Covariance identities for exponential and related distributions

Abstract

[Bobkov and Houdre (1997] proved that if ξ, η and ζ are independent standard exponential random variables, then for any two absolutely continuous functions f and g such that E|f(ξ)|²<∞ and E|g(ξ)|²<∞, the equality Cov(f(ξ),g(ξ))=Ef(ξ+η)g'(ξ+ζ) holds. We prove that the identity holds if and only if ξ, η and ζ or −ξ,−η and −ζ are standard exponential random variables.