Designed to be a sequel to Statistical Science 711. The basic five topics are: martingales, Markov chains from an advanced viewpoint, ergodic theory, Brownian motion and its applications to random walks, Donsker's theorem and the law of the iterated logarithm, and multidimensional Brownian motion, connection to PDE's. For those who have not had 711, we will prove the law of large numbers using martingales and obtain versions of the central limit theorem from Donsker's theorem. Course requires a knowledge of measure theory. Prerequisite: Statistical Science 711 or Mathematics 631.