One-shot coupling for certain stochastic recursive sequences

We consider Markov chains {[Gamma]n} with transitions of the form [Gamma]n=f(Xn,Yn)[Gamma]n-1+g(Xn,Yn), where {Xn} and {Yn} are two independent i.i.d. sequences. For two copies {[Gamma]n} and {[Gamma]n'} of such a chain, it is well known that provided E[log(f(Xn,Yn))]<0, where => is weak convergence. In this paper, we consider chains for which also [Gamma]n-[Gamma]n'-->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.