Some Characteristic Properties of the Fisher Information Matrix via Cacoullos-Type Inequalities

Some lower bounds for the variance of a function g of a random vector X are extended to a wider class of distributions. Using these bounds, some useful inequalities for the Fisher information are obtained for convolutions and linear combinations of random variables. Finally, using these inequalities, simple proofs are given of classical characterizations of the normal distribution, under certain restrictions, including the matrix analogue of the Darmois-Skitovich result.