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