Math/Stat 832 - Theory of Probability
Meetings: TR 11-12:15, VAN VLECK B131
Instructor: Benedek Valkó
Office: 409 Van Vleck
Email: valko at math dot wisc dot edu
Office hours: TR 9:30-10:30 or by appointment
Grader: Diane Holcomb
I will use the class email list to send out corrections, announcements, please check your wisc.edu email from time to time.
Richard Durrett: Probability: Theory and Examples
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 martingales, Markov chains, point processes, stationary processes and ergodic theory, Brownian motion, and diffusion processes.
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):
- Markov chains
- Point processes, Markov processes
- Ergodic theorems
- Brownian motion, stochastic integrals
Evaluation: Course grades will be based on home work assignments,
a take-home final exam at the end of the
Extra notes on the martingale central limit theorem (based on notes of S. Sethuraman).
Instructions for the homework assignments:
Homework must be handed in by the due date, either in class or by 12PM
in the instructor's mailbox. Late submissions will not be accepted.
Group work is encouraged, but you have to write up your own solution.
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.)
HOW TO MAKE THE GRADER HAPPY
Week 1. Uniform integrable random variables, regular martingales, exchangeable random variables, Hewitt-Savage 0-1 law
Week 2. De Finetti's theorem on exchangeable random variables, Martingale CLT, Markov Chains (definition, examples)
Week 3. Extensions of the Markov Property, strong Markov property, recurrence, transience
Week 4. Stationary measures, existence in the recurrent case, uniqueness
Week 5. No classes.
Week 6. Irreducible recurrent Markov chains and stationarity, return times and stationary distributions, positive recurrence, periodicity, convergence theorem
Week 7. Law of large numbers and central limit theorem for discrete Markov chains. Poisson process with non-constant intensity, continuous time birth proces.
Week 8. Continuous time discrete Markov chains. Forward and backward equations.
Week 9. Stationary sequences, ergodic sequences
Week 10. Ergodic theorems, Sub-additive ergodic theorem, longest increasing subsequence problem
Week 11. Sub-additive ergodic theorem (proof of a simpler version), Brownian motion (definition)
Week 12. Existence of Brownian motion, Kolmogorov continuity theorem, Levy's construction, path properties, Holder continuity and non-differentiability, quadratic variation, Markov property