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A Donsker theorem for Lévy measures. (English) Zbl 1310.60056
Summary: Given \(n\) equidistant realisations of a Lévy process \((L_t, t\geq 0)\), a natural estimator \(\hat{N}_n\) for the distribution function \(N\) of the Lévy measure is constructed. Under a polynomial decay restriction on the characteristic function \(\varphi\), a Donsker-type theorem is proved, that is, a functional central limit theorem for the process \(\sqrt{n}(\hat {N}_n-N)\) 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 \(F^{-1}[1/\varphi ( - \bullet )]\). 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.

MSC:
60G51 Processes with independent increments; Lévy processes
60F17 Functional limit theorems; invariance principles
60G20 Generalized stochastic processes
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