Math/Stat 734 - Theory of Probability II.

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.

Course description

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 processes.

Textbook

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.

Prerequisites

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.

Course content

 We will cover (at least) the following topics (mostly contained in chapters 6-8 of Durrett):

Evaluation

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.

Homework assignments

Please read the instructions below. 


Instructions for homework assignments


Weekly schedule

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 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