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A new pair of transformations and applications to generalized informational inequalities and Hausdorff moment problem

Razvan Gabriel Iagar, David Puertas-Centeno

TL;DR

This paper introduces two mutually inverse transformations, the up and down transforms, that map between moments and Rényi/Shannon entropies, as well as between Rényi entropies and generalized Fisher information, thereby creating a mirrored domain for informational inequalities. Using these transforms, the authors extend Stam and moment-entropy inequalities to the mirrored index domain and characterize their minimizers, which are unbounded stretched Gaussian-type densities; they also connect the Hausdorff entropic moment problem to the standard Hausdorff problem and formulate a Fisher-like moment problem. The results yield explicit transformation rules for key quantities, scaling relations, and tail behavior, together with closed-form expressions for transformed minimizers in many parameter regimes. Additionally, the work develops a sharp inequality relating Rényi and Shannon entropies in the mirrored setting and demonstrates how the transformed densities serve as a bridge between entropic reconstruction and moment-based approaches, suggesting robust tools for handling unbounded densities and extended informational bounds with potential applications in MaxEnt frameworks.

Abstract

We introduce a pair of transformations, which are mutually inverse, acting on rather general classes of probability densities in R. These transformations have the property of interchanging the main informational measures such as p-moments, Shannon and Rényi entropies, and Fisher information. We thus apply them in order to establish extensions and generalizations of the Stam and moment-entropy inequalities in a mirrored domain of the entropic indexes. Moreover, with the aid of the two transformations we establish formal solutions to the Hausdorff entropic moment problem by connecting them with the solutions of the standard Hausdorff problem. In addition, we introduce a Fisher-like moment problem and relate it to the standard Hausdorff moment problem.

A new pair of transformations and applications to generalized informational inequalities and Hausdorff moment problem

TL;DR

This paper introduces two mutually inverse transformations, the up and down transforms, that map between moments and Rényi/Shannon entropies, as well as between Rényi entropies and generalized Fisher information, thereby creating a mirrored domain for informational inequalities. Using these transforms, the authors extend Stam and moment-entropy inequalities to the mirrored index domain and characterize their minimizers, which are unbounded stretched Gaussian-type densities; they also connect the Hausdorff entropic moment problem to the standard Hausdorff problem and formulate a Fisher-like moment problem. The results yield explicit transformation rules for key quantities, scaling relations, and tail behavior, together with closed-form expressions for transformed minimizers in many parameter regimes. Additionally, the work develops a sharp inequality relating Rényi and Shannon entropies in the mirrored setting and demonstrates how the transformed densities serve as a bridge between entropic reconstruction and moment-based approaches, suggesting robust tools for handling unbounded densities and extended informational bounds with potential applications in MaxEnt frameworks.

Abstract

We introduce a pair of transformations, which are mutually inverse, acting on rather general classes of probability densities in R. These transformations have the property of interchanging the main informational measures such as p-moments, Shannon and Rényi entropies, and Fisher information. We thus apply them in order to establish extensions and generalizations of the Stam and moment-entropy inequalities in a mirrored domain of the entropic indexes. Moreover, with the aid of the two transformations we establish formal solutions to the Hausdorff entropic moment problem by connecting them with the solutions of the standard Hausdorff problem. In addition, we introduce a Fisher-like moment problem and relate it to the standard Hausdorff moment problem.

Paper Structure

This paper contains 12 sections, 20 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Up and down transformations: basic properties
  3. The down transformation
  4. The up transformation
  5. Up/down transformations and informational measures
  6. Moments, Shannon and Rényi entropy and Fisher information
  7. Dealing with the Hausdorff entropic moment problem
  8. Stretched Gaussian densities and up and down transforms
  9. Mirrored moment-entropy and Stam like inequalities
  10. Extended Stam inequalities
  11. Mirrored domain for the moment-entropy inequality
  12. Conclusions

Key Result

Proposition 2.1

Let $\alpha\in\mathbb{R}\setminus\{2\}$ and $\kappa\in\mathbb{R}^+$. Then, for any probability density $f$ satisfying the requirements of Definition def:down, we have When $\alpha=2$ one finds

Theorems & Definitions (59)

  • Definition 2.1
  • Remark 2.1: Probability density and supports
  • Remark 2.2: Canonical election
  • Remark 2.3: Supports
  • Remark 2.4: Divergence on the border
  • Remark 2.5: Derivatives and composition
  • Remark 2.6
  • Proposition 2.1: Scaling changes and down transformation
  • proof
  • Proposition 2.2
  • ...and 49 more