Several characterizations of the Generalized Inverse Gaussian (GIG) distribution on the positive real line have been proposed in the literature, especially over the past two decades. These characterization theorems are surveyed, and two new characterizations are established, one based on maximum likelihood estimation and the other is a Stein characterization.
"Characterizations of GIG laws: A survey." Probab. Surveys 11 161 - 176, 2014. https://doi.org/10.1214/13-PS227