Given a low-frequency sample of the infinitely divisible moving average random field

Consider a stationary infinitely divisible indepently scattered random measure Λ whose Lévy density is denoted by

A wide class of spatio-temporal processes with the spectral representation (

We point out that there is a large number of literature concerning estimation of the Lévy density

In this paper, we investigate asymptotic properties of the linear functional

It turns out that under certain regularity assumptions on

From a practical point of view, a naturally arising question is wether a proposed model for

This paper is organized as follows. In Section

Throughout this paper, we use the following notation.

By

In this section, we briefly recall some basic facts about

Let

A sequence of finite sets

Regular growth of a family

The following result that connects regularly and VH-growing sequences can be found in [

In what follows, denote by

Suppose that

Let

The random variable

For every

Now one can define the stochastic integral with respect to the infinitely divisible random measure Λ in the following way:

Let

A measurable function

A useful characterization of the Λ-integrability of a function

For details on the theory of infinitely divisible measures and fields we refer the interested reader to [

Let the random field

Assume that

The linear operator

A random field

Compactness of

Let

Fix

Throughout this paper, for any numbers

Let the function

Suppose that

In this section we introduce an estimator for the function

Taking derivatives in (

Now, consider for any

If

Choosing

In order to explain Assumption

The compact support property in Assumption

Assumption

As a consequence of Proposition

Let us give some examples of Λ and

Fix

In this section, we give an upper bound for the estimation error

Notice that

With the previous notations we now derive an upper bound for

A proof of Lemma

Notice that condition

In order to deduce the convergence rate in Theorem

The condition

Under the conditions of Theorem

We close this section with the following example, showing that the functions

Fix

Provided the assumptions of Theorem

Let Assumption

The parameters

Assumption

Clearly, the lower bound for

It immediately follows from formula (

For any

With the previous notation, we now can formulate the main result of this section.

A proof of Theorem

Unfortunately, we could not provide a rate for the convergence

Suppose

In order to prove Theorem

We start with the following lemma.

Let

By Assumption

In order to prove Theorem

Subsequently, we give a step by step proof for Theorem

We first show that the deterministic term

Taking into account that

Next, we observe that

From the proofs of Theorem

Since

In the sequel, we assume that

Using the

For the second integral, by the triangle inequality we observe that for any

In this section we show the asymptotic normality of the main stochastic term. For this purpose, let

In order to prove Theorem

Since

The following lemma justifies the asymptotic variance

Let

We now can give a proof of Theorem

If

Now, assume that

In this section, we show that the remainder term

In order to prove Theorem

For any

Since

We use the same idea as in the proof of [

By Lemma

Fix

The following corollary is an immediate consequence of Theorem

Fix

By Lemma

Suppose

Now we can give the proof for Theorem

First of all, observe that