THE DETERMINANT OF THE 2ND DERIVATIVE OF A LAPLACE TRANSFORM IS A LAPLACE TRANSFORM

If mu is a positive measure on R(n) with Laplace transform L(mu), we s how that there exists a positive measure nu on R(n) such that Det L(mu )'' = L(nu). We deduce various corollaries from this result, in partic ular the Rao-Blackwell estimator of the determinant of the variance of a natural exponential family on R(n) for n + 1 observations, as well as a new proof and extensions of Lindsay's [6] results.