A Donsker Theorem for Lévy Measures
Given n equidistant realisations of a Lévy process (Lt; t >= 0), a natural estimator for the distribution function N of the Lévy measure is constructed. Under a polynomial decay restriction on the characteristic function, a Donsker-type theorem is proved, that is, a functional central limit theorem for the process in the space of bounded functions away from zero. The limit distribution is a generalised Brownian bridge process with bounded and continuous sample paths whose covariance structure depends on the Fourier-integral operator. The class of Lévy processes covered includes several relevant examples such as compound Poisson, Gamma and self-decomposable processes. Main ideas in the proof include establishing pseudo-locality of the Fourier-integral operator and recent techniques from smoothed empirical processes.
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| Length: | 26 pages |
| Date of creation: | Jan 2012 |
| Date of revision: | |
| Handle: | RePEc:hum:wpaper:sfb649dp2012-003 |
| Contact details of provider: | Postal: Phone: +49-30-2093-5708 Fax: +49-30-2093-5617 Web page: http://sfb649.wiwi.hu-berlin.deEmail: More information through EDIRC
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- Denis Belomestny & Markus Reiß, 2006.
"Spectral calibration of exponential Lévy Models [2],"
SFB 649 Discussion Papers
SFB649DP2006-035, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
- Denis Belomestny & Markus Reiß, 2006. "Spectral calibration of exponential Lévy models," Finance and Stochastics, Springer, vol. 10(4), pages 449-474, December.
- Belomestny, Denis, 2011. "Spectral estimation of the Lévy density in partially observed affine models," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1217-1244, June.
- Denis Belomestny & Markus Reiß, 2006.
"Spectral calibration of exponential Lévy Models [1],"
SFB 649 Discussion Papers
SFB649DP2006-034, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
- Denis Belomestny & Markus Reiß, 2006. "Spectral calibration of exponential Lévy models," Finance and Stochastics, Springer, vol. 10(4), pages 449-474, December.
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