Deduction of global minorization from invariant-measure structure

Determine whether a global minorization condition for a two-variable Gibbs sampler can be deduced solely from the form of its invariant distribution.

Background

The paper develops a framework for proving hyper-$V$ uniform ergodicity of Markov chains using a uniform drift condition and a local minorization condition. For two-variable Gibbs samplers constructed through data augmentation, it further shows that qualitative hyper-$V$ uniform ergodicity can sometimes be inferred directly from the structure of the invariant distribution, without analyzing the associated Markov transition kernel.

The authors explicitly distinguish this result from the stronger classical global-minorization property. They state that it is not generally clear whether the invariant distribution alone contains enough information to imply such a global minorization condition, leaving the deduction of global minorization from invariant-measure structure unresolved.