Consider a stationary sequence of random variables with infinitely divisible marginal law, characterized by its Lévy density. We analyse the behaviour of a so-called cumulant Mestimator, in case this Lévy density is characterized by a Euclidean (finite dimensional) parameter. Under mild conditions, we prove consistency and asymptotic normality of the estimator. The estimator is considered in the situation where the data are increments of a subordinator as well as the situation where the data consist of a discretely sampled Ornstein-Uhlenbeck (OU) process induced by the subordinator. We illustrate our results for the Gamma-process and the Inverse-Gaussian OU process. For these processes we also explain how the estimator can be computed numerically. © Board of the Foundation of the Scandinavian Journal of Statistics 2006.