A Donsker theorem for Lévy measures

Given n equidistant realisations of a Lévy process (Lt, t 0), a natural estimator ˆNn 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( ˆNn − 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. © 2012 Elsevier Inc. All rights reserved