Hyper-V uniform ergodicity of Markov chains

2026-08-12

AI Summary

This work addresses the challenges in verifying the global minorization condition and obtaining initial-state-independent geometric convergence guarantees in Markov chain convergence analysis. To overcome these difficulties, the authors propose a novel framework that integrates a uniform drift condition with a local minorization condition, thereby establishing a stronger notion of hyper-V uniform ergodicity. This ensures that the deviations of all functions dominated by a Lyapunov function V converge to the invariant measure at a geometric rate. The approach circumvents the need to verify traditional global minorization conditions and instead leverages the structure of the invariant measure to directly infer qualitative hyper-V uniform ergodicity for two-variable Gibbs samplers. Moreover, it yields minimax-optimal convergence bounds. Empirical studies on Pólya–Gamma and Kolmogorov–Gamma Gibbs samplers demonstrate the effectiveness and superiority of the proposed framework.

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.

Problem

• uniform ergodicity
• Markov chains
• hyper-V uniform ergodicity
• drift condition
• minorization

Innovation

• hyper-V uniform ergodicity
• uniform drift condition
• local minorization
• geometric convergence
• Gibbs sampler