Here are some Lecture notes on conditioning relevant to the material and homework for weeks 8 and 9.
| Week | start | topics | Billingsley | Durrett |
|---|---|---|---|---|
| 1 | Aug 30 | Fields, sigma-fields, measurable functions, measures | 2,10,13 | A1 |
| 2 | Sep 6 | Lebesgue measure, distribution functions, coin-tossing, abstract integration | 3,12,15 | 1.1, A2,3,4 |
| 3 | Sep 13 | Probability spaces, random variables, expectation, inequalities | 4,5,20 | 1.2,1.3, A5 |
| 4 | Sep 20 | Independence, WLLN, Bernstein's theorem | 6,20 | 1.4, 1.5 |
| 5 | Sep 27 | a.s. limit theorems (max of exponentials, head-runs), modes of convergence, dominated convergence, Fatou, 4'th moment SLLN | 6,21 | 1.6 |
| 6 | Oct 4 | Glivenko-Cantelli, SLLNs, maximal inequality, convergence of random series | 22 | 1.7, 1.8 |
| 7 | Oct 11 | Renewal SLLN, stopping times, Wald's equation; Kolmogorov 0-1 law; large deviations | 22, 9 | 3.1, 1.9 |
| 8 | Oct 18 | Radon-Nikodym; joint distributions correspond to marginals and a kernel | 32, 33 | 4.1, A8 |
| 9 | Oct 25 | Product measure, Fubini's theorem and examples. Kolmogorov consistency theorem. Conditional expectation | 18, 34, 36 | 4.1, A6,7 |
| 10 | Nov 1 | Conditional expectation (continued). Definition and examples of martingales | 35 | 4.2 |
| 11 | Nov 8 | Convexity, optional sampling, MG analog of Wald, RW examples. | 35 | 4.3, 4.7 |
| 12 | Nov 15 | Maximal and upcrossing inequalities, MG convergence theorems, Levy 0-1 law, L_p convergence, play red | 35 | 4.3, 4.4, 4.5 |
| 13 | Nov 22 | Proof of Radon-Nikodym, a.s convergence of sums, reversed MGs and SLLN, de Finetti's theorem, Azuma's inequality. | 35 | 4.3, 4.7 |
| 14 | Nov 29 | Brownian motion (continues next week) | 37 | 7.1 - 7.5 |