In-Class Exam: Wednesday, December 2. Begins at 6 PM. Do arrive on time because once somebody has left no one can be admitted into the exam. The room is Van Vleck B139 and we have it until 9:00. Please bring your pencil and paper and your 3 sheets of notes. No calculators or other electronic gadgets. The exam asks you to hand in 3 problems, some with (a) and (b) part, that you choose out of 4. You will find the problems quite basic and uncomplicated. They cover the main topics: Borel-Cantelli lemmas, laws of large numbers, weak convergence, random walk and stopping times, conditional expectations, and martingales.
Class rescheduling: No class on Tuesday December 15.
Here is the syllabus for the course.
Check out the Probability Seminar for talks on topics that might interest you.
| Week | Tuesday | Thursday |
|---|---|---|
| 1 | ----- | 1.1-1.3 Probability spaces, random variables. |
| 2 | 1.3 Expectations, types of convergence, σ-algebras as information. 1.4 Definition of independence. | (Homework 1 due.) 1.4 π-λ-theorem and consequences. |
| 3 | 1.4 Independence and product measures. Kolmogorov's Extension Theorem. | 1.5 Coupon collector's problem, weak laws of large numbers. |
| 4 | (Homework 2 due.) 1.5 Weak law for the St. Petersburg game. 1.6 Borel-Cantelli lemmas. | 1.7 Strong law of large numbers |
| 5 | 1.7 Glivenko-Cantelli theorem. 1.8 Kolmogorov's 0-1 law, Kolmogorov's inequality, an application to random series. | (Homework 3 due.) 2.2 Weak convergence, portmanteu theorem. |
| 6 | 2.2 Weak convergence on R in terms of cdf's, Helly's selection theorem, tightness. | 2.3 Characteristic functions, continuity theorem. |
| 7 | (Homework 4 due.) 2.4 Lindeberg-Feller theorem. | Discussion of CLT, Berry-Esseen theorem, and local limit theorem. 2.6 Poisson convergence. |
| 8 | 2.9 Multivariate CLT. Discussion about stable laws, Donsker's invariance principle, definition of Brownian motion. | 2.3.e Moment problem. 3.1 Exchangeable σ-algebra. |
| 9 | (Homework 5 due.) 3.1 Hewitt-Savage 0-1 law. Stopping times. Strong Markov property for random walk. | 3.1 Wald's identity. Gambler's ruin. 3.2 Recurrence of SSRW in dimensions 1 and 2. |
| 10 | 3.2 Transience of SSRW in dimensions 3 and higher. 4.1 Conditional expectations. | 4.1 Further properties of conditional expectations. Conditional probability distributions. |
| 11 | 4.1 Generalization of Fubini's theorem with stochastic kernels. 4.2 First properties of martingales. | 4.2 Upcrossing lemma, martingale convergence theorem. |
| 12 | 4.3 Polya's urn. 4.4 Doob's inequality, Lp convergence of martingales. | 4.5 Uniform integrability and L1 convergence. |
| 13 | (Homework 7 due.) 4.7 Optional stopping, applications to asymmetric random walk. 4.6 Definition and limit theorem for backward martingales. | Thanksgiving Day |
| 14 | ||
| 15 |