On the determinant of the second derivative of a Laplace transform

Abstract

If $\mu$ is a positive measure on $\mathbb{R}^n$ with Laplace transform $L_{\mu}$ , we show that there exists a positive measure $\mu$ on $\mathbb{R}^n$ such that det $L_{\mu}^'' = L_{\nu}$. We deduce various corollaries from this result and, in particular, we obtain the Rao-Blackwell estimator of the determinant of the variance of a natural exponential family on $\mathbb{R}^n$ based on $(n + 1)$ observations. A new proof and extensions of Lindsay's results on the determinants of moment matrices are also given. Finally we give a characterization of the Gaussian law in $\mathbb{R}^n$.