In this paper, we give quantitative bounds on the f-total variation distance from convergence of a Harris recurrent Markov chain on a given state space under drift and minorization conditions implying ergodicity at a subgeometric rate. These bounds are then specialized to the stochastically monotone case, covering the case where there is no minimal reachable element. The results are illustrated with two examples, from queueing theory and Markov Chain Monte Carlo theory.
"Computable convergence rates for sub-geometric ergodic Markov chains." Bernoulli 13 (3) 831 - 848, August 2007. https://doi.org/10.3150/07-BEJ5162