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.