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 α-stable Lévy 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 datasets of precipitable water vapor of the Western Tropical Pacific.
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How close are time series to power tail Lévy diffusions?
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July 2017
Research Article|
July 20 2017
How close are time series to power tail Lévy diffusions?

Jan M. Gairing;
Jan M. Gairing
a)
1
Institut für Mathematik, Humboldt-Universität zu Berlin
, Berlin, Germany
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Michael A. Högele;
Michael A. Högele
b)
2
Departamento de Matemáticas, Universidad de los Andes
, Bogotá, Colombia
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Tania Kosenkova
;
Tania Kosenkova
c)
3
Institut für Mathematik, Universität Potsdam
, Potsdam, Germany
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Adam H. Monahan
Adam H. Monahan
d)
4
School of Earth and Ocean Sciences, University of Victoria
, Victoria, British Columbia V8W 3V6, Canada
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Chaos 27, 073112 (2017)
Article history
Received:
April 19 2017
Accepted:
June 04 2017
Connected Content
A companion article has been published:
Coupling of statistics and math models identifies power tail jump patterns in time series
Citation
Jan M. Gairing, Michael A. Högele, Tania Kosenkova, Adam H. Monahan; How close are time series to power tail Lévy diffusions?. Chaos 1 July 2017; 27 (7): 073112. https://doi.org/10.1063/1.4986496
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