This paper analyzes the Gibbs sampler applied to a standard variance component model, and considers the question of how many iterations are required for convergence. It is proved that for $K$ location parameters, with $J$ observations each, the number of iterations required for convergence (for large $K$ and $J$) is a constant times $(1 + \log K/\log J)$. This is one of the first rigorous, a priori results about time to convergence for the Gibbs sampler. A quantitative version of the theory of Harris recurrence (for Markov chains) is developed and applied.
"Rates of Convergence for Gibbs Sampling for Variance Component Models." Ann. Statist. 23 (3) 740 - 761, June, 1995. https://doi.org/10.1214/aos/1176324619