Statistics 330/600: Advanced Probability

Time:  Tues., Thurs., 2:30-3:45  

Prerequisite: a first course in probability, some knowledge of real
analysis.

Instructor: Joseph Chang.  
24 Hillhouse Ave., 432-0642, chang@stat.yale.edu

Teaching assistants: Yuhong Yang and Qun Xie.

Text: The required text is "An Exposition of Probability," by David
Pollard.  This is a manuscript for a forthcoming book to be published
by Springer-Verlag.  It is not in the coop; I'll tell you how to get
it during our first class meeting (it will probably be at Tycho's).
The coop has also ordered two recommended supplementary texts:
"Probability with Martingales" by D. Williams and "Probability: Theory
and Examples" by R. Durrett.

Grading: Grades will be based on weekly homework assignments.
   
********************************************************************* 

This course in measure-theoretic probability is intended to provide a
solid foundation for those who intend to do advanced work in
probability, stochastic processes, statistics, and other mathematical
fields that use probability (engineering, computer science, economics,
finance...).  The aim is to develop the mathematical foundations of
probability systematically.  These concepts and techniques are needed
in order to prove the fundamental results in probability theory, and
even in order to understand what they are saying.  They also provide a
powerful conceptual unity, which (i) eliminates artificial distinctions
made in elementary probability courses, such as the division of the
theory into "discrete" and "continuous" cases, and (ii) allows the same
theory to be applied in more abstract settings, such as stochastic
processes. 

The basic topics are:

1. The measure-theoretic framework of probability:  probability spaces,
sigma-fields, random variables, measures as linear functionals, some
standard facts such as the dominated convergence theorem, product
measures, independence.
 
2. Weak and strong laws of large numbers: convergence in probability,
almost-sure convergence, Borel-Cantelli lemmas, truncation.

3. Conditioning:  Disintegrations and conditional expectations with
respect to sigma-fields.

4. Martingales:  Submartingales and supermartingales, optional stopping
theorems, martingale convergence theorems, applications.

5. Weak convergence and Central Limit Theorems.

Other topics will be discussed as time permits.


********************************************************************* 

If you have questions or would like to discuss the course, feel free to
send email to chang@stat.yale.edu or call me at 432-0642.