Stat 930-931 / Math 648-649 - graduate level introduction to probability theory.

GENERAL DESCRIPTION:

Graduate level means ``with proofs, based on theory of 
Lebesgue mesaure and integration''.  It does not mean 
``without intuition'' nor ``purely formal''.  

Compare to Stat 510.  There they use Ross.  He does not 
even prove our first result (Strong Law of Large Numbers) 
except in finite variance case.  Furthermore, even if 
you don't care about proofs, he does not even state many 
(most!) critical results on which classical statistical 
tests are based, e.g., LIL, large deviations, any theorem 
or criterion for Poisson approximation to be valid, etc.

HOW TO DECIDE IF YOU HAVE THE PRE-REQUISITES FOR THIS COURSE: 

Students sometimes need some help deciding whether this is 
the right course for them.  If you are in doubt, have a look 
at the textual materials and the first homework (already posted) 
and see whether you can envision being comfortable with that 
material within a few weeks.  In general, facility with writing 
mathematical proofs at the level of math 360-361 is going to be 
much more important than any specific mathematical or statistical 
knowledge.  Mathematical analysis at the level of Penn's Math 508-509 
is recommended and if you've had measure theory you'll be glad of it, 
but many students have not had this and in principle an 
undergraduate level analysis course such as Penn's Math 360-361 
should be sufficient, PROVIDED YOU LEARNED IT WELL.  Most students 
have had undergraduate probability but those who have not usually 
do fine.  Students who have not learned analysis well at the 
basic level (Penn 360-361) will struggle mightily.  

ALTERNATIVE COURSE: Math 546 (Applied probability modeling)

Also provides rigorous, measure-based probability.  Emphasis,
however, is on probability modeling.  Proofs are provided but
are not the focus of the course.  Rather, we focus on how to
construct probability models, how to understand fundamental
constructions such as product spaces, weak limits and point 
processes.  We also focus on achieving precise understanding 
of classical results such as laws of large numbers, Borel-Cantelli,
Martingale results and limit theorems (laws of large numbers,
central limit theorems, Poisson limit theorems, arcsine laws, etc.).

Students will still need a prior course in real analysis but do
not need it at the level of 508-509.  Students should be prepared
to do a final project.  The course is meant to be of particular value
to students with existing applied math interests.  

The main drawback is that, as of now, we can offer only one semester.
This covers roughly the first 60% of Math 648-649, through martingale
theory.