Spring 2013

- Instructor: David Anderson
- Office: Van Vleck 617, phone: 608-263-4943
- email: anderson at math dot wisc dot edu
- Office hours: Mondays from 9-10AM and Wednesdays from 2:30 - 3:30pm.
- Lectures: Tuesdays and Thursdays, 11:00 - 12:15 PM, Room B131 in Van Vleck.

Required Text: Probability: Theory and Examples, by Rick Durrett. We will use the fourth edition, but earlier editions should be fine. Just make sure you are completing the correct homework assignments.

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 will cover chapters 5 through 8 of Durrett's book.

Prerequisites: Some familiarity with key parts from the first semester, such as measure-theoretic foundations of probability, laws of large numbers, central limit theorem.

Evaluation : Course grades will be based on homework.

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

Homework Assignments:

- Due Tuesday, January 29th. HW1.
- Due Thursday, February 7th. HW2.
- Due Tuesday, February 19th. HW3.
- Due Tuesday, March 5th. HW4.
- Due Thursday, March 21st. HW 5.
- Due Tuesday, April 23rd. HW 6.
- Due Thursday, May 16th at 12pm in my mailbox
in Van Vleck. HW 7.

Week |
Tuesday |
Thursday |

1 |
Jan. 22nd Topic. Review of 831: Chapters 1 - 3. WLLN, SLLN, CLTs. Readings: Chapters 1 - 3. Slides. |
Jan. 24th Topic. Begin review/lectures on Sections 5.1 and 5.2. Readings: Sections 5.1, 5.2. |

2 |
Jan. 29th HW 1
Due. Topic. Martingales: definition, basic properties, convergence. Readings: Sections 5.1, 5.2. |
Jan. 31st Topics. Martingales: Doob's decomposition theorem, examples, optional stopping theorem (not general). Readings: Sections 5.2, 5.3, and 5.4. |

3 |
Feb. 5th Topic. Martingales: Maximal inequalities and convergence in Lp, p>1. Begin considering uniform integrability. Readings: Section 5.4. Begin 5.5. |
Feb. 7th HW 2 Due (changed). Topic. Uniform integrability, convergence in L1. Readings: Sections 5.5. |

4 |
Feb. 12th Topic. Finish U.I., convergence in L1, Levy and Kolmogorov 0-1 laws. Readings. Section 5.5. |
Feb. 14th Topics. Finish Chapter 5: Optional Stopping theorem, backwards martingales. Readings: Sections 5.6 and 5.7. |

5 |
Feb. 19th HW 3
Due. Topic: Markov chains: construction, uniqueness, examples. Sections: 6.1 |
Feb. 21st Topics: Strong Markov property. Beginning recurrence and transience of countable state space chains. Readings: 6.2, 6.3. |

6 |
Feb. 26th No class. |
Feb. 28th Readings: 6.3 - 6.4. |

7 |
March
5th HW 4 Due. Topic: Recurrence and transience of Markov chains. Readings: 6.4. |
March 7th Topic: Stationary distributions. Readings: 6.5. |

8 |
March 12th Readings: 6.5 and 6.6 |
March 14th Readings: 6.6. |

9 |
March 19th Topic: Coupling and strong law for additive functionals. Begin chapter 7: stationary sequences. Readings: 6.6. 7.1. |
March 21st HW 5
Due. Topic: Stationary sequences. Readings: 7.1. |

10 |
March 26th No class due to spring break. |
March 28th No class due to spring break. |

11 |
April 2nd Topic: Birkhoff's Egodic Theorem. Benford's law. Link to radio lab show. Readings: 7.2. |
April 4th No Class. |

12 |
April 9th Topic: Definition and first properties of Brownian Motion. Maybe start Levy's construction. Readings: 8.1 |
April 11th Topic: Levy's construction of Brownian motion. Readings: 8.1 |

13 |
April 16th Topic: Path properties of Brownian motion. Readings: 8.1. |
April 18th Topic: Path properties of Brownian motion. Markov property. Readings: 8.1, 8.2. |

14 |
April 23rd HW 6 Due. Topic: Markov property. Blumenthal 0-1 law and consequences. Readings 8.2. |
April 25th Topic: Stopping times and the strong Markov property. Readings: 8.3. |

15 |
April 30th Topic: Martingales and Brownian motion. Readings: 8.4. |
May 2nd Topic: Integration with respect to Brownian motion. |

16 |
May 7th Topic: Integration with respect to Brownian motion. |
May 9th Topic: Integration with respect to Brownian motion. |

17 |
May 14th No Class: Finals week. |
May 16th HW 7
Due. No Class: Finals week. |

Instructions for Homework

- Homework must be handed in at class time by the due date
- 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.
- 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.