Absolute moments of the variance-gamma distribution

TL;DR

This paper derives exact formulas for the absolute moments of the variance-gamma distribution, including special cases like symmetric distributions and half-integer shape parameters, with applications to related distributions.

Contribution

It provides new explicit formulas for absolute moments of the variance-gamma distribution, including closed-form expressions for specific parameter cases.

Findings

Exact formulas involving special functions for absolute moments.

Simplified formulas for symmetric variance-gamma distribution.

Applications to Laplace distribution and normal variable products.

Abstract

We obtain exact formulas for the absolute raw and central moments of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. When the skewness parameter is equal to zero (the symmetric variance-gamma distribution), the infinite series reduces to a single term. Moreover, for the case that the shape parameter is a half-integer (in our parameterisation of the variance-gamma distribution), we obtain a closed-form expression for the absolute moments in terms of confluent hypergeometric functions. As a consequence, we deduce new exact formulas for the absolute raw and central moments of the asymmetric Laplace distribution and the product of two correlated zero mean normal random variables, and more generally the sum of independent copies of such random variables.