  In this paper, we provide sufficient conditions for the existence of the
invariant distribution and for subgeometric rates of convergence in Wasserstein
distance for general state-space Markov chains which are (possibly) not
irreducible. Compared to previous work, our approach is based on a purely
probabilistic coupling construction which allows to retrieve rates of
convergence matching those previously reported for convergence in total
variation. Our results are applied to establish the subgeometric ergodicity in
Wasserstein distance of non-linear autoregressive models and of the
pre-conditioned Crank-Nicolson Markov chain Monte Carlo algorithm in Hilbert
space.
