On the irreducibility of a Markov chain defined on a space of genotype configurations by a sampling scheme

Techniques from image processing are now being considered for use in pedigree analysis. Although the method that we develop here was motivated by considering it as an analogue to the Gibbs sampler, we justify it independently using the ergodic theorem for aperiodic, irreducible, finite Markov chains. While it is trivial to show that aperiodicity holds, irreducibility is a more interesting condition. Proofs of irreducibility are provided for some special cases and counterexamples are provided for others. Where irreducibility does not hold, an alternative method using relaxed genetic parameters and rejection is recommended and justified.