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

Abstract

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