Geodesic
Convergence of the Poincar\'{e} constant
Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 3, pp. 615-620

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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. 
@article{TVP_2003_48_3_a12,
     author = {O. Johnson},
     title = {Convergence of the {Poincar\'{e}} constant},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {615--620},
     publisher = {mathdoc},
     volume = {48},
     number = {3},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2003_48_3_a12/}
}
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O. Johnson. Convergence of the Poincar\'{e} constant. Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 3, pp. 615-620. http://geodesic.mathdoc.fr/item/TVP_2003_48_3_a12/