# 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 You can download the exam here

**Final exam:** December 21, 2:45PM - 4:45PM, the room is the same we have been using (5231 Social Sciences). Please bring pencil and paper. No calculators or other electronic gadgets.
You will be able to use two (letter sized) sheets of handwritten notes.

Old final exams: Final exam 2008, Final exam 2009

**Homework:**

- Homework 1 due September 16
- Homework 2 due September 30
- Homework 3 due October 14
- Homework 4 due October 28
- Homework 5 due November 18
- Homework 6 due December 9

**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
- Week 7. Strong law of large numbers in renewal theory, Glivenko-Cantelli lemma, Central Limit Theorem (De Moivre-Laplace version), convergence in distribution, properties and various characterizations. Sections 1.7, 2.1-2
- Week 8. Tightness, Characteristic functions, properties, examples. Sections 2.2-2.3
- Week 9. Characteristic functions and weak convergence, moments and characteristic functions, Lindeberg-Feller theorem. Sections 2.3-2.4
- Week 10. Applications of the Lindeberg-Feller theorem, Local CLT, Moment problem, Poisson approximation. Sections 2.3-2.5
- Week 11. Poisson approximation with coupling, Stable and infinitely divisible distributions, limit theorems in R^d. Sections 2.6-9
- Week 12. Conditional expectation. Section 4.1
- Week 13. Properties of conditional expectation, regular conditional distribution, martingales, sub/supermartingales, upcrossing inequality. Sections 4.1,4.2
- Week 14. Martingale convergence theorem, Doob's decomposition, applications (martingales with bounded increments, branching process, Polya urn), Optional stopping theorem, Gambler's ruin problem. Sections 4.2-4.3, 4.7
- Week 15. More examples: asymmetric gambler's ruin problem, monkey in front of the typewriter.
- Week 16. Maximal inequalities, convergence in [math]\displaystyle{ L^p }[/math] Section 4.4