  This article presents a new and easily implementable method to quantify the
so-called coupling distance between the law of a time series and the law of a
differential equation driven by Markovian additive jump noise with heavy-tailed
jumps, such as $\alpha$-stable L\'evy flights. Coupling distances measure the
proximity of the empirical law of the tails of the jump increments and a given
power law distribution. In particular they yield an upper bound for the
distance of the respective laws on path space. We prove rates of convergence
comparable to the rates of the central limit theorem which are confirmed by
numerical simulations. Our method applied to a paleoclimate time series of
glacial climate variability confirms its heavy tail behavior. In addition this
approach gives evidence for heavy tails in data sets of precipitable water
vapor of the Western Tropical Pacific.
