Absolute Moments of GH Distribution: An Alternative Approach

In this work, we focus on deriving the absolute moments of the generalised hyperbolic (GH) distribution through its mean–variance mixture structure. Conditioning on the GIG mixing variable reduces the problem to Gaussian moment calculations and known GIG power moments, avoiding direct integration with Bessel functions. The variance-gamma (VG), normal inverse Gaussian (NIG), hyperbolic, Normal, and Student’s t distributions emerge as natural special or limiting cases of the generalised hyperbolic family. This approach offers a substantially simpler and more transparent derivation, facilitating both implementation and verification.