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.
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References
Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2009)
del Barrio, E., Deheuvels, P., van de Geer, S.: Lectures on Empirical Processes. EMS Series of Lectures in Mathematics. EMS, Zürich (2007)
Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics. Wiley, New York (1999)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)
Csörgö, M., Horváth, C.: On the distributions of Lp norms and weighted quantile processes. Annales de l’I.H.P. Section B 26 (1), 65–85 (1990)
Ditlevsen, P.D.: Observation of a stable noise induced millennial climate changes from an ice-core record. Geophys. Res. Lett. 26 (10), 1441–1444 (1999)
Feller, W.: An Introduction to Probability Theory and its Applications. Geophysical Research Letters, vol. II. John Wiley & Sons, New York (1971)
Gairing, J.: Speed of convergence of discrete power variations of jump diffusions. Diplom thesis, Humboldt-Universität zu Berlin (2011)
Gairing, J., Högele, M., Kosenkova, T., Kulik, A.: Coupling distances between Lévy measures and applications to noise sensitivity of SDE. Stoch. Dyn. 15 (2), 1550009-1–1550009-25 (2014). doi:10.1142/S0219493715500094, online available
Gairing, J., Imkeller, P.: Stable CLTs and rates for power variation of α-stable Lévy processes. Methodol. Comput. Appl. Probab. 17, 1–18 (2013)
Debussche, A., Högele, M., Imkeller, P.: The Dynamics of Non-linear Reaction-Diffusion Equations with Small Lévy Noise. Springer Lecture Notes in Mathematics, vol. 2085. Springer, Cham (2013)
Hein, C., Imkeller, P., Pavlyukevich, I.: Limit theorems for p-variations of solutions of SDEs driven by additive stable Lévy noise and model selection for paleo-climatic data. Interdisciplinary Math. Sci. 8, 137–150 (2009)
Hein, C., Imkeller, P., Pavlyukevich, I.: Simple SDE dynamical models interpreting climate data and their meta-stability. Oberwolfach Reports (2008)
Högele, M., Pavlyukevich, I.: The exit problem from the neighborhood of a global attractor for heavy-tailed Lévy diffusions. Stoch. Anal. Appl. 32 (1), 163–190 (2013)
Imkeller, P., Pavlyukevich, I.: First exit times of SDEs driven by stable Lévy processes. Stoch. Process. Appl. 116 (4), 611–642 (2006)
Imkeller, P., Pavlyukevich, I., Wetzel, T.: First exit times for Lévy-driven diffusions with exponentially light jumps. Ann. Probab. 37 (2), 530–564 (2009)
Members NGRIP: NGRRIP data. Nature 431, 147–151 (2004)
Kosenkova, T., Kulik, A.: Explicit bounds for the convergence rate of a sequence of Markov chains to a Lévy-type process. (in preparation)
Pavlyukevich, I.: First exit times of solutions of stochastic differential equations with heavy tails. Stoch. Dyn. 11, Exp.No.2/3:1–25 (2011)
Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems. Vol. I: Theory, Vol. II.: Applications. Probability and its Applications. Springer, New York (1998)
Samoradnitsky, G., Taqqu, M.: Stable non-Gaussian random processes. Chapman& Hall, Boca-Raton/London/New York/Washington, DC (1994)
van der Vaard, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York (1996)
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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