This article is cited in 5 papers

Short Communications

Convergence of the Poincaré constant

O. Johnson

Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge

Abstract: The Poincaré constant $R_Y$ of a random variable $Y$ relates the $L^2(Y)$-norm of a function $g$ and its derivative $g'$. Since $R_Y - D(Y)$ is positive, with equality if and only if $Y$ is normal, it can be seen as a distance from the normal distribution. In this paper we establish the best possible rate of convergence of this distance in the central limit theorem. Furthermore, we show that $R_Y$ is finite for discrete mixtures of normals, allowing us to add rates to the proof of the central limit theorem in the sense of relative entropy.

Keywords: Poincaré constant, spectral gap, central limit theorem, Fisher information.

Received: 05.01.2001
Revised: 24.06.2002

Language: English

DOI: 10.4213/tvp276

 English version: Theory of Probability and its Applications, 2004, 48:3, 535–541

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