| LEC # | TOPICS | KEY DATES |
|---|---|---|
| 1 | Probability spaces, properties of probability | |
| 2-3 | Random variables and their properties, expectation | |
| 4 | Kolmogorov's theorem about consistent distributions | |
| 5 | Laws of large numbers | |
| 6 | Bernstein's polynomials, Hausdorff and de Finetti theorems | |
| 7 | 0-1 laws, convergence of random series | |
| 8 | Stopping times, Wald's identity Markov property, another proof of SLLN | Problem set 1 out |
| 9-10 | Convergence of laws, selection theorem | Problem set 1 due in Lec #9 |
| 11 | Characteristic functions, central limit theorem on the real line | |
| 12 | Multivariate normal distributions and central limit theorem | |
| 13 | Lindeberg's central limit theorem Levy's equivalence theorem, three series theorem | |
| 14 | Levy's continuity theorem Levy's equivalence theorem, three series theorem (cont.) Conditional expectation | Problem set 2 out |
| 15-16 | Martingales, Doob's decomposition Uniform integrability | Problem set 2 due in Lec #15 |
| 17 | Optional stopping, inequalities for Martingales | |
| 18-19 | Convergence of Martingales | Problem set 3 out in Lec #19 |
| 20-21 | Convergence on metric spaces, Portmanteau theorem Lipschitz functions | Problem set 3 due in Lec #20 |
| 22 | Metrics for convergence of laws, empirical measures | |
| 23 | Convergence and uniform tightness | |
| 24-25 | Strassen's theorem, relationship between metrics | |
| 26-27 | Kantorovich-Rubinstein theorem | |
| 28-29 | Prekopa-Leindler inequality, entropy and concentration | Problem set 4 out in Lec #29 |
| 30 | Stochastic processes, Brownian motion | Problem set 4 due |
| 31 | Donsker invariance principle | |
| 32-33 | Empirical process and Kolmogorov's chaining | |
| 34-35 | Markov property of Brownian motion, reflection principles | |
| 36 | Laws of Brownian motion at stopping times Skorohod's imbedding | |
| 37 | Laws of the iterated logarithm |