Meetings: TR 11-12:15 Van Vleck B113 |
Instructor: Timo Seppäläinen |
Office: 419 Van Vleck. Office hours after class on Tuesdays, or any time by appointment |
Phone: 263-2812 |
E-mail: seppalai at math dot wisc dot edu |
In-Class Exam: Sunday, December 4, Van Vleck B113, begins at 12 noon. Do arrive on time because once somebody has left no one can be admitted into the exam. 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 that you choose out of 5. (Some have (a) and (b) part.) You will find the problems quite basic and uncomplicated, covering the main topics: Borel-Cantelli lemmas, laws of large numbers, weak convergence, characteristic functions, normal and Poisson limits, and conditional expectations. |
This is a graduate level introductory course on mathematical probability theory. Here is the Mathematics Department's information page on the course. The material will be based on the book
Richard Durrett: Probability: Theory and Examples. (The fourth edition is the newest but any edition should work.)
There are numerous good books on probability and it may be helpful to look at other books besides Durrett. For example, these authors have written graduate texts: Patrick Billingsley, Leo Breiman, Kai Lai Chung, Richard M. Dudley, Bert Fristedt and Lawrence Gray, Olav Kallenberg, Sidney Resnick, Albert Shiryaev, Daniel Stroock.
We cover selected portions of Chapters 2-5 of Durrett. These are the main topics:
Foundations, existence of stochastic processes |
Independence, 0-1 laws, strong law of large numbers |
Characteristic functions, weak convergence and the central limit theorem |
Conditional expectations |
Martingales |
Week | Tuesday | Thursday |
---|---|---|
2 | 1.1-1.7 Measure theory, probability spaces, random variables. | 1.1-1.7 Measure theory, probability spaces, random variables. |
3 | 1.1-1.7 Computing expectations. | 2.1 Independence |
4 | (Homework 1 due.) 2.1 Proof of π-λ-theorem. Independence and product measures. Definition of convolution. | 2.1 Distribution of a sum of independent random variables. Construction of stochastic processes via Kolmogorov's extension theorem. 2.2 Types of convergence. Chebyshev's inequality. |
5 | (Homework 2 due.) 2.2 Coupon collector's problem. Weak law of large numbers for a triangular array. WLLN. | 2.2 St. Petersburg paradox. 2.3 Borel-Cantelli Lemma. SLLN under a finite fourth moment. Jensen's inequality. |
6 | 2.3 Relationship between a.s. convergence and convergence in probability. 2nd Borel-Cantelli lemma. | (Homework 3 due.) 2.4 Strong law of large numbers. |
7 | 2.4 Glivenko-Cantelli theorem. 2.5 Kolmogorov's 0-1 law, Kolmogorov's inequality, an application to random series. | Midwest Probability Colloquium. No class. |
8 | (Homework 4 due.) 3.2 Weak convergence, portmanteau theorem. | 3.2 Finish portmanteau theorem. Weak convergence on R in terms of cdf's, Scheffé's theorem, continuous mapping theorem, joint versus marginal convergence. |
9 | Philip M. Wood lectures. | Class rescheduled. |
10 | (Homework 5 due.) 3.2 Helly's selection theorem, tightness. 3.3 Definition of the characteristic function. | 3.3 Properties of characteristic functions. Continuity theorem. |
11 | 3.4 Central limit theorem. | 3.4 Lindeberg-Feller theorem. 3.5 Discussion of a local limit theorem. 3.6 Poisson limit. |
12 | 3.6 Poisson process. 3.7-3.8 Brief discussion of stable and infinitely divisible laws. 3.9 Continuity theorem in d dimensions. Multivariate normal. | (Homework 6 due.) 3.9 Multivariate CLT. Brief discussion about Brownian motion, diffusions, and Donsker's invariance principle. 5.1 Definition and first examples of conditional expectation. |
13 | 5.1 Existence and properties of conditional expectation. | Thanksgiving. |
14 | 5.1 Further properties of conditional expectation. Conditional probability distributions. | 5.1 Existence of conditional probability distributions on R. Generalization of Fubini's theorem with stochastic kernels. |
15 | 5.2 Martingales. | 5.2 Stopping times, upcrossing lemma. |
16 | 5.2 Doob decomposition, martingale convergence theorem, applications to random walk. | 5.3 Polya's urn, Galton-Watson branching process. |