Parametric estimation for subordinators and induced OU-processes

Key words and Phrases: cumulant, empirical characteristic function,
Lévy process, self-decomposable distribution, stationary process.

Consider a stationary sequence of random variables with infinitely divisible
marginal law, characterized by its Lévy density. We analyze the behavior
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 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.