马尔可夫链的超V一致遍历性
Hyper-V uniform ergodicity of Markov chains
AI总结：
该研究提出超V一致遍历性概念，通过新的漂移与局部极小化条件推导更强收敛保证，确立最优收敛界，应用于两类吉布斯采样器并简化二元采样器收敛分析。
AI中文摘要：
我们提出了一种新的一致漂移条件和局部极小化条件，该条件可推导出马尔可夫链的一种更强的加权形式的一致遍历性，我们将其命名为超V一致遍历性。该收敛保证了，对于由支配函数V控制的所有函数，偏差向不变测度的几何衰减与初始化无关。该方法的一个关键优势在于，它无需建立全局极小化条件，而全局极小化条件在实践中通常难以验证，同时该方法能产生比全局极小化更强的收敛保证。我们确立了该框架在极小极大意义下的最优收敛界。通过将该框架应用于P'olya-Gamma和Kolmogorov-Gamma吉布斯采样器，证明了该框架的实用性。我们还表明，二元吉布斯采样器的定性超V一致遍历性收敛可通过不变测度的形式推断，从而完全绕过了收敛分析。
英文摘要：
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.
