Math/Stat 831 Theory of Probability

Fall 2009

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.

Homework 1 due September 10. First solve this question. Suppose f is a measurable mapping from one measurable space S to another measurable space U. If A is a measurable subset of S, does it follow that the image f(A) is a measurable subset of U? Then from Chapter 1 these exercises: 1.9 (answer also the converse question: if Y is uniform, does it follow that F is continuous?), 2.1, 2.8, 3.8, 3.15. Notes: In 2.8 you may assume the conclusion of 2.7 since this is basic measure theory. Instead of 2.8 you may do 2.9. The purpose is to get the result which is important. If you are able, do not assume X is real-valued but rather takes its values in some abstract space.
Homework 2 due September 22. Solutions.
Homework 3 due October 1. Solutions.
Homework 4 due October 13. Solutions.
Homework 5 due October 27. Solutions.
Homework 6 due November 12. Solutions.
Homework 7 (last one) due November 24.

Check out the Probability Seminar for talks on topics that might interest you.

Fall 2009 Schedule

(Section numbers refer to Durrett's book.)

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, L^p 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

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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, L^p 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
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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.
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.