One-shot coupling for certain stochastic recursive sequences

We consider Markov chains {Γn} with transitions of the form Γn=f(Xn,Yn)Γn−1+g(Xn,Yn), where {Xn} and {Yn} are two independent i.i.d. sequences. For two copies {Γn} and {Γn′} of such a chain, it is well known that L(Γn)−L(Γn′)⇒0 provided E[log(f(Xn,Yn))]<0, where ⇒ is weak convergence. In this paper, we consider chains for which also ||Γn−Γn′||→0, where ||·|| is total variation distance. We consider in particular how to obtain sharp quantitative bounds on the total variation distance. Our method involves a new coupling construction, one-shot coupling, which waits until time n before attempting to couple. We apply our results to an auto-regressive Gibbs sampler, and to a Markov chain on the means of Dirichlet processes.