A Donsker theorem for Lévy measures

Given n equidistant realisations of a Lévy process (Lt,t≥0), a natural estimator Nˆn for the distribution function N of the Lévy measure is constructed. Under a polynomial decay restriction on the characteristic function φ, a Donsker-type theorem is proved, that is, a functional central limit theore...

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Bibliographic Details
Published in:Journal of functional analysis 2012-11, Vol.263 (10), p.3306-3332
Main Authors: Nickl, Richard, Reiß, Markus
Format: Article
Language:English
Subjects:
Jump measure
Nonlinear inverse problem
Pseudo-differential operators
Smoothed empirical processes
Uniform central limit theorem
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Summary:Given n equidistant realisations of a Lévy process (Lt,t≥0), a natural estimator Nˆn for the distribution function N of the Lévy measure is constructed. Under a polynomial decay restriction on the characteristic function φ, a Donsker-type theorem is proved, that is, a functional central limit theorem for the process n(Nˆn−N) 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 F−1[1/φ(−•)]. 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.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2012.08.012