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This article presents a new and easily implementable method to quantify theso-called coupling distance between the law of a time series and the law of adifferential equation driven by Markovian additive jump noise with heavy-tailedjumps, such as $\alpha$-stable L\'evy flights. Coupling distances measure theproximity of the empirical law of the tails of the jump increments and a givenpower law distribution. In particular they yield an upper bound for thedistance of the respective laws on path space. We prove rates of convergencecomparable to the rates of the central limit theorem which are confirmed bynumerical simulations. Our method applied to a paleoclimate time series ofglacial climate variability confirms its heavy tail behavior. In addition thisapproach gives evidence for heavy tails in data sets of precipitable watervapor of the Western Tropical Pacific.

How close are time series to power tail L\'evy diffusions?