Math 521

• Probability sample spaces, random variables, distribution functions
• Expectation and moments
• Independence
• Weak and strong laws of Large Numbers
• Weak convergence, characteristic functions
• Central Limit Theorems
• Poisson Approximation
• Infinitely Divisible and Stable Distributions
• Random Walks, stopping times, transience and recurrence

Math 522

• Martingales, almost sure convergence, L^p convergence, backward martingales, optional stopping theorems
• Markov chains: Markov property, recurrence and transience, stationary distribution, convergence theorem, Gibbs sampler, Metropolis algorithm, and exact sampling with coupled Markov chains
• Birkoff's ergodic theorems

Math 523

• Brownian motions: construction and basic properties
• Skorokhod's theorem on embedding random walk into Brownian motion, Central limit theorems for martingales and for dependent variables
• Empirical distributions and Brownian bridge
• Laws of the iterated logarithm for Brownian motion and random walks with finite variance
• Topics on stochastic calculus and its application in mathematical finance.

Textbook: Probability: Theory and Examples, 3rd Ed., Rick Durrett.
Additional reference: A Course in Probability Theory, 3rd ed. by K. L. Chung, 2001.

Prerequisites: A knowledge of measure theory and Lebesgue integration, such as covered in the Real Analysis course Math 524/5/6 or Math 424/5/6. A brief review of measure theory will be given in Math 521.