Title: Compound Poisson approximation to estimate the Lévy density,
Abstract: We construct an estimator of the Lévy density of a pure jump Lévy
process, possibly of infinite variation, from the discrete observation of one
trajectory at high frequency. The novelty of our procedure is that we directly
estimate the Lévy density relying on a pathwise strategy, whereas existing
procedures rely on spectral techniques. By taking advantage of a compound
Poisson approximation of the Lévy density, we circumvent the use of spectral
techniques and in particular of the Lévy-Khintchine formula. A linear wavelet
estimators is built and its performance is studied in terms of $L_p$ loss
functions, $p\geq 1$, over Besov balls. The resulting rates are minimax-optimal
for a large class of Lévy processes. We discuss the robustness of the
procedure to the presence of a Brownian part and to the estimation set getting
close to the critical value 0.