Hyper-V uniform ergodicity of Markov chains

Abstract

We develop a new uniform drift condition and local minorization that implies a stronger weighted form of uniform ergodicity for Markov chains we call hyper-V uniform ergodicity. The convergence guarantees geometric decay of the bias towards the invariant measure independently of the initialization for all functions controlled by a dominating function V. A key advantage of the approach is that it bypasses the need to establish a global minorization condition, which is often substantially more difficult to verify in practice, while yielding stronger convergence guarantees than global minorization. Optimal convergence bounds in a minimax sense of the framework are established. The utility of the framework is demonstrated through applications to the P'olya-Gamma and Kolmogorov-Gamma Gibbs samplers. We also show qualitative hyper-V uniform ergodicity convergence for two-variable Gibbs samplers can be inferred by the form of the invariant measure, bypassing convergence analysis entirely.

Disclosure

“esulting conclusions are qualitative, such meth- ods have the potential to become practical tools for applied statisticians seeking to establish uniform ergodicity without undertaking a detailed convergence analysis. Statement of AI use GPT 5.6 Sol was used to assist in the development of proof techniques appearing in Section 3, and the resulting proofs were independently verified by the authors. GPT 5.6 Sol was also used to assist with the Python code used for the simulation stu”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T05:12:43.207599Z

Structural counts