Advanced Probability Revision Summary

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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