A Donsker theorem for Lévy measures - Richard Nickl; Markus Reiß (Humboldt University of Berlin, Collaborative Research Center 649 - Economic Risk, 2012)
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.
Keywords: C14; C22; 330; uniform central limit theorem; nonlinear inverse problem; smoothed empirical processes; pseudo-differential operators; jump measure; Stochastischer Prozess; Schätztheorie; Theorie