Throughout this paper, we use the following notation.

By $\mathcal{B}({\mathbb{R}^{d}})$ we denote the Borel σ-field on the Euclidean space ${\mathbb{R}^{d}}$. The Lebesgue measure on ${\mathbb{R}^{d}}$ is denoted by ${\nu _{d}}$ and we shortly write ${\nu _{d}}(dx)=dx$ when we integrate w.r.t. ${\nu _{d}}$. For any measurable space $(M,\mathcal{M},\mu )$ we denote by ${L^{\alpha }}(M)$, $1\le \alpha <\infty $, the space of all $\mathcal{M}|\mathcal{B}(\mathbb{R})$-measurable functions $f:M\to \mathbb{R}$ with ${\textstyle\int _{M}}|f{|^{\alpha }}(x)\mu (dx)<\infty $. Equipped with the norm $||f|{|_{{L^{\alpha }}(M)}}={\left({\textstyle\int _{M}}|f{|^{\alpha }}(x)\mu (dx)\right)^{1/\alpha }}$, ${L^{\alpha }}(M)$ becomes a Banach space and, in the case $\alpha =2$, even a Hilbert space with scalar product ${\left\langle f,g\right\rangle _{{L^{\alpha }}(M)}}={\textstyle\int _{M}}f(x)g(x)\mu (dx)$, for any $f,g\in {L^{2}}(M)$. With ${L^{\infty }}(M)$ (i.e. if $\alpha =\infty $) we denote the space of all real-valued bounded functions on M. In the case $(M,\mathcal{M},\mu )=(\mathbb{R},\mathcal{B}(\mathbb{R}),{\nu _{1}})$ we denote by the Sobolev space of order $\delta >0$ equipped with the Sobolev norm $||f|{|_{{H^{\delta }}(\mathbb{R})}}=||{\mathcal{F}_{+}}f(\cdot ){(1+{\cdot ^{2}})^{\delta /2}}|{|_{{L^{2}}(\mathbb{R})}}$, where ${\mathcal{F}_{+}}$ is the Fourier transform on ${L^{2}}(\mathbb{R})$. For $f\in {L^{1}}(\mathbb{R})$, ${\mathcal{F}_{+}}f$ is defined by ${\mathcal{F}_{+}}f(x)={\textstyle\int _{\mathbb{R}}}{e^{itx}}f(t)dt$, $x\in \mathbb{R}$. Throughout the rest of this paper $(\Omega ,\mathcal{A},\mathbb{P})$ denotes a probability space. Note that in this case ${L^{\alpha }}(\Omega )$ is the space of all random variables with finite α-th moment. For an arbitrary set A we introduce the notation $\text{card}(A)$ or briefly $|A|$ for its cardinality. Let $\operatorname{supp}(f)=\{x\in {\mathbb{R}^{d}}:f(x)\ne 0\}$ be the support set of a function $f:{\mathbb{R}^{d}}\to \mathbb{R}$. Denote by $\text{diam}(A)=\sup \{\| x-y{\| _{\infty }}:x,y\in A\}$ the diameter of a bounded set $A\subset {\mathbb{R}^{d}}$.

In this section, we briefly recall some basic facts about regularly growing sets. For a more detailed investigation on this topic, see, e.g., [7].

Let $a=({a_{1}},\dots ,{a_{d}})\in {\mathbb{R}^{d}}$ be a vector with positive components. In the sequel, we shortly write $a>0$ in this situation. Moreover, let and define for any $j\in {\mathbb{Z}^{d}}$ the shifted block ${\Pi _{j}}(a)$ by Clearly $\{{\Pi _{j}},\hspace{2.5pt}j\in {\mathbb{Z}^{d}}\}$ forms a partition of ${\mathbb{R}^{d}}$. For any $U\subset {\mathbb{Z}^{d}}$, introduce the sets A sequence of sets ${U_{n}}\subset {\mathbb{R}^{d}}$ ($n\in \mathbb{N}$) tends to infinity in Van Hove sense or shortly is VH-growing, if for any $a>0$ For a finite set $A\subset {\mathbb{Z}^{d}}$, define by $\partial A=\{j\in {\mathbb{Z}^{d}}\setminus A,\hspace{2.5pt}\operatorname{dist}(j,A)=1\}$ its boundary, where $\operatorname{dist}(j,A)=\inf \{\| j-x{\| _{\infty }},\hspace{2.5pt}x\in A\}$.

A sequence of finite sets ${A_{n}}\in {\mathbb{Z}^{d}}$ ($n\in \mathbb{N}$) is called regularly growing (to infinity), if

The following result that connects regularly and VH-growing sequences can be found in [7, p.174].

In what follows, denote by ${\mathcal{E}_{0}}({\mathbb{R}^{d}})$ the collection of all bounded Borel sets in ${\mathbb{R}^{d}}$.

Suppose that $\Lambda =\{\Lambda (A);\hspace{2.5pt}A\in {\mathcal{E}_{0}}({\mathbb{R}^{d}})\}$ is an infinitely divisible random measure on some probability space $(\Omega ,\mathcal{A},P)$, i.e. a random measure with the following properties:

For every $A\in {\mathcal{E}_{0}}({\mathbb{R}^{d}})$, let ${\varphi _{\Lambda (A)}}$ denote the characteristic function of the random variable $\Lambda (A)$. Due to the infinite divisibility of the random variable $\Lambda (A)$, the characteristic function ${\varphi _{\Lambda (A)}}$ has a Lévy–Khintchin representation which can, in its most general form, be found in [21, p. 456]. Throughout the rest of the paper we make the additional assumption that the Lévy–Khintchin representation of $\Lambda (A)$ is of a special form, namely with where ${\nu _{d}}$ denotes the Lebesgue measure on ${\mathbb{R}^{d}}$, ${a_{0}}$ and ${b_{0}}$ are real numbers with $0\le {b_{0}}<\infty $ and ${v_{0}}:\mathbb{R}\to \mathbb{R}$ is a Lévy density, i.e. a measurable function which fulfills ${\textstyle\int _{\mathbb{R}}}\min \{1,{x^{2}}\}{v_{0}}(x)dx<\infty $. The triplet $({a_{0}},{b_{0}},{v_{0}})$ will be referred to as the Lévy characteristic of Λ. It uniquely determines the distribution of Λ. This particular structure of the characteristic functions ${\varphi _{\Lambda (A)}}$ means that the random measure Λ is stationary with the control measure $\lambda :\mathcal{B}(\mathbb{R})\to [0,\infty )$ given by

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

A useful characterization of the Λ-integrability of a function f is given in [21, Theorem 2.7]. Now let $f:{\mathbb{R}^{d}}\to \mathbb{R}$ be Λ-integrable; then the function $f(t-\cdot )$ is Λ-integrable for every $t\in {\mathbb{R}^{d}}$ as well. We define the moving average random field $X=\{X(t),\hspace{2.5pt}t\in {\mathbb{R}^{d}}\}$ by Recall that a random field is called infinitely divisible if its finite dimensional distributions are infinitely divisible. The random field X above is (strictly) stationary and infinitely divisible and the characteristic function ${\varphi _{X(0)}}$ of $X(0)$ is given by where K is the function from (2.1). The argument ${\textstyle\int _{{\mathbb{R}^{d}}}}K(uf(s))\hspace{0.2222em}\mathrm{d}s$ in the above exponential function can be shown to have a similar structure as $K(t)$; more precisely, we have where ${a_{1}}$ and ${b_{1}}$ are real numbers with ${b_{0}}\ge 0$ and the function ${v_{1}}$ is the Lévy density of $X(0)$. The triplet $({a_{1}},{b_{1}},{v_{1}})$ is again referred to as the Lévy characteristic (of $X(0)$) and determines the distribution of $X(0)$ uniquely. A simple computation shows that the triplet $({a_{1}},{b_{1}},{v_{1}})$ is given by the formulas where $\operatorname{supp}(f):=\{s\in {\mathbb{R}^{d}}:\hspace{2.5pt}f(s)\ne 0\}$ denotes the support of f and the function U is defined via Note that the Λ-integrability of f immediately implies that $f\in {L^{1}}({\mathbb{R}^{d}})\cap {L^{2}}({\mathbb{R}^{d}})$. Hence, all integrals above are finite.

For details on the theory of infinitely divisible measures and fields we refer the interested reader to [21].

Let the random field $X=\{X(t),\hspace{2.5pt}t\in {\mathbb{R}^{d}}\}$ be given as in Section 2.3 and define the function $u:\mathbb{R}\to \mathbb{R}$ by $u(x)=x$. Suppose further that an estimator $\widehat{u{v_{1}}}$ for $u{v_{1}}$ is given. In our recent preprint [13], we provided an estimation approach for $u{v_{0}}$ based on relation (2.4) which we briefly recall in this section. Therefore, quite a number of notations are required.

Assume that f satisfies the integrability condition and define the operator $\mathcal{G}:{L^{2}}(\mathbb{R})\to {L^{2}}(\mathbb{R})$ by Moreover, define the isometry $\mathcal{M}:{L^{2}}(\mathbb{R})\to {L^{2}}({\mathbb{R}^{\times }},\frac{dx}{|x|})$ by and let the functions ${m_{f,\pm }}:{\mathbb{R}^{\times }}\to \mathbb{C}$ and ${\mu _{f}}:{\mathbb{R}^{\times }}\to \mathbb{C}$ be given by Multiplying both sides in (2.4) by u leads to the equivalent relation Suppose $u{v_{1}}\in {L^{2}}(\mathbb{R})$ and assume that for some $\beta \ge 0$, Then the unique solution $u{v_{0}}\in {L^{2}}(\mathbb{R})$ to equation (2.6) is given by cf. [13, Theorem 3.1]. Based on this relation, the paper [13] provides the estimator for $u{v_{0}}$, where ${({a_{n}})_{n\in \mathbb{N}}}\subseteq (0,\infty )$ is an arbitrary sequence, depending on the sample size n, that tends to 0 as $n\to \infty $, and the mapping $\frac{1}{{\mu _{f,n}}}:\mathbb{R}\to \mathbb{C}$ is defined by $\frac{1}{{\mu _{f,n}}}:=\frac{1}{{\mu _{f}}}{\mathbb{1}_{\{|{\mu _{f}}|>{a_{n}}\}}}$. Here ${\mathcal{F}_{\times }}:{L^{2}}({\mathbb{R}^{\times }},\frac{dx}{|x|})\to {L^{2}}({\mathbb{R}^{\times }},\frac{dx}{|x|})$ denotes the Fourier transform on the multiplicative group ${\mathbb{R}^{\times }}$ which is defined by for all $u\in {L^{1}}({\mathbb{R}^{\times }},\frac{dx}{|x|})\cap {L^{2}}({\mathbb{R}^{\times }},\frac{dx}{|x|})$, with $\delta :{\mathbb{R}^{\times }}\to \mathbb{R}$ given by $\delta (x)={\mathbb{1}_{(-\infty ,0)}}(x)$ (cf. [13, Section 2.2]). A more detailed introduction to harmonic analysis on locally compact abelian groups can be found, e.g., in [12].

A random field $X=\{X(t),\hspace{2.5pt}t\in T\}$, $T\subseteq {\mathbb{R}^{d}}$, defined on some probability space $(\Omega ,\mathcal{A},\mathbb{P})$ is called m-dependent if for some $m\in \mathbb{N}$ and any finite subsets U and V of T the random vectors ${(X(u))_{u\in U}}$ and ${(X(v))_{v\in V}}$ are independent whenever for all $u={({u_{1}},\dots ,{u_{d}})^{\top }}\in U$ and $v={({v_{1}},\dots ,{v_{d}})^{\top }}\in V$.

Let $\Lambda =\{\Lambda (A),\hspace{2.5pt}A\in {\mathcal{E}_{0}}({\mathbb{R}^{d}})\}$ be a stationary infinitely divisible random measure defined on some probability space $(\Omega ,\mathcal{A},\mathbb{P})$ with characteristic triplet $({a_{0}},0,{v_{0}})$, i.e. Λ is purely non-Gaussian. For a known Λ-integrable function $f:{\mathbb{R}^{d}}\to \mathbb{R}$ let $X=\{X(t)={\textstyle\int _{{\mathbb{R}^{d}}}}f(t-x)\Lambda (dx),\hspace{2.5pt}t\in {\mathbb{R}^{d}}\}$ be the infinitely divisible moving average random field defined in Section 2.3.

Fix $\Delta >0$ and suppose X is observed on a regular grid $\Delta {\mathbb{Z}^{d}}=\{j\Delta ,\hspace{2.5pt}j\in {\mathbb{Z}^{d}}\}$ with the mesh size Δ, i.e. consider the random field Y given by For a finite subset $W\subset {\mathbb{Z}^{d}}$ let ${({Y_{j}})_{j\in W}}$ be a sample drawn from Y and denote by n the cardinality of W.

Throughout this paper, for any numbers a, $b\ge 0$, we use the notation $a\lesssim b$ if $a\le cb$ for some constant $c>0$.

Suppose that $\widehat{u{v_{1}}}$ is an estimator for $u{v_{1}}$ (which we will precisely define in the next section) based on the sample ${({Y_{j}})_{j\in W}}$. Then, using the notation in Section 2.4, we introduce the linear functional The purpose of this paper is to investigate asymptotic properties of ${\hat{\mathcal{L}}_{W}}$ as the sample size $|W|=n$ tends to infinity.

In this section we introduce an estimator for the function $u{v_{1}}$. Therefore, let ψ denote the characteristic function of $X(0)$. Then, by Assumption 3.1, (2), together with formula (2.3), we find that ψ can be rewritten as for some $\gamma \in \mathbb{R}$ and the Lévy density ${v_{1}}$ given in (2.4). We call γ the drift parameter or shortly drift of X. As a consequence of representation (3.3), the random field X is purely non-Gaussian. It is subsequently assumed that the drift γ is known.

Taking derivatives in (3.3) leads to the identity Neglecting γ for the moment, this relation suggests that a natural estimator $\widehat{{\mathcal{F}_{+}}[u{v_{1}}]}$ for ${\mathcal{F}_{+}}[u{v_{1}}]$ is given by with and $\hat{\psi }(t)={\textstyle\sum _{j\in W}}{e^{it{Y_{j}}}}$, $\hat{\theta }(t)={\textstyle\sum _{j\in W}}{Y_{j}}{e^{it{Y_{j}}}}$ being the empirical counterparts of ψ and $\theta =-i{\psi ^{\prime }}$.

Now, consider for any $b>0$ a function ${K_{b}}:\mathbb{R}\to \mathbb{R}$ with the following properties: Then, for any $b>0$, we define the estimator $\widehat{u{v_{1}}}$ for $u{v_{1}}$ by

In order to explain Assumption 3.1, we prepend the following proposition whose proof can be found in Appendix.

The compact support property in Assumption 3.1, (1) ensures that the random field ${({Y_{j}})_{j\in {\mathbb{Z}^{d}}}}$ is m-dependent with $m>{\Delta ^{-1}}\text{diam}(\operatorname{supp}(f))$ (cf. Lemma 2.4). In particular, m increases when the grid size Δ of the sample is decreasing. Moreover, compact support of f together with $f\in {L^{2+\tau }}(\mathbb{R})$ implies that $f\in {L^{q}}(\mathbb{R})$ for all $0<q\le 2+\tau $. Consequently, f fulfills the integrability condition (2.5). In contrast, if f does not have compact support, the Λ-integrability only ensures $f\in {L^{2}}(\mathbb{R})$.

Assumption 3.1, (3) is a moment assumption on Λ. More precisely, it is satisfied if and only if for all $A\in {\mathcal{E}_{0}}({\mathbb{R}^{d}})$, cf. [22]. By Proposition 3.3, (b), this assumption also implies $\mathbb{E}|X(0){|^{2+\tau }}<\infty $ in our setting.

As a consequence of Proposition 3.3, (a) and (c), Assumption 3.1, (4) ensures regularity of $u{v_{1}}$ whereas (5) yields the polynomial decay of ψ. It was shown in [13, Theorem 3.10] that ψ and $u{v_{1}}$ are connected via the relation hence, more regularity of $u{v_{1}}$ ensures slower decay rates for $|\psi (x)|$ as $x\to \pm \infty $. Further results on the polynomial decay of infinitely divisible characteristic functions as well as sufficient conditions for this property to hold can be found in [23].

Let us give some examples of Λ and f satisfying Assumption 3.1, (1)–(5).

In this section, we give an upper bound for the estimation error $\mathbb{E}|{\hat{\mathcal{L}}_{W}}v-\mathcal{L}v|$ that allows to derive conditions under which ${\hat{\mathcal{L}}_{W}}$ is consistent for the linear functional $\mathcal{L}:{L^{2}}(\mathbb{R})\to \mathbb{R}$ given by With the notations from Section 2.4, we have that the adjoint operator ${\mathcal{G}^{-1\ast }}:\text{Image}(\mathcal{G})\to {L^{2}}(\mathbb{R})$ of ${\mathcal{G}^{-1}}$ is given by where ${\bar{\mu }_{f}}$ denotes the complex conjugate function of ${\mu _{f}}$. Moreover, the adjoint ${\mathcal{G}_{n}^{-1\ast }}:{L^{2}}(\mathbb{R})\to {L^{2}}(\mathbb{R})$ of ${\mathcal{G}_{n}^{-1}}$ writes as with $\frac{1}{{\bar{\mu }_{f,n}}}=\frac{1}{{\bar{\mu }_{f}}}{\mathbb{1}_{\{|{\bar{\mu }_{f}}|>{a_{n}}\}}}$. Notice that ${\mathcal{G}_{n}^{-1\ast }}$ is a bounded operator whereas ${\mathcal{G}^{-1\ast }}$ is unbounded in general.

With the previous notations we now derive an upper bound for $\mathbb{E}|{\hat{\mathcal{L}}_{W}}v-\mathcal{L}v|$. Therefore, recall condition (${\mathbf{U}_{\beta }}$) from Section 2.4.

A proof of Lemma 3.6 as well as of Theorem 3.7 below can be found in Appendix.

We close this section with the following example, showing that the functions ${g_{t}}$ considered in [20, p. 3309] may belong to the range of ${\mathcal{G}^{-1\ast }}$.

Provided the assumptions of Theorem 3.7 are satisfied, is bounded in mean. In this section, we give conditions under which ${\operatorname{err}_{W}}(v)$ is asymptotically Gaussian. For this purpose, introduce the following notation.

For any $j\in W$ and any admissible function $v\in \mathcal{U}(\xi ,{\beta _{2}})$, introduce the random variables In the sequel, it is assumed that the random field Y introduced in (3.1) is observed on a sequence ${({W_{k}})_{k\in \mathbb{N}}}$ of regularly growing observation windows (cf. Section 2.2). To avoid longer notations, we drop the index k in this notation and shortly write W instead of ${W_{k}}$. Moreover, we denote by n $(=n(k))$ the cardinality of W.

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

A proof of Theorem 3.12 can be found in Section 4.

In this section we show the asymptotic normality of the main stochastic term. For this purpose, let ${P_{n}}:\mathcal{B}(\mathbb{R})\to [0,1]$ be the empirical measure given by where ${\delta _{x}}:\mathcal{B}(\mathbb{R})\to \{0,1\}$ denotes the Dirac measure concentrated in $x\in \mathbb{R}$. Further, for any $v\in \mathcal{U}(\xi ,{\beta _{2}})$, define the random fields ${({Z_{j,v,n}^{(k)}})_{j\in {\mathbb{Z}^{d}}}}$, $k=1,2$, by and The following theorem is the main result of this section.

In order to prove Theorem 4.5, we first show some auxiliary statements. We begin with the following representation for the main stochastic term.

The following lemma justifies the asymptotic variance ${\sigma ^{2}}$ in Theorem 3.12.

We now can give a proof of Theorem 4.5.

In this section, we show that the remainder term ${E_{2}}$ is stochastically negligible as the sample size n tends to infinity.

In order to prove Theorem 4.8, some auxiliary statements are required. Therefore, we introduce the following notation.

For any $t\in \mathbb{R}$, $j\in {\mathbb{Z}^{d}}$, let the centered random variables ${\xi _{j}^{(l)}}(t)$, ${\tilde{\xi }_{j}^{(l)}}(t)$, $l=1,2$, be given by Then $\hat{\psi }-\psi $ and $\hat{\theta }-\theta $ can be rewritten as In the sequel, we shortly write ${\xi ^{(l)}}(t)$, ${\tilde{\xi }^{(l)}}(t)$ for the random fields ${({\xi _{j}^{(l)}}(t))_{j\in {\mathbb{Z}^{d}}}}$ and ${({\tilde{\xi }_{j}^{(l)}}(t))_{j\in {\mathbb{Z}^{d}}}}$, $l=1,2$. Moreover, for any $K>0$, we define the random fields ${\bar{\xi }_{K}^{(l)}}(t)={({\bar{\xi }_{j,K}^{(l)}}(t))_{j\in {\mathbb{Z}^{d}}}}$, $l=1,2$, by For any finite subset $V\subset {\mathbb{Z}^{d}}$ and any random field $Y={({Y_{j}})_{j\in {\mathbb{Z}^{d}}}}$, let

The following corollary is an immediate consequence of Theorem 4.11.

Now we can give the proof for Theorem 4.8.

It remains to show that the result of Theorem 3.12 still holds true if γ is assumed to be arbitrary. For this purpose, consider the sample ${({\tilde{Y}_{j}})_{j\in W}}$ given by ${\tilde{Y}_{j}}={Y_{j}}-\gamma $, $j\in W$. Moreover, let ${\psi _{\ast }}(t)=\mathbb{E}[{e^{it{\tilde{Y}_{0}}}}]$ be the characteristic function of ${\tilde{Y}_{0}}$ and write ${\hat{\psi }_{\ast }}$ for its empirical counterpart, i.e. ${\hat{\psi }_{\ast }}(t)=\frac{1}{n}{\textstyle\sum _{j\in W}}{e^{it{\tilde{Y}_{j}}}}$. Then, with the notation we have for any $t\in \mathbb{R}$, where ${\theta _{\ast }}(t)=\mathbb{E}[{\tilde{Y}_{0}}{e^{it{\tilde{Y}_{0}}}}]$ and ${\hat{\theta }_{\ast }}(t)=\frac{1}{n}{\textstyle\sum _{j\in W}}{\tilde{Y}_{j}}{e^{it{\tilde{Y}_{j}}}}$. For any $v\in \mathcal{U}(\xi ,{\beta _{2}})$, consider the decomposition As W is regularly growing to infinity, the first summand on the right-hand side of the last equation tends to a Gaussian random variable since ${\psi _{\ast }}$ is an infinitely divisible characteristic function without drift component. For the second summand, we find that Hence, by (K3) and Definition 3.10, (iii), we obtain where the last term tends to zero as $n\to \infty $, since ${b_{n}}=o({n^{-1/2}})$. Moreover, Taking into account that $|\psi (x)|=|{\hat{\psi }_{\ast }}(x)|$, relation (4.2) with $p=1/2$ yields Applying again (4.2) with $p=1$ implies as $n\to \infty $; thus, by dominated convergence, we have All in all, this shows that Theorem 3.12 holds for any fixed $\gamma \in \mathbb{R}$.