Convergence of the Poincare constant

The Poincare 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.