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Step Size in Stein's Method of Exchangeable Pairs

Published online by Cambridge University Press:  10 August 2009

NATHAN ROSS
NATHAN ROSS*
Affiliation:
Department of Mathematics, University of Southern California, 3620 South Vermont Avenue, KAP 108, Los Angeles, California 90089-2532, USA (e-mail: nathanfr@usc.edu)
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Abstract

Stein's method of exchangeable pairs is examined through five examples in relation to Poisson and normal distribution approximation. In particular, in the case where the exchangeable pair is constructed from a reversible Markov chain, we analyse how modifying the step size of the chain in a natural way affects the error term in the approximation acquired through Stein's method. It has been noted for the normal approximation that smaller step sizes may yield better bounds, and we obtain the first rigorous results that verify this intuition. For the examples associated to the normal distribution, the bound on the error is expressed in terms of the spectrum of the underlying chain, a characteristic of the chain related to convergence rates. The Poisson approximation using exchangeable pairs is less studied than the normal, but in the examples presented here the same principles hold.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 6 , November 2009 , pp. 979 - 1017
Copyright
Copyright © Cambridge University Press 2009

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