Sharp informational inequalities involving Kullback-Leibler and Rényi divergences and a family of scaling-invariant relative Fisher measures
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| Format: | Preprint |
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2025
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| _version_ | 1866908462244429824 |
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| author | Iagar, Razvan Gabriel Puertas-Centeno, David Toranzo, Elio V. |
| author_facet | Iagar, Razvan Gabriel Puertas-Centeno, David Toranzo, Elio V. |
| contents | We introduce a new transformation called \emph{relative differential-escort}, which extends the usual differential-escort transformation by relating the change of variable to a reference probability density. As an application of it, we define a biparametric family of \emph{relative Fisher measures} presenting significant advantages with respect to the pre-existing ones in the literature: invariance under scaling changes and, consequently, sharp inequalities between the new relative Fisher measure and the well established Kullback-Leibler and Rényi divergences. We also introduce a biparametric family of \emph{relative cumulative moment-like measures} and we establish sharp lower bounds of these new measures by the Kullback-Leibler and Rényi divergences. The optimal bound and the minimizing densities are given. We also construct a family of inequalities for an arbitrary and fixed minimizing density in which the so-called generalized trigonometric functions plays a central role, providing thus one more interesting application of the newly introduced inequalities and measures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_17408 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp informational inequalities involving Kullback-Leibler and Rényi divergences and a family of scaling-invariant relative Fisher measures Iagar, Razvan Gabriel Puertas-Centeno, David Toranzo, Elio V. Mathematical Physics We introduce a new transformation called \emph{relative differential-escort}, which extends the usual differential-escort transformation by relating the change of variable to a reference probability density. As an application of it, we define a biparametric family of \emph{relative Fisher measures} presenting significant advantages with respect to the pre-existing ones in the literature: invariance under scaling changes and, consequently, sharp inequalities between the new relative Fisher measure and the well established Kullback-Leibler and Rényi divergences. We also introduce a biparametric family of \emph{relative cumulative moment-like measures} and we establish sharp lower bounds of these new measures by the Kullback-Leibler and Rényi divergences. The optimal bound and the minimizing densities are given. We also construct a family of inequalities for an arbitrary and fixed minimizing density in which the so-called generalized trigonometric functions plays a central role, providing thus one more interesting application of the newly introduced inequalities and measures. |
| title | Sharp informational inequalities involving Kullback-Leibler and Rényi divergences and a family of scaling-invariant relative Fisher measures |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2507.17408 |