We will cover the basics of stochastic processes from Le Gall’s book on Measure Theory, Probability and Stochastic Processes and then we will focus on discrete time Markov chains and cover topics related to their mixing time from Levine and Peres’ book.
To understand the term “mixing time of a Markov chain”, let us consider a (the) classical example: shuffling a deck of n cards using a riffle shuffle. A configuration of the deck is given by a permutation of length \(n\). Clearly the riffle shuffle is a Markov chain; i.e., the probability distribution of the configuration after a shuffle depends only on the current configuration. How many riffle shuffles does one need to completely mix the deck? The answer due to Bayer and Diaconis (1992) is that you need at most 7 shuffles. We will make this statement rigorous, and consider many other such examples and techniques for estimating the mixing time. This approach will let us survey many of the chains and models that are areas of active research in modern (discrete) probability theory.
Arjun Krishnan
12:30pm - 1.45pm Mondays and Wednesdays in Hylan 1106A.
In addition to Le Gall and Peres and Levine mentioned above, I will also refer to Durrett’s Probability v5
| Midterm | 50% |
| Final | 50% |
I will assign homework, but you do not have to turn them in.
We will have one midterm and one final, one in-class and other take home. The idea is that you should study and internalize the material at the halfway point and at the end of the class.
Midterm: TBA
Final: TBA
If you are having difficulty seek help immediately - do not wait until it is too late to recover from falling behind. So please attend office hours or schedule an appointment to meet with me.
All assignments and activities associated with this course must be performed in accordance with the University of Rochester’s Academic Honesty Policy. More information is available at: www.rochester.edu/college/honesty. This resource is great, and has a great FAQ section. One of the policies is that if you’re found cheating, the instructor is required to report you. I intend to follow this policy.
The University of Rochester respects and welcomes students of all backgrounds and abilities. In the event you encounter any barrier(s) to full participation in this course due to the impact of disability, please contact the Office of Disability Resources. The access coordinators in the Office of Disability Resources can meet with you to discuss the barriers you are experiencing and explain the eligibility process for establishing academic accommodations. You can reach the Office of Disability Resources at: (585)275-9049; 1-154 Dewey Hall. Or, visit Center for Excellence in Teaching and Learning.
To be granted alternate testing accommodations, you must fill out forms with CETL at least seven days before each and every exam. These forms are not sent automatically.