Course Information for STOR 831 (Fall 2009)
 
Advanced Probability

 

Class meetings: MONDAY and WEDNESDAY 2-3:15pm, in HANES 1.  

Prerequisites:  Good working knowledge of measure theoretic probability, and real analysis at the level of Rudin's book ``Principles of Real Analysis''. 

Registration: Enrollment and registration for the course is handled by Charlotte Rogers in the Department Statistics and Operations Research.  Ms. Rogers can be reached at 962.2307, or by email at crogers@email.unc.edu.  

Instructor:  Andrew B. Nobel, Department of Statistics and Operations Research

Office: Hanes 308   Phone: 962-1352.

Office Hours: TBA and by appointment.
 

Course Overview: STOR 831 is a PhD level course in advanced probability that is appropriate for graduate students in Statistics and Mathematics, and for mathematically oriented graduate students in other fields of study.  The course will cover basic results from several areas in probability.  It will focus on results that are of potential use in probabilisitic and statistical research.  Results will be presented in a mathematically rigorous fashion.  Whenever possible we will present complete proofs that illustrate important techniques and ideas.   The course will be self-contained: the majority of the results and applications will be derived from first principles.


Texts:
There is no required textbook for the course.  Depending on the topic, lectures will be based on textbooks, lecture notes, or research papers.  The instructor will provide pointers to the relevant sources for each topic.


Syllabus: 
The following is a tentative list of topics (as of August 25) and will be updated as needed.  The final list of topics will depend on available time and, to some extent, student preferences.

    Large deviation theory: Cramer's theorem in R^1 and R^d and applications
           Sanov's theorem for finite alphabets
           Varadhan's Integral Lemma
           Long rare segments in random walks

     Introduction to coupling with applications
          Poisson approximation for independent indicators
          Existence of stationary distributions for Harris Markov Chains

    Poisson approximation using the Chen-Stein method, with emphasis on the case of dependent indicators
         Combinatorial and probabilisitic applications

    Basic comparision theorems for Gaussian random variables. 

    Concentration inequalities
         The Efron-Stein inequality
         Gaussian concentration via log Sobolev inequalities
         Borell's inequality for Gaussian processes
         Inequalities for bounded differences and self-bounding functions

    Ergodic theory: basic definitions, notions of mixing, ergodic and subadditive ergodic theorems


Grading: Reading assignments and homework problems will be set periodically throughout the semester.  In addition, students will be asked to read and offer written or oral comments on a paper in the literature.  More details on the project will be provided later in the semester.


Lecture references:

"Large Deviations Techniques and Applications", second edition, by A. Dembo and O. Zeitouni, Springer, 1998.

"Lectures on the Coupling Method'', by T. Lindvall, Wiley, 1992.

"Poisson Approximation", by A.D. Barbour, L. Holst and S. Janson, Oxford Science, 1992.

"Concentration Inequalities", by S. Boucheron, O. Bousquet, and G. Lugosi, 2004.  Lecture notes available from G. Lugosi's web page. 

"Probability: Theory and Examples", by R. Durrett.  Preliminary versions of the 4th edition are available from the author's web page.

Background and further references:

"Probability and Measure", third edition, P. Billingsley, Wiley, 1995.

"The Cauchy-Schwartz Master Class'', J.M. Steele, Cambridge, 2004.  (A nice book on basic inequalities that can be read with relatively little background.  Contains many of problems, with solutions.)

"Principles of Mathematical Analysis", third edition, W. Rudin, McGraw Hill. (A good reference for real analysis.)

"Probability", L. Breiman, SIAM, 1992.

"Real Analysis and Probability", R.M. Dudley, Chapman and Hall, 1989.  

"The Concentration of Measure Phenomenon'', M. Ledoux, AMS Mathematical Surveys, 2000.

Honor Code: Students are expected to adhere to the UNC honor code at all times.