Asymptotic equivalence for pure jump L\'evy processes with unknown L\'evy density and Gaussian white noise

TL;DR

This paper proves that, under certain smoothness conditions, observing a Lévy process either discretely or continuously becomes statistically equivalent to observing a Gaussian white noise process as the observation time grows large.

Contribution

It establishes a global asymptotic equivalence between Lévy process experiments and Gaussian white noise experiments, with explicit Markov kernels for reproducing one from the other.

Findings

Asymptotic equivalence holds under smoothness conditions on the Lévy density.

Explicit Markov kernels are constructed for experiment reproduction.

Results apply as observation time T tends to infinity.

Abstract

The aim of this paper is to establish a global asymptotic equivalence between the experiments generated by the discrete (high frequency) or continuous observation of a path of a L{\'e}vy process and a Gaussian white noise experiment observed up to a time T, with T tending to $\infty$. These approximations are given in the sense of the Le Cam distance, under some smoothness conditions on the unknown L{\'e}vy density. All the asymptotic equivalences are established by constructing explicit Markov kernels that can be used to reproduce one experiment from the other.