Given a low frequency sample of an infinitely divisible moving average random field { R d f (x−t)Λ(dx); t ∈ R d } with a known simple function f , we study the problem of nonparametric estimation of the Lévy characteristics of the independently scattered random measure Λ. We provide three methods, a simple plug-in approach, a method based on Fourier transforms and an approach involving decompositions with respect to L 2 -orthonormal bases, which allow to estimate the Lévy density of Λ. For these methods, the bounds for the L 2 -error are given. Their numerical performance is compared in a simulation study.An Inverse Problem for ID Moving Averages with Lévy characteristics (a 1 , b 1 , v 1 ), where f = n k=1 f k 1I ∆ k is a simple function. Suppose a sample (X(t 1 ), . . . , X(t N )) from X is available. The problem studied in this paper is the nonparametric estimation of (a 0 , b 0 , v 0 ). For any simple function f with congruent sets ∆ k , X(t) in (1) has the same distribution as a linear combination of i.i.d. infinitely divisible random variables. Therefore, existence and uniqueness of a characteristic triplet (a 0 , b 0 , v 0 ) with the property that a certain linear combination of independent random variables with the corresponding infinitely divisible distribution leading to a random variable with Lévy characteristics (a 1 , b 1 , v 1 ) becomes a characterization problem for such distributions. For certain distributions, namely the Poisson and the Gaussian one as well as a mixture of both, all possible distributions for the summands in the linear combination can be described (see e.g. [1]). The disadvantage of those characterization theorems is that they do not give any information about the involved parameters (expectation and variance of each summand) and so it is not possible to derive sufficient conditions for the existence of a solution in terms of the kernel function f . Therefore, to solve the inverse problem, we prefer to use concrete relations between the characteristic triplets of X and Λ (Section 3) given in terms of f . The recent preprint [2] covers the case d = 1 estimating the Lévy density v 0 of the integrator Lévy process {L s } of a moving average processThe estimate is based on the inversion of the Mellin transform of the second derivative of the cumulant of X(0). A uniform error bound as well as the consistency of the estimate are given. It is not assumed that f is simple, however, main results are subject to a number of quite restricting integrability assumptions onto x 2 v 0 (x) and f as well as mixing properties of {L s } that are tricky to check. Additionally, the logarithmic convergence rate shown there (cf. [2, Corollary 1]) is too slow.In our approach, we develop the ideas of [3] and use Banach fixed-point theorem combined with a recursive iteration procedure (Theorem 4.1) to give sufficient conditions for the existence of a (unique) solution of our (generally speaking, illposed) inverse problem v 1 → v 0 . We consider simple functions f since 1. in applications, f is mainly discr...

1 This study was designed to follow changes in plasma catecholamine concentrations during ,B-adrenoceptor blockade using doses of antagonists which decreased the mean arterial pressure (MAP) by about 15 mm Hg. Noradrenaline, adrenaline and dopamine were radioenzymatically determined in 34 patients with moderate essential hypertension during an 8 week course of treatment with either pindolol (with intrinsic sympathomimetic activity, ISA) or propranolol (without ISA). Plasma catecholamines were determined before and 1, 7, 28 and 56 days after commencement of treatment and 1 week after discontinuation of treatment. 2 After one day of pindolol therapy plasma catecholamine concentrations were decreased, but no decrease in MAP was observed. After one day of propranolol therapy, however, MAP was decreased, but except for increased levels of adrenaline, plasma catecholamines showed no changes. 3 On the 56th day of therapy both 0-adrenoceptor blockers had decreased the MAP. Pindolol therapy had caused a decrease in all three catecholamines whereas propranolol had caused no change except for decreased dopamine levels. 4 One week after cessation of propranolol therapy catecholamines were decreased but the MAP had begun to return to initial values; after cessation of pindolol therapy however, the MAP remained decreased. 5 The dissimilar relationships between blood pressure and catecholamine concentration during pindolol and propranolol therapy are evidence for multiple and different modes of action for P-adrenoceptor blockers with and without ISA. This study demonstrates that catecholamine concentrations were generally decreased during low-dose P-adrenoceptor blocker therapy, with lower catecholamine levels during pindolol treatment than during propranolol treatment.

We introduce basic statistical methods for the extrapolation of stationary random fields. For square integrable fields, we set out basics of the kriging extrapolation techniques. For (non-Gaussian) stable fields, which are known to be heavy tailed, we describe further extrapolation methods and discuss their properties. Two of them can be seen as direct generalizations of kriging.

Given a low-frequency sample of the infinitely divisible moving average randomand v 0 being the Lévy density of the integrator random measure Λ. In this paper, we study asymptotic properties of the linear functional L 2 (R) ∋ v → v, uv 0 L 2 (R) , if the (known) kernel function f has compact support. We provide conditions that ensure consistency (in mean) and prove a central limit theorem for it.

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