A Donsker Theorem for Lévy Measures

Abstract

Given $n$ equidistant realisations of a Lévy process $(L_t,\,t\ge 0)$, 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 $ϕ$, 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 ${\cal F}^{-1}[1/ϕ(-\cdot)]$. 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.