# Math/Stat 733 - Theory of Probability I.

Fall 2020

Meetings: TR 1pm-2:15pm (online)
Instructor: Benedek Valkó
Email: valko at math dot wisc dot edu
Office hours: Tu 4-5pm, F 3-4pm, or by appointment (online)

This page gives an overview of the course, all relevant information and resources will be provided in the official Canvas page.

### 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 the basics of probability 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, 5th edition, 2019

There are several good textbooks on probability and it might help to have a look around. Here is a list of textbooks that could be used for extra reading:
• Olav Kallenberg: Foundations of Modern Probability. 2nd edition, Springer, 2002
• William Feller. An introduction to probability theory and its applications. Vol. I. Third edition. John Wiley and Sons Inc., New York, 1968.
• David Williams. Probability with martingales. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge, 1991.
• Patrick Billingsley. Probability and measure. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1995.

### Prerequisites

Measure theory is a basic tool for this course. A suitable background can be obtained from Math 629 or Math 721. Chapter 1 in Durrett covers the measure theory needed. We will very briefly review some measure theory at the beginning of the semester, but it will be expected that all students in the course are familiar with the basics. Prior exposure to elementary probability theory could be useful, but it is not required.

### Course content

We cover selected portions of Chapters 1-4 of Durrett. This is a rough course outline:

 Weeks 1-2: Foundations, properties of probability spaces Weeks 3-5: Independence, 0-1 laws, strong law of large numbers Weeks 6-10: Characteristic functions, weak convergence and the central limit theorem Weeks 11-15: Conditional expectation, Martingales

The course continues in the spring semester as Math 734 covering topics such as Markov chains, stationary processes, ergodic theory, and Brownian motion.

### Evaluation

Course grades will be based on biweekly home work assignments (25%), class participation (15%), a midterm exam (30%) and the final exam (30%). (See the Canvas page for more information.)

Piazza

We will be using Piazza for class discussion.  The system is catered to getting you help fast and efficiently from classmates and myself.  Rather than emailing all questions to me, I encourage you to post your questions on Piazza. If you have any problems or feedback for the developers, email team@piazza.com.
You can access our Piazza page from the Canvas page of the course.

### Instructions for homework assignments

• Assignments will be posted and collected on the Canvas page.
• No late assignments will be accepted.
• Observe rules of academic integrity. Handing in plagiarized work, whether copied from a fellow student or off the web, is not acceptable. Plagiarism cases will lead to sanctions.
• Neatness and clarity are essential. Write one problem per page except in cases of very short problems. Staple you sheets together. You are strongly encouraged to use LaTeX (or Overleaf) to typeset your solutions.
• 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 separate some of the technical details as "Lemmas" and put them at the end of the solution. A good rule of thumb is if the grader needs to pick up a pencil to check your assertion, you should have proved it. The grader can deduct points in such cases.
• You can use basic facts from analysis and measure theory in your homework, and the theorems we cover in class without reproving them. If you find a helpful theorem or passage in another book, do not copy the passage but use the idea to write up your own solution. If you do use other literature for help, cite your sources properly. However, it is better to attack the problems with your own resources instead of searching the literature or the internet. The purpose of the homework is to strengthen your problem solving skills, not literature search skills.
• It is valuable to discuss ideas for homework problems with other students. But 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. Similarly, finding solutions on the internet is tantamount to cheating. It is the same as copying someone else's solution.