Math/Stat 734 - Theory of Probability II.
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)
page gives an overview of the course, all relevant
information and resources will be provided in the official
This is the second semester of an
introductory course on graduate level mathematical probability
theory. 734 covers core topics in discrete-time and
continuous-time stochastic processes. This includes martingales,
Markov chains, point processes, stationary processes and ergodic
theory, Brownian motion.
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,
- 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
We cover selected
portions of Chapters 4-8 of Durrett. This is a rough course
|Weeks 1-2: Martingales (review, Doob's inequality, uniform
integrability, backwards martingales)
|Weeks 3-7: Markov chains
|Weeks 8-9: Stationary sequences, ergodicity, subadditive
|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.)
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 firstname.lastname@example.org.
You can access our Piazza page from the Canvas page of the course.
Instructions for homework
- 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
- 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
- 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.
If you would like to learn more
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.