Math/Stat 831 Probability I

Fall 2011

Meetings: TR 11-12:15 Van Vleck B113

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.

	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.

Prerequisites

Measure theory is a basic tool for this course. A suitable background can be obtained from Math 629 or Math 721 (possibly concurrently). Chapter 1 in Durrett covers the measure theory needed. If desired some measure theory can be reviewed at the start. Prior exposure to elementary probability theory is also necessary.

Course Content

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

The course continues in the Spring Semester on topics such as Markov chains, stationary processes and ergodic theory, and Brownian motion. Requests for particular topics for the spring term are welcome.

Exams and Grades

Course grades will be based on take-home work and one in-class exam where you can bring 3 sheets of notes. Homework assignments, updates on rescheduling classes, and other matters will appear on this course homepage.

Homework and Exams

Homework 1 due September 20.
Homework 2 due September 27.
Homework 3 due October 6.
Homework 4 due October 18.
Homework 5 due November 1.
Homework 6 due November 17.
Exam
Homework 7 due December 13.

Related Seminars

Check out the Probability Seminar and the Statistics Seminar for talks that might interest you.

Schedule

This schedule tracks our progress. Section numbers refer to the 4rd edition of Durretts' text.

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.

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.

Instructions for Homework

Homework must be handed in by the due date, either in class or by 3 PM in the instructor's office or mailbox. Late submissions cannot be accepted.
Neatness and clarity are essential. Write one problem per page except in cases of very short problems. Staple you sheets together. You are welcome to use LaTeX to typeset your solutions.
It is not trivial to learn to write solutions. You have to write enough to show that you understand the flow of ideas and that you are not jumping to unjustified conclusions, but not too much to get lost in details. If you are unsure of the appropriate level of detail to include, you can separate some of the technical details as "Lemmas" and put them at the end of the solution. A good rule of thumb is if the grader needs to pick up a pencil to check your assertion, you should have proved it. The grader can deduct points in such cases.
You can use basic facts from analysis and measure theory in your homework, and the theorems we cover in class without reproving them. If you find a helpful theorem or passage in another book, do not copy the passage but use the idea to write up your own solution. If you do use other literature for help, cite your sources properly. However, it is better to attack the problems with your own resources instead of searching the literature or the internet. The purpose of the homework is to strengthen your problem solving skills, not literature search skills. (In particular: do not use Lyapunov's theorem as a substitute for Lindeberg-Feller unless we have covered Lyapunov's theorem or it has been proved in a homework.)
It is valuable to discuss ideas for homework problems with other students. But it is not acceptable to write solutions together or to copy another person's solution. In the end you have to hand in your own personal work.