Lecture 1. Regular conditional distributions and probabilities, definition of martingale, martingale transform, martingale decomposition, Azuma-Hoeffding inequality.
Lecture 2. Doob's maximal inequalities, stopping time, upcrossing inequality, almost sure martingale convergence, second Borel-Cantelli lemma, Polya's urn.
Lecture 3. L_p martingale convergence theorems, Levy's 0-1 law, Doob's decomposition, quadratic variation process.
Lecture 4. Non-negative martingales as changes of measure, optional stopping theorem, backwards/reversed martingales.
Lecture 5. Markov chains, the Markov and strong Markov property, irreducibility, transience and recurrence.
Lecture 6. Criteria for transience and recurrence, simple random walks, birth-death chains, harmonic functions, Dirichlet problem and Poisson equation.
Lecture 7. Existence and uniqueness of stationary measures and convergence.
Lecture 8. Periodic Markov chains, Perron-Frobenius theorem, reversible Markov chains.
Lecture 9. Ergodic theorems.
Lecture 10. Ergodic decomposition, structure of stationary Markov chains, Harris chains.
Lecture 11. Brownian motion: characterization, invariance, path properties.
Lecture 12. Brownian motion: Markov and strong Markov property, Blumenthal's 0-1 law, reflection principle, martingale property, recurrence and transience.
Lecture 13. Weak convergence, Lindeberg's CLT, Donsker's invariance principle.
Lecture notes on Limit Theorems by S.R.S. Varadhan.
Probability: Theory and Examples by Richard Durrett.
Probability Theory -- A Comprehensive Course by Achim Klenke.
Foundations of Modern Probability by Olav Kallenberg.
Probability by Leo Breiman.
An Introduction to Probability Theory and Its Applications, Vol II by William Feller.
A Course in Probability Theory by Kai Lai CHUNG.