Spring 2016: Probability Theory II (G6106)

This class is the continuation of Probability Theory I (STAT G6105) for Statistics PhD students.

Our main topic will be stochastic processes. Distributions of stochastic processes are essentially probability measures on infinite-dimensional spaces, and we will have to learn basic analysis and measure theory on such spaces as we go along. Other topics include conditioning and martingales.

Many of these concepts are also of great importance in statistics—infinite-dimensional spaces are the natural habitat of nonparametric models, conditioning is the principal tool of Bayesian statistics, and so forth—and I will try to highlight connections between probability and statistics wherever possible.


This class is for PhD students only. We will not make exceptions.
The target audience are PhD students in the statistics program. PhD students from other departments are welcome, but require instructor's permission (please contact me).

Course Specs

Time: Mondays and Wednesdays, 10:10-11:25am.
Room: SSW 903.
Requirements: Probability Theory I.

Class Notes


Textbooks and Syllabus

For a tentative syllabus, have a look at last year's class notes (although I will probably make some adjustments, since the content of Probability Theory I has changed somewhat).

No textbook is required; the relevant reference are the class notes.

Here are some books you may find useful that are available online (if you are connecting from a Columbia IP address): You may already own a copy of the Jacod/Protter textbook. Most of our topics are not covered by this book, but it may be a useful reference for results covered in Probability I:
  • Probability Essentials.
    Jean Jacod and Philip Protter.
    Springer, 2000.

    [Available online]
Another good general-purpose reference that covers more topics and in more depth is: A great reference on analysis, measures, Borel fields etc is the book by Aliprantis and Border. This is quite simply one of the best math books I know.
  • Infinite-Dimensional Analysis.
    C. D. Aliprantis and K. C. Border.
    Springer, 2006.

    [Available online]
My favorite book on probability, though perhaps a little daunting as an introductory text:
  • Foundations of Modern Probability.
    Olav Kallenberg.
    Springer, 2001.

    [Available online]

Spring 2015 web page

The class web page for the previous installment of this class is still available here.