A Direct Route to Markov Chain Convergence via Asymptotic Equivalence with the Target

For a Markov kernel $T$ with an invariant probability measure $\pi$, 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 $\pi$ for every starting point $x$: 1.) asymptotic absolute continuity: the singular mass sing$(T^{n}_{x}\mid\pi)$ tends to $0$; 2.) asymptotic domination of the target: the singular mass sing$(\pi\mid T^{n}_{x})$ tends to $0$, as $n \to \infty$. This criterion, on countably generated measurable spaces, is both sufficient and necessary for the Markov chain convergence. A density version of this criterion is verified on general measurable spaces in three cases: (i) $T$ has a positive transition density wrt $\pi$; (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 most results, the state space is a general measurable space, which carries no structure beyond a $\sigma$-algebra. Countable generation is only assumed where the density-free form of the criterion is stated. None of the theorems proved here is new; what is offered is a short route to a single, widely applicable Markov chain convergence criterion, which is both sufficient and necessary.