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Suppose that a compound Poisson process is observed discretely in time and assume that its jump distribution is supported on the set of natural numbers. In this paper we propose a non-parametric Bayesian approach to estimate the intensity of the underlying Poisson process and the distribution of the jumps. We provide a MCMC scheme for obtaining samples from the posterior. We apply our method on both simulated and real data examples, and compare its performance with the frequentist plug-in estimator proposed by Buchmann and Gr\"ubel. On a theoretical side, we study the posterior from the frequentist point of view and prove that as the sample size $n\rightarrow\infty$, it contracts around the `true', data-generating parameters at rate $1/\sqrt{n}$, up to a $\log n$ factor., Comment: 27 pages, 7 figures

Key words and Phrases: cumulant, empirical characteristic function,Lévy process, self-decomposable distribution, stationary process.Consider a stationary sequence of random variables with infinitely divisiblemarginal law, characterized by its Lévy density. We analyze the behaviorof a so-called cumulant M-estimator, in case this Lévy density ischaracterized by a Euclidean (finite-dimensional) parameter. Under mildconditions, we prove consistency and asymptotic normality of the estimator.The estimator is considered in the situation where the data are incrementsof a subordinator as well as the situation where the data consist of adiscretely sampled Ornstein Uhlenbeck process induced by the subordinator.We illustrate our results for the Gamma-process and theInverse-Gaussian-OU-process. For these processes we also explain how theestimator can be computed numerically.