A Donsker Theorem for L\'evy Measures

TL;DR

This paper establishes a functional central limit theorem for a natural estimator of the Lévy measure's distribution function, extending Donsker's theorem to a broad class of Lévy processes under certain decay conditions.

Contribution

It proves a Donsker-type theorem for the estimator of the Lévy measure distribution function, including a general limit process and covering various important Lévy process classes.

Findings

The estimator converges in distribution to a generalized Brownian bridge.

The limit process's covariance depends on a Fourier-integral operator.

The theorem applies to compound Poisson, Gamma, and self-decomposable Lévy processes.

Abstract

Given $n$ equidistant realisations of a L\'evy process $(L_t,\,t\ge 0)$, a natural estimator $\hat N_n$ for the distribution function $N$ of the L\'evy measure is constructed. Under a polynomial decay restriction on the characteristic function $\phi$, 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 ${\cal F}^{-1}[1/\phi(-\cdot)]$. The class of L\'evy 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.