Convergence rates for a hierarchical Gibbs sampler

We establish results for the rate of convergence in total variation of a particular Gibbs sampler to its equilibrium distribution. This sampler is for a Bayesian inference model for a gamma random variable, whose only complexity lies in its multiple levels of hierarchy. Our results apply to a wide range of parameter values when the hierarchical depth is 3 or 4. Our method involves showing a relationship between the total variation of two ordered copies of our chain and the maximum of the ratios of their respective coordinates. We construct auxiliary stochastic processes to show that this ratio converges to 1 at a geometric rate.