Functional equations for multivariate exponential distributions

A large number of characterizations of univariate exponential distributions are known; these often lead to a functional equation, relatively few of which have been extended to the multivariate case. This paper is an exposition of multivariate extensions and relatives of the functional equation which stems from the following characterization: let Wk be the minimum of k independent copies of X. Then (i) kWk has the same distribution as X for k = 1,2, ... if and only if (ii) X has an exponential distribution.