| Instructor: | David Pollard |
| Email: | david.pollard@yale.edu |
| When: | Tuesday, Friday 10:30--11:45 |
| Where: | 24 Hillhouse Avenue |
| Office hours: | To be arranged with each group |
Markov chains on general state spaces; diffusions; Markov random fields; Gibbs measures; percolation. After STAT 600.
Intended audience
Students who have already taken a measure theoretic probability course
Text
No single book. References to books and the literature will be given for each subtopic, as I become more organized. The class materials for an introductory stochastic processes course (Stat 251, Fall 2004) might be helpful. I will also prepare some handouts.
Topics (tentative)
- Markov chains on countable state spaces. The classical theory leading to existence of a stationary distribution and conditions for convergence to that distribution.
- Markov chains on general state spaces. Exploration of how the theory for countable state spaces can be extended.
References:
- Steven Orey, "Limit theorems for Markov chain transition probabilities". (My copy was published by Van Nostrand in 1971.)
- Very concise. Contains many of the main ideas, but without the recent refinements.
- Esa Nummelin, "General irreducible Markov chains and non-negative operators", Cambridge University Press 1984. (Paperback 2004.)
- Less concise than Orey. Describes the splitting technique for creating an artificial atom.
- S.P. Meyn and R.L. Tweedie, "Markov chains and stochastic stability", Springer 1993.
- Clear but it takes a lot of reading to reach the main ideas. Many examples. I started with this book then moved back to Nummelin then Orey.
- Persi Diaconis & David Freedman, Technical reports from http://www.stat.berkeley.edu/tech-reports/index.html
- 501. (December 1, 1997) On Markov Chains with Continuous State Space and 497. (November 24, 1997) On the Hit & Run Process
- Markov random fields for finitely or countably many variables. A self-contained introduction to Gibbs measures.
References:
- Ross Kindermann and J. Laurie Snell, "Markov Random Fields and Their Applications", available for free from http://www.ams.org/online_bks/conm1/ .
- Hans-Otto Georgii, "Gibbs Measures and Phase Trasitions", de Gruyter 1988.
- Tough reading.
- Random trees. An introduction to some of the work by Aldous, Steele, and others.
References:
- Russell Lyons and Yuval Peres, "Probability on Trees and Networks", available from http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html.
- Interesting material on branching processes and random trees.
- J. Michael Steele and David Aldous, "The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence,", available from http://www-stat.wharton.upenn.edu/~steele/Publications/
- Percolation, if time permits.
Grading
I hope that students will work in groups to flesh out arguments sketched in class. I will meet with each group regularly to help. I will explain in the first lecture how this method of learning can work.