We consider four aspects of what Meyn and Tweedie (1993) call `stability'
of Markov chains: 
-irreducibility, small sets, Harris recurrence,
and geometric ergodicity. They are generally proved in that order,
since the simpler notions must be established before the more complex
can be considered. For the point process samplers in Algorithm A
we really need only two proofs. The same
argument shows both -irreducibility and that every bounded set
is small, and a second argument shows both Harris recurrence and geometric
ergodicity.
-Irreducibility