ON THE GENERALIZED VARIANCE

Let mu be a positive measure on an Euclidean real finite dimensional l inear space E and let F (mu) = {P(theta, mu); theta epsilon Theta} be the natural exponential family generated by mu. The aim of this Note i s to show that if mu is infinitely divisible then the generalized vari ance of mu, i.e., the determinant of the covariance operator of P (the ta, mu), is the Laplace transform of a positive measure rho (mu) on E. We then investigate the effect of the transformation mu --> rho (mu).