The course covers the following topics: Construction of measures, integration, transformation, product spaces, distributions, expectation, Borel-Cantelli lemmas, characteristic functions, weak convergence, independence, weak law of large numbers, strong law of large numbers, central limit theorem, conditional expectation, Markov chains, stopping times and renewal times, martingales, martingale convergence Theorems, Doob’s decomposition theorem, up-crossing inequality, and Birkhoff’s ergodic theorem. The focus will be on mathematical rigor and development of all the tools to prove the results formally.

Advisory: Students are expected to have basic background in probability and real analysis at undergraduate level.