A Kotz and Steutel type of characterization of the gamma family

Let X and Y be independent identically distributed (i.i.d.) nondegenerate and positive random variables with a common absolutely continuous distribution function F(x). We use the notation Z?=?max(X, Y) and W?=?min(X, Y). In the present paper, we prove that ${\frac{(Z - W)}{(Z + W)}}$ and (Z +?W) are independent if and only if X and Y have gamma distribution.