  In any Markov chain Monte Carlo analysis, rapid convergence of the chain to
its target probability distribution is of practical and theoretical importance.
A chain that converges at a geometric rate is geometrically ergodic. In this
paper, we explore geometric ergodicity for two-component Gibbs samplers which,
under a chosen scanning strategy, evolve by combining one-at-a-time updates of
the two components. We compare convergence behaviors between and within three
such strategies: composition, random sequence scan, and random scan. Our main
results are twofold. First, we establish that if the Gibbs sampler is
geometrically ergodic under any one of these strategies, so too are the others.
Further, we establish a simple and verifiable set of sufficient conditions for
the geometric ergodicity of the Gibbs samplers. Our results are illustrated
using two examples.
