Convergence rate bounds for iterative random functions using one-shot coupling

One-shot coupling is a method of bounding the convergence rate between two copies of a Markov chain in total variation distance. The method is divided into two parts: the contraction phase, when the chains converge in expected distance and the coalescing phase, which occurs at the last iteration, when there is an attempt to couple. The method closely resembles the common random number technique used for simulation. In this paper, we present a general theorem for finding the upper bound on the Markov chain convergence rate that uses the one-shot coupling method. Our theorem does not require the use of any exogenous variables like a drift function or minorization constant. We then apply the general theorem to two families of Markov chains: the random functional autoregressive process and the randomly scaled iterated random function. We provide multiple examples of how the theorem can be used on various models including ones in high dimensions. These examples illustrate how theorem's conditions can be verified in a straightforward way. The one-shot coupling method appears to generate tight geometric convergence rate bounds.