Weighted CKP Inequalities Involving Rényi Divergence Powers

TL;DR

This paper develops weighted Pinsker-type (CKP) inequalities using Rényi divergence powers $D_\alpha$ and their associated Tsallis distances $T_\alpha$, deriving explicit bounds for the weighted total variation $\|w(\nu-\mu)\|_{TV}$ in terms of $\|w\|_\beta$ and $T_\alpha$. It introduces a linearization framework and characterizes the unique extremizer for the associated variational problem, giving an explicit density form and a defining constant $c$. The authors establish Hölder-type inequalities for densities, derive sharp constants across regimes $\alpha\ge2$ and $1<\alpha\le2$, and connect these results to Pearson–Vajda distances, yielding transport-entropy bounds under moment conditions. The work provides a unified, quantitative bridge between weighted total variation, Rényi divergences, and transport-entropy inequalities with concrete constants and extremal structures, broadening the toolkit for probabilistic and information-theoretic analyses under weaker moment assumptions.

Abstract

Pinsker-type inequalities are considered for the weighted total variation distance between probability measures in terms of the Rényi divergence powers. They are applied in derivation of transport-entropy inequalities under moment-type conditions.