Math 6710: Probability Theory I

Fall 2012


Course overview

This will be a fairly standard introduction to measure-theoretic probability theory at the graduate level. We will begin with a brief review of abstract measure theory, with a view to its use in probability: random variables, expectation, measurability and σ-fields. We will discuss the idea of independence and prove the strong law of large numbers and some related results. Next we will attack the central limit theorem, which will require some discussion of weak convergence (aka convergence in distribution) and characteristic functions (aka Fourier transforms). Finally, to begin a transition to stochastic processes, we will talk about random walk, martingales, and possibly Markov chains, all in discrete time.


R. Durrett, Probability: Theory and Examples, 4th edition. Here is an unofficial errata list. Let me know if you find other errors and I will add them to this list (and pass them along to Rick.)

Also, several other texts are on reserve at the Mathematics Library (Malott Hall, 4th floor). You can check them out for 24 hours at a time. See here for the list.

Some other potentially useful references:

Lecture notes

Caution! These lecture notes are very rough. They are mainly intended for my own use during lecture. They almost surely contain errors and typos. In many cases details, precise statements, and proofs are left to Durrett's text, homework or presentations. But perhaps these notes will be useful as a reminder of what was done in lecture. If I do something that is substantially different from Durrett I will put it in here.

Anyway here they are. 6710-lecture-notes.pdf

Homework assignments

  1. (Due Thu, Aug 30) hw01.pdf (Correction posted August 26)
  2. (Due Thu, Sep 6) hw02.pdf
  3. (Due Thu, Sep 13) hw03.pdf
  4. (Due Thu, Sep 20) hw04.pdf (Updated September 17 with minor typo fixes)
  5. (Due Thu, Sep 27) hw05.pdf (Updated September 26 to remove problem 1)
  6. (Due Thu, Oct 4) hw06.pdf
  7. There is no HW 7. Enjoy fall break!
  8. (Due Thu, Oct 18) hw08.pdf
  9. (Due Thu, Oct 25) hw09.pdf
  10. (Due Thu, Nov 1) hw10.pdf
  11. (Due Thu, Nov 8) hw11.pdf
  12. (Due Thu, Nov 15) hw12.pdf
  13. There is no HW 13. Enjoy Thanksgiving break!
  14. (Due Thu, Nov 29) hw14.pdf

Homework policies

Written homework will be due in class on Thursdays.

Late homework policy: You will have 3 automatic extensions which you can use any time during the semester to turn in an assignment on the Tuesday after it is due. You can use these for any reason you like; if you have an emergency, travel, or just want more time to work on the assignment. You don't have to tell me in advance; just hand in the assignment on Tuesday and write "extension" on it. You cannot use multiple extensions on the same assignment. Note that homework turned in with an extension may not be graded until the following week. Other than this, no late homework will be accepted without a compelling reason.

Group work policy: The purpose of homework in this class is to gain experience in working with the ideas studied in class; this is usually where the most learning takes place. You are encouraged to discuss the homework problems with classmates or colleagues on a high level, but you should write your solutions by yourself.

Student presentations

Since it's important to be able to communicate your mathematical work to others, we will have brief student presentations throughout the course. A few times during the semester, each student will be given a homework-like problem or a theorem from the text, and asked to prepare it and present it during class. These will generally take about 5-10 minutes.

When it is your turn, I will send you an email at least 6 days ahead of the presentation date. Dates and assigned material are negotiable if there are problems. You are always welcome to come and discuss or practice your presentation with me ahead of time.



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