Timo Seppäläinen, Fall 2017:
This is the first semester of a two-semester graduate-level introduction to probability theory. It also serves as a stand-alone introduction to the subject. The course will focus on the following topics: foundational material (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).
FOR MATHEMATICS STUDENTS: In addition to serving students interested in majoring in probability theory, this course goes well together with many areas of mathematics as probability is becoming more and more connected with other parts of mathematics. At the same time the importance of probability in science, technology and industry is increasing, so knowledge of probability can enhance the employment prospects of a math student.
FOR STUDENTS IN OTHER PROGRAMS: Graduate probability is particularly popular among statistics, economics and engineering students. A PhD Math Minor requires four courses, and 733 or 733-734 serve well for that purpose.
PREREQUISITES: Comfort with rigorous analysis and some elementary probability are necessary. Probability theory operates in a measure-theoretic framework, so it is important to know basic measure theory. Depending on the needs of the audience, we can start with a review of measure theory.
TEXTBOOK: Rick Durrett: Probability: Theory and Examples http://www.amazon.com/Probability-Cambridge-Statistical-Probabilistic-Ma...