On multidimensional generalized Cramér-Rao inequalities, uncertainty relations and characterizations of generalized q-Gaussian distributions

摘要

In the present work, we show how the generalized Cramer-Rao inequality for the estimation of a parameter, presented in a recent paper, can be extended to the multidimensional case with general norms on Rn, and to a wider context. As a particular case, we obtain a new multidimensional Cramer-Rao inequality which is saturated by generalized q-Gaussian distributions. We also give another related Cramer-Rao inequality, for a general norm, which is saturated as well by these distributions. Finally, we derive uncertainty relations from these Cramer-Rao inequalities. These uncertainty relations involve moments computed with respect to escort distributions, and we show that some of these relations are saturated by generalized q-Gaussian distributions. These results introduce extended versions of Fisher information, new Cramer-Rao inequalities, and new characterizations of generalized q-Gaussian distributions which are important in several areas of physics and mathematics.