox. Late submissions will not be accepted. Group work is encouraged, but you have to write up your own solution.
You can use basic facts from analysis and measure theory and also the results we cover in class. If you use other literature for help, cite your sources properly. (Although you should always try to solve the problems on your own before seeking out other resources.)

**Schedule:**

- Week 1. Definition of probability space, examples, Section 1.1
- Week 2. Properties of probability measures, random variables, distribution and distribution function of random variables, density function, functions of random variables, limits of random variables Section 1.2
- Week 3. Definition and properties of the expectation, theorems about expectation of limits and limits of expectations, independence, Sections 1.3-1.4
- Week 4. Sufficient conditions for independence, Kolmogorov's 0-1 law, independence and expectation, existence of independent random variables with specified distributions, Kolmogorov's extension theorem, Section 1.4
- Week 5. A.s. convergence, convergence in probability and L_p convergence. Weak law of large numbers with second moment. Borel-Cantelli lemmas. Strong law of large numbers with fourth moment. Description of convergence in probability using subsequences. Sections 1.5-1.6
- Week 6. Examples: coupon collector problem, number of cycles in random permutations, longest head run, St. Petersburg paradox. Weak and strong law of large numbers with truncation. Sections 1.5-1.7