Stat 530 / Math 546 -- Probability Theory
text: "Probability: theory and examples", 3rd Edition, by R. Durrett.

The required background is (1) enough math background to understand
proof techniques in real analysis (closed sets, uniform convergence,
fourier series, etc.) and (2) some exposure to probability theory
at an intuitive level (a course at the level of Ross's probability
text or some exposure to probability in a statistics class).  

Homeworks: there will be five homework sets in the fall and four in 
the spring, as well as a take-home final exam in each semester.

Semester 1: 

After a summary of the necessary results from mesaure theory, we 
will learn the probabist's lexicon (random variables, independence, etc.).
We will then develoop the necessary techniques (Borel Cantelli lemmas,
estimates on sums of independent random variables and  truncation techniques) 
to prove the classical laws of large numbers.  Next come Fourier
techniques and the Central Limit Theorem, followed by combinatorial
techniques and the study of random walks. 

Semester 2 (Stat 531 / Math 547):

The first task will be to understand conditional expectation.  This is 
harder than you think.  From there, we will define and study martingales.  
We will discuss Markov chains, not giving a comprehensive treatment 
but hitting the basics and some highlights.  We will construct 
Brownian motion and just barely scratch the surface as to the 
properties of this strange object.  I will end with several lectures
on diffusion processes.