Math/Stat 831 - Theory of Probability
Fall 2008
Meetings: TR 11-12:15, B123 Van Vleck
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 first semester of a two-semester
graduate-level introduction to probability theory and it
also serves as a stand-alone introduction to the subject.
The course will focus on discrete-time stochastic processes
and cover at least the following topics:
foundations (probability spaces and existence
of processes), independence, zero-one laws, laws of large numbers,
weak convergence and the central limit theorem,
conditional expectations and their properties, and
martingales (convergence theorem and basic properties).
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 the previous year's course) is:
Olav Kallenberg: Foundations of Modern Probability.
Prerequisites: Probability
theory operates in a measure-theoretic framework, so it is
important to know basic measure theory. A suitable background can be obtained from Math 629 or Math 721. The appendix of Durrett covers the measure theory we need. If needed, some aspects of measure theory can be reviewed at the beginning.
Comfort with rigorous analysis and some elementary probability are also necessary.
Course Content: we will cover (at least) the following topics (mostly contained in the first four chapters of Durrett):
- foundations (probability spaces and existence
of processes)
- independence, zero-one laws
- laws of large numbers
- weak convergence and the central limit theorem
- conditional expectations and their properties
- martingales (convergence theorems and basic properties)
Evaluation: Course grades will be based on home work assignments, a take-home midterm exam and an in-class exam at the end of the semester.
The date and time for the final exam is Friday, December 12, 2008, 4:30pm, Room B139.
You will be allowed to bring three sheets of handwritten notes to the final exam. Textbooks, calculators, electronic gadgets won't be allowed.
The final will be cumulative, but it will put more weight on the second half of the semester.
Homework:
Instructions for the homework assignments:
Homework must be handed in by the due date, either in class or by 3PM 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.)
Schedule:
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Week 1. Definition of the probability space, basic properties, examples, random variables, distribution, distribution function
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Week 2. Expectation of random variables, basic properties, inequalities, connection to limits.
Independence of events, random variables, σ-fields, sufficient condition for independence of σ-fields
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Week 3. Independent random variables and product measures, expectation and independence,
Kolmogorov's 0-1 Law, construction of independent random variables, Kolmogorov's Extension Theorem
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Week 4. Types of convergences, Weak Law of Large Numbers with second moment, Weak LLN using cutoff, Borel-Cantelli lemma,
Strong LLN with fourth moment
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Week 5. Characterization of convergence in probability using a.s. convergence, the second Borel-Cantelli lemma, Strong LLN with existence of expectation
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Week 6. Uniform integrability, Coupon collector's problem, a.s. limit in renewal processes,
Glivenko-Cantelli thm, weak convergence of distributions
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Week 7. Properties of weak convergence, equivalent definitions, Helly's selection thm, tightness, characteristic functions
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Week 8. Inversion formula for the characteristic function, examples, Continuity theorem
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Week 9. Central Limit Theorem, Lindeberg-Feller theorem
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Week 10. Speed of convergence in the CLT, local CLT, Poisson approximation
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Week 11. Poisson process, stable and infinitely divisible distributions, conditional expectation
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Week 12. Properties of conditional expectation, martingales, sub- and supermartingales
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Week 13. Properties of martingales, stopping times, predictable sequences.
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Week 14. The martingale convergence theorem, applications.
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Week 15. Optional stopping, regular martingales