Abstract
This article aims at the statistical assessment of time series with large fluctuations in short time, which are assumed to stem from a continuous process perturbed by a Lévy process exhibiting a heavy tail behavior. We propose an easily implementable procedure to estimate efficiently the statistical difference between the noise process generating the data and a given reference jump measure in terms of so-called coupling distances introduced in Gairing et al. (Stoch Dyn 15(2):1550009-1–1550009-25, 2014). After a short introduction to Lévy processes and coupling distances we recall basic statistical approximation results and derive asymptotic rates of convergence. In the sequel the procedure is elaborated in detail in an abstract setting and eventually applied in a case study with simulated and paleoclimate proxy data. Our statistic indicates the dominant presence of a non-stable heavy-tailed jump Lévy component for some tail index α > 2 in the paleoclimatic record.
Keywords
- Time series with heavy tails
- Index of stability
- Goodness-of-fit
- Empirical Wasserstein distance
- Limit theorems
- Empirical quantile process
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Acknowledgements
The authors would like to thank Sylvie Roelly and University Potsdam for constant hospitality and support. J. Gairing and M. Högele would like to thank the IRTG 1740 Berlin-São Paulo: “Dynamical Phenomena in Complex Networks” and the Berlin Mathematical School (BMS) for infrastructure support. The DFG Grant Schi 419/8-1 and the joint DFFD-RFFI project No. 09-01-14 are gratefully acknowledged by A. Kulik.
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Gairing, J., Högele, M., Kosenkova, T., Kulik, A. (2016). On the Calibration of Lévy Driven Time Series with Coupling Distances and an Application in Paleoclimate. In: Ancona, F., Cannarsa, P., Jones, C., Portaluri, A. (eds) Mathematical Paradigms of Climate Science. Springer INdAM Series, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-39092-5_7
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DOI: https://doi.org/10.1007/978-3-319-39092-5_7
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