| Week | start | topics | Durrett |
|---|---|---|---|
| 1 | Jan 19 | Measure-theory background to Markov chains. Examples and elementary properties | 5.1, 5.2 |
| 2 | Jan 26 | Markov chains: . Strong Markov property. Classification of states; recurrence and transience. | 5.2, 5.3 |
| 3 | Feb 2 | Existence and convergence results for invariant measures. | 5.2, 5.3 |
| 4 | Feb 9 | Coupling. a.s. ergodic theorem; mixing times and coupling. | 5.4 |
| 5 | Feb 16 | Markov chains and martingales; iterated function systems. | . |
| 6 | Feb 23 | Martingales. (review 205A material on convergence theorems.). Levy 0-1 law. Randon-Nikodym theorem. Conditional Borel-Cantelli. Kakutani dichotomy. Galton-Watson processes. | 4.3 |
| 7 | Mar 2 | Maximal inequalities. Boundary crossing inequalities. General forms of optional sampling theorem. Azuma's inequality. | 4.7 |
| 8 | Mar 9 | IID large deviation theorem. Entropy. Statement of Sanov's Theorem. | 1.9 |
| 9 | Mar 16 | Ergodic theorem; applications to RW | 6.1, 6.2, 6.3 |
| 10 | Mar 30 | Entropy; subadditive ergodic theorem and applications. | 6.5, 6.6 |
| 11 | Apr 6 | Brownian motion: definition, existence, continuity and non-differentiability of paths; associated martingales. | Chap. 7 |
| 12 | Apr 13 | Calculations via martungales; calculations via reflection principle; maxima and hitting times, last zero before 1 and first zero after 1; Brownian bridge. | 7.5, 7.4, 7.3 |
| 13 | Apr 20 | Law of iterated logarithm. Skorokhod embedding. Donsker's invariance principle. | 7.6 |
| 14 | Apr 27 | Applications of Donsker's invariance principle. Martingale case. | 7.7 |
| 15 | May 4 | Empirical distributions and Brownian bridge. Local time at zero. | 7.8 |
Last homework assignment. There's some bizarre problem with the PDF not appearing, so here it is
205B Homework, due Thursday May 7
Durrett Chapter 7 problems 5.4, 6.1, 6.3, 7.1