Differential-escort transformations and the monotonicity of the LMC-R\'enyi complexity measure

Escort distributions have been shown to be very useful in a great variety of fields ranging from information theory, nonextensive statistical mechanics till coding theory, chaos and multifractals. In this work we give the notion and the properties of a novel type of escort density, the differential-escort densities, which have various advantages with respect to the standard ones. We highlight the behavior of the differential Shannon, R\'enyi and Tsallis entropies of these distributions. Then, we illustrate their utility to prove the monotonicity property of the LMC-R\'enyi complexity measure and to study the behavior of general distributions in the two extreme cases of minimal and very high LMC-R\'enyi complexity. Finally, this transformation allows us to obtain the Tsallis q-exponential densities as the differential-escort transformation of the exponential density.