Normal scale mixtures and dual probability densities

Abstract

Let p be a probability density function representable as a scale mixture of normal probability densities. Whenever p is bounded, there is a dual density [pcirc] proportional to the characteristic function of p. We present numerous examples for pairs of dual probability densities representable as normal scale mixtures, among them generalized hyperbolic, Student's t and McKay'sBessel function, symmetric stable and exponential power, generalized logistic and generalized hyperbolic secant, α-Laplace or Linnik and generalized Cauchy densities.