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Parametric Estimation for Subordinators and Induced OU Processes

Abstract.  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 M‐estimator, 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.