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:**

"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.

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