Mixture representations of noncentral distributions

TL;DR

This paper explores mixture representations of noncentral distributions derived from symmetric distributions, extending classical results like the noncentral chi-squared case to logistic and hyperbolic secant distributions, with implications in statistics and stochastic processes.

Contribution

It introduces new mixture representations for noncentral distributions beyond the normal case, including logistic and hyperbolic secant, and relates these to Poisson mixtures and the Ray-Knight theorem.

Findings

Mixture representations for logistic and hyperbolic secant distributions.

Alternative representations for chi-squared distributions.

Connections to the generalized second Ray-Knight theorem.

Abstract

With any symmetric distribution $\mu$ on the real line we may associate a parametric family of noncentral distributions as the distributions of $(X+\delta)^2$, $\delta\not=0$, where $X$ is a random variable with distribution $\mu$. The classical case arises if $\mu$ 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.