**Spring 2021**

**Meetings:** TR 1pm-2:15pm (online)

**Instructor:** Benedek Valkó

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

**Office hours:** To be announced (see Canvas page)

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

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.

Basic knowledge of measure theory and comfort with rigorous analysis is important for this course. We will rely on the theory introduced in the first semester of the course, so a review of the basic notions and definitions might be useful

Weeks 1-2: Martingales (review, Doob's inequality, uniform
integrability, backwards martingales) |

Weeks 3-7: Markov chains |

Weeks 8-9: Stationary sequences, ergodicity, subadditive
ergodic theorem |

Weeks 10-14: Brownian motion |

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.

- 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.

Check out the Probability Seminar, the Graduate Probability Seminar and the Statistics Seminar for talks that might interest you. Have a look at the wiki page of our probability group to learn more about the probabilists at UW-Madison.