Adaptive nonparametric estimation for compound Poisson processes robust to the discrete-observation scheme

A compound Poisson process whose jump measure and intensity are unknown is observed at finitely many equispaced times and a purely data-driven wavelet-type estimator of the L\'evy density $\nu$ is constructed through the spectral approach. Assuming minimal tail assumptions, it is shown to estimate $\nu$ at the best possible rate of convergence over Besov balls under the losses $L^p(\mathbb{R})$, $p\in[1,\infty]$, and robustly to the observation regime (high- and low-frequency). The adaptive estimator is obtained by applying Lepski\u{i}'s method and, thus, novel exponential-concentration inequalities are proved including one for the uniform fluctuations of the empirical characteristic function. These are of independent interest, as are the proof-strategies employed to depart from the ubiquitous quadratic structure and to show robustness to the observation scheme without polynomial-moment conditions.