#### Statistics 606 (Spring 2006)

 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)

1. Markov chains on countable state spaces. The classical theory leading to existence of a stationary distribution and conditions for convergence to that distribution.
2. 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
3. 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.
4. 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/
5. Percolation, if time permits.