Small sets and Markov transition densities

The theory of general state-space Markov chains can be related strongly to the case of discrete state-space by use of Doeblin’s notion of small sets and associated minorization conditions. The general theory shows that small sets exist for all Markov chains on state-spaces with countably generated -algebras, though the minorization provided by the theory concerns n-step transition kernels for some unspecified n. Partly motivated by the growing importance of small sets for Markov chain Monte Carlo and Coupling from the Past, we show that in general one cannot take n = 1 even if the kernel is assumed to have a density function (though of course one can take n = 1 if the kernel density is continuous). However n = 2 will suffice for kernels with densities (integral kernels), and in fact small sets of order 2 abound in the technical sense that the kernel density can be expressed as a countable sum of separable summands based on small sets. This can be exploited to produce a representation using a latent discrete Markov chain; indeed one might say, inside every Markov chain with measurable transition density there is a discrete state-space Markov chain struggling to escape. We conclude by discussing complements to these results, including their relevance to Harris-recurrent Markov chains and relationships of the counterexample to Turán problems for bipartite graphs.