One-shot coupling is a method of bounding the convergence rate between two
copies of a Markov chain in total variation distance, which was first
introduced by Roberts and Rosenthal and generalized by Madras and Sezer. 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. One-shot coupling does not
require the use of any exogenous variables like a drift function or a
minorization constant. In this paper, we summarize the one-shot coupling method
into the One-Shot Coupling Theorem. We then apply the theorem to two families
of Markov chains: the random functional autoregressive process and the
autoregressive conditional heteroscedastic (ARCH) process. We provide multiple
examples of how the theorem can be used on various models including ones in
high dimensions. These examples illustrate how the theorem's conditions can be
verified in a straightforward way. The one-shot coupling method appears to
generate tight geometric convergence rate bounds