The use of Gibbs samplers driven by improper posteriors has been a controve
rsial issue in the statistics literature over the last few years. It has re
cently been demonstrated that it is possible to make valid statistical infe
rences through such Gibbs samplers. Furthermore, theoretical and empirical
evidence has been given to support the idea that there are actually computa
tional advantages to using these nonpositive recurrent Markov chains rather
than more standard positive recurrent chains. These results provide motiva
tion for a general study of the behavior of the Gibbs Markov chain when it
is not positive recurrent. This article concerns stability relationships am
ong the two-variable Gibbs sampler and its subchains. We show that these th
ree Markov chains always share the same stability; that is, they are either
all positive recurrent, all null recurrent, or all transient. In addition,
we establish general results concerning the ways in which positive recurre
nt Markov chains can arise from null recurrent and transient Gibbs chains.
Six examples of varying complexity are used to illustrate the results.