Introduction to Probability using Measure Theory
This course will be an introduction to the issues and techniques of probability theory, at the graduate level. The topics covered will include: (i) The measure theoretic framework of modern probability theory; probability spaces and random variables; (ii) Independence and zero-one laws; (iii) Laws of large numbers and Kolmogorov's three series theorem; (iv) Convergence in distribution and the Central Limit Theorem; (v) Conditional Expectation; (vi) An introduction to martingales in discrete-time and applications to Markov chains. Time permitting, we will try to give brief introduction to large deviations and Brownian motion.
Probability with martingales by David Williams
Real Analysis (640:501 or an equivalent) is an essential prerequisite. Students should also have had an undergraduate course, at the level of Ross's text, A First Course in Probability, so that they have a basic familiarity with elementary, combinatorial probability and with the binomial, Poisson, exponential, and normal distributions.