Math/Stat 734 - Theory of Probability II.
Meetings: TR 1PM-2:15PM, Van Vleck B131
Instructor: Benedek Valkó
Office: 409 Van Vleck
Email: valko at math dot wisc dot edu
Office hours: by appointment
I will use the class email list to send out
corrections, announcements, please check your wisc.edu email
from time to time.
This is the second semester of an introductory course on graduate
level mathematical probability theory. 832 covers core topics in
discrete-time and continuous-time stochastic processes. This
includes Markov chains, point processes, stationary processes
and ergodic theory, Brownian motion, and if time permits diffusion
Richard Durrett: Probability:
Theory and Examples
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 831-832 recently) is:
Olav Kallenberg: Foundations of Modern Probability.
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.
We will cover (at least) the following topics (mostly
contained in chapters 6-8 of Durrett):
- Markov chains
- Point processes, Markov processes
- Ergodic theorems
- Brownian motion, stochastic integrals
Course grades will be based on biweekly home work assignments
(25%), a midterm exam (25%) and the final exam (50%). We will have
an in-class midterm exam on March 6. You can use one page of
hand-written (not photocopied!) notes.
The final exam will be in-class on Tuesday, May 6. You may use
two pages of hand-written notes.
Please read the instructions below.
Instructions for homework
- 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
- That being said, group work is encouraged, but you have to
write up your own solution. Identical solutions will get no
- Homework must be handed in by the due date at the beginning
of the class or via email (LaTeX edited solutions are
encouraged). Late submissions will not be accepted.
- Put problems in the correct order and staple your pages
- Do not use paper torn out of a binder.
- Be neat. There should not be text crossed out.
- Recopy your problems. Do not hand in your rough draft or
- Papers that are messy, disorganized or unreadable cannot be
- 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
1. week (1/21, 1/23): Quick
review of martingales. The `Monkey problem'.
2. week (1/28, 1/30): A
central limit theorem for martingales. Markov chains: introduction,
transition probability function
3. week (2/4, 2/6):
Extensions of the Markov property, Chapman-Kolmogorov identity, the
strong Markov property, recurrent and transient states
4. week (2/11, 2/13):
Invariant measures and stationary distributions, asymptotic
5. week (2/18, 2/20):
Convergence theorem, LLN and CLT for Markov chains
6. week (2/25, 2/27):
Conditioning Markov chains, an optimization problem. The Poisson
7. week (3/4, 3/6): Brief
introduction to continuous time Markov chains. In-class midterm
8. week (3/11, 3/13):
Stationary sequences, ergodicity
9. week (3/25, 3/27):
Ergodicity, Birkhoff's ergodic theorem
10. week (4/1, 4/3):
Subadditive ergodic theorem, applications, the longest increasing
subsequence problem, stationary Gaussian processes
11. week (4/8, 4/10):
Brownian motion, existence, Holder continuity
12. week (4/15, 4/17):
Markov property of the Brownian motion, the reflection principle
13. week (4/22, 4/24):
Brownian martingales, estimates on hitting times, multidimensional
Brownian motion, Donsker's theorem
14. week (4/29, 5/1): The
invariance principle, Brownian bridge