Duality for real and multivariate exponential families

Consider a measure mu on R-n generating a natural exponential family F(mu) with variance function V-F(mu)(m) and Laplace transform exp(l(mu)(s)) = integral(Rn) exp(-< x, s >mu(dx)). A dual measure mu(*) satisfies -l(mu*)'(-l(mu)'(s)) = s. Such a dual measure does not always exist. One important property is l(mu*)"(m) = (V-F(mu)(m))(-1), leading to the notion of duality among exponential families (or rather among the extended notion of T exponential families TF obtained by considering all translations of a given exponential family F). (C) 2021 Elsevier Inc. All rights reserved.