Convolution-invariant subclasses of generalized hyperbolic distributions

Abstract

It is rigorously shown that the generalized Laplace distributions and the normal inverse Gaussian distributions are the only subclasses of the generalized hyperbolic distributions that are closed under convolution. The result is obtained by showing that the corresponding two classes of variance mixing distributions—gamma and inverse Gaussian—are the only convolution-invariant classes of the generalized inverse Gaussian distributions.

Keywords:

• Bessel function distribution
• Gamma variance normal mixture
• Generalized inverse Gaussian distribution
• Generalized asymmetric Laplace distribution
• Inverse gamma distribution
• Variance–mean normal mixture

Mathematics Subject Classification:

• 60E07
• 60E10

Funding

The authors acknowledge the support of the Swedish Research Council Grant 2008-5382.