Convergence of the Poincaré constant

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.