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 firstname.lastname@example.org.
Instructor: Andrew B. Nobel, Department of Statistics and Operations Research
Office: Hanes 308 Phone: 962-1352.
Hours: TBA and by appointment.
Overview: STOR 831 is a PhD level course in
advanced probability that is appropriate for graduate
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
be presented in a mathematically
Whenever possible we will present complete proofs that illustrate
important techniques and
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
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
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
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
Deviations Techniques and Applications", second edition, by A. Dembo
and O. Zeitouni, Springer, 1998.
and further references:
and Measure", third edition, P. Billingsley, Wiley,
"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.)
L. Breiman, SIAM, 1992.
Analysis and Probability", R.M. Dudley, Chapman and Hall, 1989.
are expected to adhere to the UNC honor code at all