Math/Stat 832 - Theory of Probability II.
Winter 2009
Meetings: TR 13-14:15, Van Vleck B119
Instructor: Benedek Valkó
Office: 409 Van Vleck
Phone: 263-2782
Email: valko at math dot wisc dot edu
Office hours: after class on Tuesdays, or 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 martingales, Markov chains, point processes,
stationary processes and ergodic theory,
Brownian motion, and diffusion processes.
Textbook:
Richard Durrett: Probability: Theory and Examples
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 (plan to) cover the following topics (mostly contained in chapters 5-7 of Durrett):
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Markov chains
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Point processes, Markov processes
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Ergodic theorems
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Brownian motion, stochastic integrals
Evaluation: Course grades will be based on home work assignments, and a take-home final exam.
Homework:
Instructions for the homework assignments:
Homework must be handed in by the due date by the beginning of the
class. It can be handed in by person in class, in the instructor's
mailbox or by email. 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.)
Schedule:
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Week 1. Doob's maximal inequality, another proof for the (sub)martingale convergence theorem, backwards martingales
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Week 2. U-statistics, Markov chains in discrete time
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Week 3. Examples, extension of the Markov property, Strong Markov property
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Week 4. Recurrent and transient states, Stationary measures
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Week 5. Stationary measures (existence and uniqueness), positive and null recurrent states, convergence of average number of visits
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Week 6. Convergence to the stationary measures (coupling proof,
algebraic proof), periodic case, Law of Large Numbers, Central Limit
Theorem
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Week 7. Continuous time Markov processes (the exponential clock construction), Stationary sequences, Ergodicity
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Week 8. Ergodicity, Birkhoff's Ergodic Theorem, Recurrence
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Week 9. Subadditive ergodic theorem: examples and proof
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Week 10. Subadditive ergodic theorem: proof, Brownian motion