Fall 2012

- Instructor: David Anderson
- Office: Van Vleck 617, phone: 608-263-4943
- email: anderson at math dot wisc dot edu
- Office hours: Tuesdays and Fridays: 1:00 - 2:30 PM.
- Lectures: Tuesdays and Thursdays, 11:00 - 12:15 PM, Room B139 in Van Vleck.
- The syllabus can be found here.

Required Text: Probability: Theory and Examples, by Rick Durrett. We will use the fourth edition. For background on measure theory, I recommend Gerald Folland's Real Analysis.

Course content: This is a graduate level introductory course on mathematical probability theory. Here is the Mathematics Department's information page on the course.

We cover selected portions of Chapters 1-5 of Durrett. These are the main topics: foundations (probability spaces and existence of processes), independence, zero-one laws, laws of large numbers, weak convergence and the central limit theorem, conditional expectations and their properties, and martingales (convergence theorem and basic properties).

The course continues in the Spring Semester on topics such as Markov chains, stationary processes and ergodic theory, and Brownian motion.

Prerequisites: Probability theory operates in a measure-theoretic framework, so it is important to know basic measure theory. A suitable background can be obtained from Math 629 or Math 721. Comfort with rigorous analysis and some elementary probability are also necessary.

Evaluation : Course grades will be based on home work assignments, a take-home midterm exam and an in class final exam at the end of the semester.

- Homework: 40%
- Midterm: 25%
- Final Exam: 35%

Final exam information. Topics covered: entire course.

Date: 12/19/12

Time: 5:05PM - 7:05PM

Place: Social Sciences 6102 (NOTE THAT THIS IS NOT OUR NORMAL ROOM!!!)

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

Homework Assignments:

- Here is assignment 1. Due date is Thursday, September 13th. The raw latex is here.
- Here is assignment 2. Due date is Thursday, September 27th.
- HW 3: Exercises 2.2.2 and 2.2.3 from text. Due Thursday, October 4th.
- HW 4. Due Tuesday, October 16th.
- HW 5. Due Thursday, November 1st.
- HW 6. Due Thursday, December 6th.
- HW 7, suggested problems. 5.1.9, 5.2.1,
5.2.4. Let S_n be a random symmetric random walk (i.e.
there are \xi_i, taking values plus and minus one with
probability 1/2 and S_n = \sum_{i=1}^{n} \xi_i). Find a
cubic polynomial g so that g(S_n) is a martingale.

Fall 2012 Schedule: This schedule is tentative and is subject to change. Section numbers refer to Durrett's book.

Week |
Tuesday |
Thursday |

1 |
Sept. 4th Topic: Measure Theory/Probability. Distributions. Readings: 1.1-1.2. |
Sept. 6th Topic: More on Random variables. Integration and expectations. Readings: 1.2-1.4. |

2 |
Sept. 11th Topic: Integration and expectations. Readings: 1.4-1.7. |
Sept.
13th HW 1 Due 1.4-1.7. |

3 |
Sept. 18th 2.1, 2.1.1. |
Sept. 20th 2.1.2 - 2.1.4, 2.2 |

4 |
Sept. 25th 2.2 |
Sept. 27th HW 2
Due 2.2, 2.3 |

5 |
Oct. 2nd 2.3. |
Oct.
4th HW 3 Due 2.3, 2.4. Includes renewal theory. |

6 |
Oct. 9th 2.4 (Renewal Theory and Glivenko-Cantelli theorem). |
Oct. 11th 3.1. |

7 |
Oct. 16th 3.2 Matlab code from lecture is here. |
Oct. 18th HW 4
Due 3.2 |

8 |
Oct. 23rd 3.2, 3.3 |
Oct. 25th 3.3. Characteristic functions and the inversion formula. |

9 |
Oct. 30th No class. |
Nov. 1st Midterm Exam Handed out. HW 5 Due. 3.3, 3.4. |

10 |
Nov. 6th 3.4: Lindeberg-Feler and applications. |
Nov. 8th Midterm
Exam Collected. Finish 3.4, Berry-Esseen bound, start section 3.6. |

11 |
Nov. 13th Section 3.6. |
Nov. 15th Chapter 3.6. |

12 |
Nov. 20th Poisson processes. Slides. |
Nov. 22nd NO CLASS: THANKSGIVING |

13 |
Nov. 27th Section 3.6 and 3.9. |
Nov. 29th Topic : Conditional expectation. Readings : Section 5.1. |

14 |
Dec. 4th Topic : Martingales and their convergence. Readings : Section 5.2. |
Dec. 6th HW 6
Due. Section 5.3. |

15 |
Dec. 11th Chapter 5. |
Dec. 13th Chapter 5. |

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.

- 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.