Covariance identities for exponential and related distributions

Bobkov and Houdre (1997) proved that if xi, eta and zeta are independent standard exponential random variables, then for any two absolutely continuous functions f and g such that E\f(xi)\(2) < infinity and E\g(xi)\(2) < infinity, the equality Cov(f(xi),g(xi)) = Ef'(xi + eta)g'(xi + zeta) holds. We prove that the identity holds if and only if xi, eta and zeta or -xi, -eta and -zeta are standard exponential random variables. (C) 1999 Elsevier Science B.V. All rights reserved.