MATHEMATICS 563

Probability Theory

Fall 2006

Class Notes

Overview. Probability Theory is a one semester course. During the semester, we will take a measure theoretic approach to the basic issues in probability - random varibles, expectations, modes of convergence, stochastic independence, laws of large numbers, and the central limit theorem. If time permits, we will also look at infinitely divisible distributions as limits of sums of independent random variables. Throughout the course, I will try to maintain a dialogue between the abstract issues and the practical examples of probability theory.

We will not be using a text, but rather notes will be posted on-line.

Day to Day Operations. The course meets for lecture each Monday, Wednesday, and Friday from 11:00 to 11:50 in the Physics and Atmospheric Sciences Building Room 314.
If you need to contact me, write me electronically at jwatkins@math.arizona.edu, call me at 621-5245, or drop by my office Math 520.

Evaluation of Students. We will evaluate your work in the course through your performance on homework assignments. The notes have a large number of exercises - - they will be graded at a point per problem written or presented to a maximum of 60 points. The first 20 points must be completed by October 15th, the first 40 by November 15th. I will also give a take home final worth 20 points. A student may choose to replace the final with a report. Below is a list of suggested topics.

• Random sums
• Branching processes
• Geometric probability
• Feller's proof of the central limit theorem
• Rates of convergence in the central limit theorem
• The delta method
• Entropy and large deviations for finite alphabets
• 0-1 laws
• Borel's normal number theorem
• DeFinetti's theorem
• Stable laws
• A topic in the history of probability
• Simulation exercise

The course material will be taken from a variety of sources. I will be primarily be using

• Probability and Measure by Patrick Billingsley
• Probability by Leo Brieman
• A Course in Probability Theory by Kai Lai Chung
• Probability: Theory and Examples by Richard Durrett
• A Modern Approach to Probability Theory by Bert Fristedt and Lawrence Gray
• Probability and Martingales by David Williams
  and the classic
• An Introduction to Probability Theory and Its Applications, Volumes I & II by William Feller.

Here is an approximate course syllabus.

• Introduction
• Random Variables
• Expectations and Integration Theory
• Borel-Cantelli Lemmas
• Class Theorems
• Extension Theorems
• Independence
• Sums of Independent Random Variables
• Zero-One Laws
• Strong Law of Large Numbers
• Convergence in Distributions
• Central Limit Theorem
• Levy-Khintchin representation

Those who complete the course will be prepared to take Mathematics 565a, Discrete Time Stochastic Processes, and Mathematics 567a, Theoretical Statistics.

Best wishes to you for a good semester in this course and in all your other activities.

Joe Watkins