On escort distributions, q-gaussians and Fisher information

Escort distributions are a simple one parameter deformation of an original distribution p. In Tsallis extended thermostatistics, the escort-averages, defined with respect to an escort distribution, have revealed useful in order to obtain analytical results and variational equations, with in particular the equilibrium distributions obtained as maxima of Rényi-Tsallis entropy subject to constraints in the form of a q-average. A central example is the q-gaussian, which is a generalization of the standard gaussian distribution. In this contribution, we show that escort distributions emerge naturally as a maximum entropy trade-off between the distribution p(x) and the uniform distribution. This setting may typically describe a phase transition between two states. But escort distributions also appear in the fields of multifractal analysis, quantization and coding with interesting consequences. For the problem of coding, we recall a source coding theorem by Campbell relating a generalized measure of length to the Rényi-Tsallis entropy and exhibit the links with escort distributions together with pratical implications. That q-gaussians arise from the maximization of Rényi-Tsallis entropy subject to a q-variance constraint is a known fact. We show here that the (squared) q-gaussian also appear as a minimum of Fisher information subject to the same q-variance constraint.