**Spring 2014**

**Meetings:** TR 1PM-2:15PM, Van Vleck B131

**Instructor:** Benedek Valkó

**Office:** 409 Van Vleck

**Phone:** 263-2782

**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.

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:

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.

- 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.

- Homework 1 Due date: February 6.
- Homework 2 Due date: February 20.
- Homework 3 Due date: March 13.
- Homework 4 Due date: April 3.
- Homework 5 Due date: May 1.

**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.- That being said, group work is encouraged, but you have to write up your own solution. Identical solutions will get no credit!
- 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 together.
- 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 first attempt.
- Papers that are messy, disorganized or unreadable cannot be graded.
- 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.)

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 behavior, periodicity

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 process

7. week (3/4, 3/6): Brief introduction to continuous time Markov chains. In-class midterm exam.

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