发表机构
University of Amsterdam（阿姆斯特丹大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种新的马尔可夫链收敛直接准则，通过渐近等价性证明收敛，适用于一般可测空间，涵盖Metropolis--Hastings等算法，无需不可约性等传统条件。
AI 中文摘要
对于具有不变概率测度π的马尔可夫核T，我们通过一种称为与目标的渐近等价性的准则，给出了马尔可夫链收敛定理的自包含证明。该准则对每个起点x的T^n_x和π的勒贝格分解假设了两部分条件：1. 渐近绝对连续性：奇异质量sing(T^n_x | π)趋于0；2. 目标的渐近支配性：奇异质量sing(π | T^n_x)趋于0，当n→∞时。该准则在可数生成的可测空间上，对马尔可夫链收敛既是充分条件也是必要条件。我们在一般可测空间的三种情形下验证了该准则的密度版本：(i) T具有相对于π的正转移密度；(ii) T由带正转移密度的绝对连续部分与起点处的一个原子组成，这涵盖了Metropolis--Hastings算法；(iii) 转移密度仅在有限步数后为正，该步数可能依赖于起点x。为展示我们的一般准则，我们研究了随机扫描的Gibbs采样器和并行 tempering 算法。此外，我们证明在所有提及的设定下Birkhoff遍历定理均适用，从而得到强大数定律。在整篇论文中，既未使用不可约性、非周期性、常返性，也未使用耦合、分裂构造或小集。在大多数结果中，状态空间是一般可测空间，仅带有σ代数，无其他结构。可数生成性仅在陈述该准则的无密度形式时被假设。本文证明的所有定理均非新结果；所提供的是通往单一、广泛适用的马尔可夫链收敛准则的短路径，该准则兼具充分性与必要性。
英文摘要
For a Markov kernel $T$ with an invariant probability measure $π$, we give a self-contained proof of the Markov chain convergence theorem via a criterion called asymptotic equivalence with the target. It assumes two parts about the Lebesgue decompositions of $T^n_x$ and $π$ for every starting point $x$: 1.) asymptotic absolute continuity: the singular mass $\mathrm{sing}(T^n_x \mid π)$ tends to $0$; and, 2.) asymptotic domination of the target: the singular mass $\mathrm{sing}(π\mid T^n_x)$ tends to $0$, as $n \to \infty$. Assuming a jointly measurable density for the absolutely continuous part of each iterate $T^n$ w.r.t. $π$, this criterion is sufficient and necessary for convergence. A positive minorant density version of it is verified in three cases: i.) $T$ has a positive transition density w.r.t. $π$; ii.) $T$ consists of an absolutely continuous part with positive transition density together with an atom at the starting point, which covers the Metropolis-Hastings algorithm; iii.) the transition density is positive only after a finite number of steps that may depend on the starting point $x$. To demonstrate our general criterion, we investigate the Gibbs sampler with random scan and the parallel tempering algorithm. Furthermore, we show that in all mentioned settings Birkhoff's ergodic theorem applies, so as to obtain the strong law of large numbers. Throughout this paper, neither irreducibility, nor aperiodicity, nor recurrence, nor couplings, nor splitting constructions, nor small sets are used. In all results, the state space is a general measurable space with no structure beyond a $σ$-algebra. That joint measurability is assumed of the Markov kernel, not of the space; countable generation supplies it. None of the theorems proved here is new; what is offered is a short route to a single, widely applicable Markov chain convergence criterion.
