Tempered distributions and their application in computing conditional moments for normal mixtures

We offer a unified theory, which provides necessary and sufficient conditions for almost everywhere convergence of radial smooth functions using Fourier integrals on Euclidean spaces. We apply this methodology to obtain expressions for the conditional variance of scale mixtures of multivariate normal. We present some specific examples, i.e., the multivariate t-distribution and when the scale is gamma distributed. (C) 2003 Elsevier B.V. All rights reserved.