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On Stein’s method for multivariate normal approximation

Elizabeth Meckes

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The purpose of this paper is to synthesize the approaches taken by Chatterjee-Meckes and Reinert-Röllin in adapting Stein’s method of exchangeable pairs for multivariate normal approximation. The more general linear regression condition of Reinert-Röllin allows for wider applicability of the method, while the method of bounding the solution of the Stein equation due to Chatterjee-Meckes allows for improved convergence rates. Two abstract normal approximation theorems are proved, one for use when the underlying symmetries of the random variables are discrete, and one for use in contexts in which continuous symmetry groups are present. A first application is presented to projections of exchangeable random vectors in ℝn onto one-dimensional subspaces. The application to runs on the line from Reinert-Röllin is reworked to demonstrate the improvement in convergence rates, and a new application to joint value distributions of eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold is presented.

Chapter information

Christian Houdré, Vladimir Koltchinskii, David M. Mason and Magda Peligrad, eds., High Dimensional Probability V: The Luminy Volume (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2009), 153-178

First available in Project Euclid: 2 February 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

multivariate analysis normal approximation Stein’s method eigenfunctions Laplacian

Copyright © 2009, Institute of Mathematical Statistics


Meckes, Elizabeth. On Stein’s method for multivariate normal approximation. High Dimensional Probability V: The Luminy Volume, 153--178, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2009. doi:10.1214/09-IMSCOLL511.

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