Mixture representations of noncentral distributions

With any symmetric distribution μ on the real line we may associate a parametric family of noncentral distributions as the distributions of (X+δ)2,δ=0, where X is a random variable with distribution μ. The classical case arises if μ is the standard normal distribution, leading to the noncentral chi-squared distributions. It is well known that these may be written as Poisson mixtures of the central chi-squared distributions with odd degrees of freedom. We obtain such mixture representations for the logistic distribution and for the hyperbolic secant distribution. We also derive alternative representations for chi-squared distributions and relate these to representations of the Poisson family. While such questions originated in parametric statistics they also appear in the context of the generalized second Ray–Knight theorem, which connects Gaussian processes and local times of Markov processes.