Generalized Divergence Measures and Weak Convergence for the Sets of Probability Measures

TL;DR

This paper generalizes divergence measures like Kullback-Leibler and Jensen-Shannon from pairs of probability measures to sets, establishing key properties and convergence results under sublinear expectations.

Contribution

It introduces generalized divergence measures for sets of probability measures and proves fundamental properties and convergence results, extending classical divergences.

Findings

Established duality formula for generalized divergences

Proved Pinsker-type inequality for the new measures

Derived convergence results under sublinear expectations

Abstract

This paper extends the asymmetric Kullback-Leibler divergence and symmetric Jensen-Shannon divergence from two probability measures to the case of two sets of probability measures. We establish some fundamental properties of these generalized divergences, including a duality formula and a Pinsker-type inequality. Furthermore, convergence results are derived for both the generalized asymmetric and symmetric divergences, as well as for weak convergence under sublinear expectations.