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Does anybody have suggestions on what to read to learn more about couplings pertaining to statistics?

I'm working on a research project on Poisson approximations and am looking to perform a coupling on the unknown distribution. However, I cannot find much material on how to perform a coupling and the general calculations for it. I haven't had formal training on measure theory or intense probability theory, just upper level statistics courses. Any suggestions?

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I took the liberty of adding some tags –  Yemon Choi Jul 1 '11 at 2:41
    
Will, the coupling method is - in my limited understanding - a fairly well known technique or bunch of techniques in probability theory, so I'm hopeful someone who reads MO may have a good suggestion or two. –  Yemon Choi Jul 1 '11 at 4:51
    
I sent email to a friend, a retired probabilist. He may have some ideas. –  Will Jagy Jul 1 '11 at 5:31

2 Answers 2

up vote 5 down vote accepted

My friend Marty suggests the Lindvall book as well as

H. Thorisson, Coupling, Stationarity, and Regeneration. Springer, New York, 2000.

http://www.springer.com/mathematics/probability/book/978-0-387-98779-8

and points out that coupling is used now in basic textbooks in stochastic processes to prove the ergodic theorem for Markov Chains. So he recommends

Geoffrey Grimmett and David Stirzaker, Probability and Random Processes, 3rd edition, Oxford University Press, 2001.

http://ukcatalogue.oup.com/product/9780198572220.do

A monograph that presents the 1975 Stein-Chen method:

A. D. Barbour, Lars Holst, and Svante Janson, Poisson Approximation

http://ukcatalogue.oup.com/product/academic/series/mathematics/osip/9780198522355.do?sortby=bookTitleAscend

A rare accessible discussion is in chapter 2 of:

Ross, Sheldon and Peköz, Erol (2007). A second course in probability. www.ProbabilityBookstore.com. ISBN 978-0979570407.

Link to page for Peköz, which gives further link for book purchase to Amazon:

http://smgpublish.bu.edu/pekoz/

A number of useful links at

http://www.math.lsa.umich.edu/~fomin/525w07.html

and see

http://en.wikipedia.org/wiki/Stein%27s_method

http://en.wikipedia.org/wiki/Coupling_%28probability%29

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Have you looked at Lindvall's "Lectures on the Coupling Method"?

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I was thinking of mentioning this since the book is on my shelf, but as I haven't read beyond the first few pages I was leery of recommending it. –  Yemon Choi Jul 1 '11 at 4:50
    
I actually did purchase that exact book. It's been pretty difficult for me to read without having formal training in the symbols and terms. I end up spending a lot of time online looking up what confuses me! –  KatherineAnne Jul 6 '11 at 4:07

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