Independence preserving property of Kummer laws

Efoevi Angelo Koudou, Jacek Wesołowski

We prove that if $X,Y$ are positive, independent, non-Dirac random variables and if for $\alpha ,\beta \ge 0$, $\alpha \ne \beta$, $$ ψ_{α,β}(x,y)=\left(y\,\tfrac{1+β(x+y)}{1+αx+βy},\;x\,\tfrac{1+α(x+…