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A Donsker Theorem for Lévy Measures


  • Richard Nickl
  • Markus Reiß


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.

Suggested Citation

  • Richard Nickl & Markus Reiß, 2012. "A Donsker Theorem for Lévy Measures," SFB 649 Discussion Papers SFB649DP2012-003, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  • Handle: RePEc:hum:wpaper:sfb649dp2012-003

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    References listed on IDEAS

    1. Denis Belomestny & Markus Reiß, 2006. "Spectral calibration of exponential Lévy models," Finance and Stochastics, Springer, vol. 10(4), pages 449-474, December.
    2. 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.
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    Cited by:

    1. Trabs, Mathias, 2014. "On infinitely divisible distributions with polynomially decaying characteristic functions," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 56-62.
    2. Koltchinskii, Vladimir & Nickl, Richard & van de Geer, Sara & Wellner, Jon A., 2016. "The mathematical work of Evarist Giné," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3607-3622.
    3. Reiß, Markus, 2013. "Testing the characteristics of a Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2808-2828.
    4. Vetter, Mathias, 2014. "Inference on the Lévy measure in case of noisy observations," Statistics & Probability Letters, Elsevier, vol. 87(C), pages 125-133.
    5. Trabs, Mathias, 2015. "Quantile estimation for Lévy measures," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3484-3521.
    6. Hoffmann, Michael & Vetter, Mathias, 2017. "Weak convergence of the empirical truncated distribution function of the Lévy measure of an Itō semimartingale," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1517-1543.
    7. Mélina Bec & Claire Lacour, 2015. "Adaptive pointwise estimation for pure jump Lévy processes," Statistical Inference for Stochastic Processes, Springer, vol. 18(3), pages 229-256, October.

    More about this item


    uniform central limit theorem; nonlinear inverse problem; smoothed empirical processes; pseudo-differential operators; jump measure;

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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