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 infinitedimensional 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—infinitedimensional 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.
Our main topic will be stochastic processes. Distributions of stochastic processes are essentially probability measures on infinitedimensional 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—infinitedimensional 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.
Audience
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, 10:1011:25am.Room: SSW 903.
Requirements: Probability Theory I.
Class Notes
 Class Notes (Version: 9 May 2016)
Homework
 Homework 1 (Due: 3 Feb 2016)
 Homework 2 (Due: 10 Feb 2016)
 Homework 3 (Due: 17 Feb 2016)
 Homework 4 (Due: 24 Feb 2016)
 Homework 5 (Due: 2 Mar 2016)
 Homework 6 (Due: 30 Mar 2016)
 Homework 7 (Due: 6 Apr 2016)
 Homework 8 (Due: 13 Apr 2016)
 Homework 9 (Due: 20 Apr 2016)
 Homework 10 (Due: 27 Apr 2016)
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]

Probability Theory.
Achim Klenke.
Springer, 2000.
[Available online]

InfiniteDimensional Analysis.
C. D. Aliprantis and K. C. Border.
Springer, 2006.
[Available online]

Foundations of Modern Probability.
Olav Kallenberg.
Springer, 2001.
[Available online]