Rates of Convergence for Gibbs Sampler and Other Markov Chains

A Ph.D. thesis by Jeffrey S. Rosenthal.

Submitted to the Department of Mathematics, Harvard University, April 1992.

Ph.D. supervisor: Persi Diaconis.


It is a pleasure to thank the many people who made this thesis possible.

It is difficult to overstate my gratitude to my Ph.D. supervisor, Dr. Persi Diaconis. With his enthusiasm, his inspiration, and his great efforts to explain things clearly and simply, he helped to make mathematics fun for me. Throughout my thesis-writing period, he provided encouragement, sound advice, good teaching, good company, and lots of good ideas. I would have been lost without him.

I would like to thank the many people who have taught me mathematics: my high school math teachers (especially Larry Rice), my undergraduate teachers at Toronto (especially Edward Bierstone, Peter Botta, Man-Duen Choi, Andres del Junco, and Nathan Isgur), and my graduate teachers (especially Raoul Bott and Jim Fill). For their kind assistance with writing letters, giving wise advice, helping with various applications, and so on, I wish to thank in addition Robin Gottlieb, Joe Harris, John Imbrie, Jun Liu, Esther Silberstein, and Dan Stroock.

I am indebted to my many student colleagues for providing a stimulating and fun environment in which to learn and grow. I am especially grateful to Gary Baumgartner, Paul Geiger, Leonya Livshits, Gordon MacDonald, and Phil Morenz at Toronto, and to Eric Belsley, Karen Chandler, Shun-Jen Chen, Carl Dou, Martin Hildebrand, Peter Magyar, James McKernan, and Dan Rockmore at Harvard. Peter Magyar was particularly helpful mathematically, patiently teaching me the theory of Lie groups.

I wish to thank my best friend in high school (Moshin Lee), my best friend as an undergraduate (Aaron Boxer), and my best friend as a graduate student (Marc Goldman), for helping me get through the difficult times, and for all the emotional support, comraderie, entertainment, and caring they provided.

I am grateful to the secretaries and librarians in the math departments of Toronto and Harvard, for helping the departments to run smoothly and for assisting me in many different ways. Marie Bachtis and Ida Bulat at Toronto, and Donna d'Fini, Andrea McDonnell, Nancy Miller, and Sarah Warren at Harvard, deserve special mention.

I wish to thank my entire extended family for providing a loving environment for me. My brothers, my half-siblings, my sister-in-common-law, my step-mother, some uncles, and some first-cousins-once-removed were particularly supportive.

Lastly, and most importantly, I wish to thank my parents, Helen S. Rosenthal and Peter Rosenthal. They bore me, raised me, supported me, taught me, and loved me. To them I dedicate this thesis.

See front matter, introduction, chapter 2, chapter 3, chapter 4, chapter 5, conclusion, and my math genealogy.

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