| Week | start | topics | Billingsley | Durrett |
|---|
| 1 | Jan 16 | Measure-theory background to Markov chains |
| 5.1 |
| 2 | Jan 23 | Markov chains: examples and elementary
properties. Strong Markov property. | | 5.1, 5.2 |
| 3 | Jan 30 | Classification of states; recurrent
and transience, invariant measures. | | 5.2, 5.3 |
| 4 | Feb 6 | Existence and convergence results for
invariant measures; coupling. | | 5.4, 5.5 |
| 5 | Feb 13 | a.s. ergodic theorem; Metropolis; mixing times
and coupling. | | 5.5 |
| 6 | Feb 20 | Markov chains and martingales; iterated
function systems. | | . |
| 7 | Feb 27 | Martingales (continued from 205A).
Levy 0-1 law. Conditional Borel-Cantelli. Kakutani dichotomy.
Galton-Watson processes. Randon-Nikodym theorem.
| | 4.3 |
| 8 | Mar 6 | Azuma's inequality. Boundary crossing
inequalities. General forms of optional sampling theorem.
| | 4.7 |
| 9 | Mar 13 |
IID large deviation theorem. Start topics below.
| | 1.9 |
| 10 | Mar 20 |
Review characteristics functions (205A).
Infinitely divisible laws; Poisson
limits.
Weak convergence and CLT in R^d;
method of moments; convergence of types; extremal laws.
(selected topics as time permits). |
28, 29, 30, 14 | 2.6, 2.8, 2.9 |
| 11 | Apr 3 | Ergodic theorem; applications to
RW |
| 6.1, 6.2, 6.3 |
| 12 | Apr 10 | Entropy; subadditive ergodic theorem and
applications.
| | 6.5, 6.6 |
| 13 - 15 | Apr 17 | Brownian motion | |
Chap. 7 |
| 15 | May 4 | In class final | | . |