My plan to write a post a day about a likely piece of Advanced Probability bookwork has been more useful than even I anticipated. It’s surprising how often you come to write something down and realise you don’t understand half the steps. Anyway, here’s a list, in approximately chronological order through the course, of what I’ve covered:

- The existence of Conditional Expectations
- Martingales
- Martingale Convergence Theorems
- Levy’s Convergence Theorem
- Existence of Brownian Motion (Wiener’s Theorem)
- The Strong Markov Property for BM
- Blumenthal’s 0-1 Law
- BM is a.s. nowhere differentiable (not examinable I think, but interesting)
- Recurrence and transience properties of BM
- The Dirichlet Problem

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