Some properties of generalized Fisher information in the context of nonextensive thermostatistics

•We present two generalized Fisher information in nonextensive statistics.•Generalized qq-Gaussians are characterized via generalized Fisher information.•The interplay between generalized Fisher information, qq-Gaussians and qq-entropies is described.•Properties of generalized Fisher information and of their minimization are studied.

We present two extended forms of Fisher information that fit well in the context of nonextensive thermostatistics. We show that there exists an interplay between these generalized Fisher information, the generalized qq-Gaussian distributions and the qq-entropies. The minimum of the generalized Fisher information among distributions with a fixed moment, or with a fixed qq-entropy is attained, in both cases, by a generalized qq-Gaussian distribution. This complements the fact that the qq-Gaussians maximize the qq-entropies subject to a moment constraint, and yields new variational characterizations of the generalizedqq-Gaussians. We show that the generalized Fisher information naturally pop up in the expression of the time derivative of the qq-entropies, for distributions satisfying a certain nonlinear heat equation. This result includes as a particular case the classical de Bruijn identity. Then we study further properties of the generalized Fisher information and of their minimization. We show that, though non additive, the generalized Fisher information of a combined system is upper bounded. In the case of mixing, we show that the generalized Fisher information is convex for q≥1q≥1. Finally, we show that the minimization of the generalized Fisher information subject to moment constraints satisfies a Legendre structure analog to the Legendre structure of thermodynamics.