On background driving distribution functions (BDDF) for some selfdecomposable variables

TL;DR

This paper derives formulas for background driving distribution functions of various selfdecomposable variables, facilitating their simulation and potential statistical applications.

Contribution

It provides explicit formulas for BDDFs of several classical selfdecomposable distributions, enhancing their practical simulation methods.

Findings

Formulas for BDDFs of gamma, Student t, and inverse Gaussian distributions.

Representation of stochastic area under planar Brownian motion.

Potential applications in statistical modeling and simulation.

Abstract

Many classical variables (statistics) are selfdecomposable. They admit the random integral representations via L\'evy processes. In this note are given formulas for their background driving distribution functions (BDDF). This may be used for a simulation of those variables. Among the examples discussed are: gamma variables, hyperbolic characteristic functions, Student t-distributions, stochastic area under planar Brownian motions, inverse Gaussian variable, logistic distributions, non-central chi-square, Bessel densities and Fisher z-distributions. Found representations might be of use in statistical applications.