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

Geometric Ergodicity & Scanning Strategies For Two-Component Gibbs Samplers