Markov processes and martingales

1. Sigma algebras, conditional expectation and its properties
2. Martingales, stopped martingales, Doob's optional stopping theorem, applications (ABRACADABRA problem, hitting time for simple random walk)
3. Martingale convergence theorem, L² martingales, Doob decomposition, angle bracket process
4. Uniformly integrable martingales, Levy's upward and downward theorems, applications (Kolmogorov's 0-1 law, strong law of large numbers)
5. Doob's inequalities (submartingale inequality, L^p inequality), law of iterated logarithm
6. Stationary processes, ergodicity, examples (ergodicity of Markov chains), ergodic theorems
7. Central limit theorem for martingales, central limit theorem for Markov chains