Probability Reading Group

Fall 2016


News

10 June 2016: We plan to have a reading seminar on Markov chains, based lightly on the book Denumerable Markov chains by Wolfgang Woess and Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities by Steven Orey (the former is available in the library; Vadim has an electronic copy of the latter). Please email me if you are interested in giving a presentation in the reading group, either on this book or a related topic. Below, I include a list of topics that have been suggested.

5 August 2016: A course summary here. Please see the course description for grading criteria.

7 September 2016: First meeting today!


Proposed Meetings

Wednesday, September 7, at 11:30 AM in KED B015. Speaker: David McDonald. Subject: Construction of a Markov chain on a countable state space. The shift operator and the Markov property. Irreducibility, aperiodicity and recurrence. An example - The age process.

Wednesday, September 14, at 11:30 AM in KED B015. Speaker: Gail Ivanoff. Subject: Review of the convergence theory of forward and backward martingales.

Wednesday, September 21, at 11:30 AM in KED B015. Speaker: David McDonald. Subject: Invariant and tail sigma fields. Harmonic functions. Space-Time harmonic functions and the connection to the tail field. Convergence to steady state.

Wednesday, September 28, at 11:30 AM in KED B015. Speaker: David McDonald. Subject: An example - convergence of the age distribution.

Wednesday, October 5, at 11:30 AM in KED B015. Speaker: Aaron Smith. Subject: Rate of convergence to stationarity.

Wednesday, October 12, at 11:30 AM in KED B015. Speaker: David McDonald. Subject: Potential theory of absorbing chains. Yaglom limits.

Wednesday, October 19, at 11:30 AM in KED B015. Speaker: Rafal Kulik. Subject: Geometric Ergodicity.

Wednesday, October 26: Reading week - no meeting.

Wednesday, November 2, at 11:30 AM in KED B015. Speaker: Youssouph Cissokho. Subject: Geometric Ergodicity.

Wednesday, November 9, at 11:30 AM in KED B015. Speaker: Nicholas Denis. Subject: Honest Bounds on Markov chain Monte Carlo.

Wednesday, November 16, at 11:30 AM in KED B015. Speaker: Blair Drumond. Subject: TBA.

Wednesday, November 23, at 11:30 AM in KED B015. Speaker: Gustavo Valente. Subject: Random walks on groups, Poisson boundary and Mackey range.


Possible Additional Meetings (Subject to Interest)

The following is a short list of topics that might be appropriate for presentations. Please feel free to suggest other topics.

Harris recurrence (Part of chapter 9 of Meyn & Tweedie).

Drift and Regularity (Chapter 11 of Meyn & Tweedie).

Introduction to ergodicity and geometric ergodicity (Chapters 13 and 15 of Meyn & Tweedie). (2 meetings)

Introduction to geometric ergodicity from statistics literature (I suggest "Honest Exploration of Intractable Probability Distributions via Markov Chain Monte Carlo" by Jim Hobert and Galin Jones, though other work by Hobert, Khare, Rosenthal, etc is also reasonable). (1 meeting).

Recent work on renewal theory (I suggest "Honest Exploration of Intractable Probability Distributions via Markov Chain Monte Carlo" by Robert Baxendale). (1 meeting).

Techniques for estimating mixing times (could include: key definitions and an introduction to key ideas about spectral bounds, Cheeger constants, Dirichlet forms, etc from the free book by Levin, Peres & Wilmer; an application of the mixing time literature to statistics (one option from the statistical literature is the recent paper "On the computational complexity of high-dimensional Bayesian variable selection" by Yang, Wainwright and Jordan)). (Up to 2 meetings).

The section on Central Limit Theorems from Meyn and Tweedie. (1 meeting).

Read some recent advances related to geometric ergodicity. Some suggestions include "Spectral gaps for a Metropolis-Hastings algorithm in infinite dimensions" by Hairer, Stuart and Vollmer or "Subgeometric rates of convergence in Wasserstein distance for Markov chains" by Durmus, Forte and Moulines. In general, I find articles by Andrew Stuart and Matti Vihola to be very well-written, and would probably suggest other work by them. (up to 3 meetings).

Other important and essentially independent pieces of theory. This could include the skipped pieces of Meyn & Tweedie (Chapters 6 and 12). It could also include an introduction to scaling limits (e.g. "Optimal scaling for various Metropolis-Hastings algorithms" by Roberts and Rosenthal), introduction to comparison (e.g. "Ordering Monte Carlo Markov Chains" by Mira and Geyer and/or "Establishing some order amongst exact approximations of MCMCs" by Andrieu/Vihola). (1-5 meetings)

Advanced ideas in coupling theory. This could include the notions of coupling from the past or evolving sets (both have good expositions in the free book by Levin, Peres and Wilmer) or notions of duality (see "Strong Stationary Times Via a New Form of Duality" by Diaconis and Fill) or applications (see "Mixing times are hitting times of large sets" by Peres and Sousi). (Up to 2 meetings).


Links

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