The independence of $U = \min (X, Y)$ and $V = X - Y$ or $W = |X - Y|$ is studied where $X$ and $Y$ are not assumed to be independent. The bivariate exponential distribution of Marshall and Olkin is characterized as the distribution with exponential marginals where $U$ is exponential and independent of $V$.
@article{1176343905,
author = {Block, Henry W.},
title = {A Characterization of a Bivariate Exponential Distribution},
journal = {Ann. Statist.},
volume = {5},
number = {1},
year = {1977},
pages = { 808-812},
language = {en},
url = {http://dml.mathdoc.fr/item/1176343905}
}
Block, Henry W. A Characterization of a Bivariate Exponential Distribution. Ann. Statist., Tome 5 (1977) no. 1, pp. 808-812. http://gdmltest.u-ga.fr/item/1176343905/