On characterizations of the gamma and generalized inverse Gaussian distributions

Given two independent non-degenerate positive random variables X and Y, Letac and Wesolowski (Ann. Probab. 28 (2000) 1371) proved that U=(X+Y)-1 and V=X-1-(X+Y)-1 are independent if and only if X and Y are generalized inverse Gaussian (GIG) and gamma distributed, respectively. Note that X=(U+V)-1 and Y=U-1-(U+V)-1. This interesting transformation between (X,Y) and (U,V) preserves a bivariate probability measure which is a product of GIG and gamma distributions. In this work, characterizations of the GIG and gamma distributions through the constancy of regressions of Vr on U are considered.