### Markov processes and martingales

(Bálint Vető, 2019)
- Sigma algebras, conditional expectation and its properties
- Martingales, stopped martingales, Doob's optional stopping theorem,
applications (ABRACADABRA problem, hitting time for simple random walk)
- Martingale convergence theorem, L
^{2} martingales, Doob decomposition,
angle bracket process
- Uniformly integrable martingales, Levy's upward and downward theorems,
applications (Kolmogorov's 0-1 law, strong law of large numbers)
- Doob's inequalities (submartingale inequality, L
^{p} inequality), law of
iterated logarithm
- Stationary processes, ergodicity, examples (ergodicity of Markov
chains), ergodic theorems
- Central limit theorem for martingales, central limit theorem for Markov
chains