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

## TL;DR

This paper introduces differential-escort transformations, a new type of escort distribution, and demonstrates their usefulness in analyzing the monotonicity of the LMC-Rényi complexity measure and deriving Tsallis q-exponentials.

## Contribution

The work defines differential-escort densities, explores their entropy properties, and applies them to prove complexity measure monotonicity and generate Tsallis q-exponentials from exponential densities.

## Key findings

- Differential-escort densities have advantageous properties over standard escort distributions.
- They enable proof of the monotonicity of the LMC-Rényi complexity measure.
- They can generate Tsallis q-exponentials from exponential densities.

## Abstract

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.