The X + Y, X/Y Characterization of the Gamma Distribution

We prove, by elementary methods, that if X and Y are in-dependent random variables, not constant, such that X + Y is independent of X/Y then either X,Y or —X, —Y have gamma distributions with common scale parameter. This extends the result of Lukacs, who proved it for positive random variables, using differential equations for the characteristic functions. The aim here is to use more elementary methods for the X,Y positive case as well as elementary methods for proving that the restriction to positive X,Y may be removed.