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. We construct a purely data-driven estimator of the L\évy density $\nu$ through the spectral approach using general Calderon--Zygmund integral operators, which include convolution and projection kernels. Assuming minimal tail assumptions, it is shown to estimate $\nu$ at the minimax rate of estimation over Besov balls under the losses $L^p(\mathbb{R})$, $p\in[1,\infty]$, and robustly to the observation regime (high- and low-frequency). To achieve adaptation in a minimax sense, we use Lepski\u{i}'s method as it is particularly well-suited for our generality. 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 deal with general Calderon--Zygmund operators, to depart from the ubiquitous quadratic structure and to show robustness without polynomial-tail conditions. Part of the motivation for such generality is a new insight we include here too that, furthermore, allows us to unify the main two approaches to construct estimators used in related literature.