Math 546 (Stat 530-) Probability Theory

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, as well as a take-home final exam in each semester.

After a summary of the necessary results from measure theory, we will learn the probabilist's lexicon (random variables, independence, etc.).  We will then develop 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 comes Fourier techniques and the Central Limit Theorem, followed by combinatorial techniques and the study of random walks.

Texts:
"Probability: theory and examples", 3rd Edition, by R. Durrett.