On multidimensional generalized Cram\'er-Rao inequalities, uncertainty relations and characterizations of generalized $q$-Gaussian distributions

In the present work, we show how the generalized Cram\'er-Rao inequality for the estimation of a parameter, presented in a recent paper, can be extended to the mutidimensional case with general norms on $\mathbb{R}^{n}$, and to a wider context. As a particular case, we obtain a new multidimensional Cram\'er-Rao inequality which is saturated by generalized $q$-Gaussian distributions. We also give another related Cram\'er-Rao inequality, for a general norm, which is saturated as well by these distributions. Finally, we derive uncertainty relations from these Cram\'er-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 Cram\'er-Rao inequalities, and new characterizations of generalized $q$-Gaussian distributions which are important in several areas of physics and mathematics.