  In compact settings, the convergence rate of the empirical optimal transport
cost to its population value is well understood for a wide class of spaces and
cost functions. In unbounded settings, however, hitherto available results
require strong assumptions on the ground costs and the concentration of the
involved measures. In this work, we pursue a decomposition-based approach to
generalize the convergence rates found in compact spaces to unbounded settings
under generic moment assumptions that are sharp up to an arbitrarily small
$\epsilon > 0$. Hallmark properties of empirical optimal transport on compact
spaces, like the recently established adaptation to lower complexity, are shown
to carry over to the unbounded case.
