Asymptotic Behavior of the Gibbs Sampler

Mild sufficient conditions for the ergodicity of the Gibbs sampler are obtained. The geometric ergodicity of the Gibbs sampler is then discussed, where the main tool is the “drift” criterion. This criterion requires the construction of a generalized energy function so that the chain is dissipating energy, at a suitable rate, whenever it lies outside the “center” of the state-space. For the “two-variable” case, a useful recipe is provided for constructing appropriate energy functions. These results are illustrated with four examples.