A Stein characterisation of the generalized hyperbolic distribution

TL;DR

This paper develops a Stein characterization for the generalized hyperbolic distribution, unifying several known distributions and enabling new analytical tools for probabilistic approximation.

Contribution

It introduces a Stein equation for the GH distribution that generalizes existing equations for related distributions, advancing Stein's method applications.

Findings

Derived a Stein equation for the GH distribution

Unified Stein characterizations for related distributions

Facilitates probabilistic approximation techniques

Abstract

The generalized hyperbolic (GH) distributions form a five parameter family of probability distributions that includes many standard distributions as special or limiting cases, such as the generalized inverse Gaussian distribution, Student's $t$-distribution and the variance-gamma distribution, and thus the normal, gamma and Laplace distributions. In this paper, we consider the GH distribution in the context of Stein's method. In particular, we obtain a Stein characterisation of the GH distribution that leads to a Stein equation for the GH distribution. This Stein equation reduces to the Stein equations from the current literature for the aforementioned distributions that arise as limiting cases of the GH superclass.