Throughout this paper, we use the following notation.

By B(Rd) we denote the Borel σ-field on the Euclidean space Rd . The Lebesgue measure on Rd is denoted by νd and we shortly write νd(dx)=dx when we integrate w.r.t. νd . For any measurable space (M,M,μ) we denote by Lα(M) , 1≤α<∞ , the space of all M|B(R) -measurable functions f:M→R with ∫M|f|α(x)μ(dx)<∞ . Equipped with the norm ||f||Lα(M)=∫M|f|α(x)μ(dx)1/α , Lα(M) becomes a Banach space and, in the case α=2 , even a Hilbert space with scalar product f,gLα(M)=∫Mf(x)g(x)μ(dx) , for any f,g∈L2(M) . With L∞(M) (i.e. if α=∞ ) we denote the space of all real-valued bounded functions on M. In the case (M,M,μ)=(R,B(R),ν1) we denote by Hδ(R)={f∈L2(R):∫R|F+f|2(x)(1+x2)δdx<∞} the Sobolev space of order δ>0 0$]]> equipped with the Sobolev norm ||f||Hδ(R)=||F+f(·)(1+·2)δ/2||L2(R) , where F+ is the Fourier transform on L2(R) . For f∈L1(R) , F+f is defined by F+f(x)=∫Reitxf(t)dt , x∈R . Throughout the rest of this paper (Ω,A,P) denotes a probability space. Note that in this case Lα(Ω) is the space of all random variables with finite α-th moment. For an arbitrary set A we introduce the notation card(A) or briefly |A| for its cardinality. Let supp(f)={x∈Rd:f(x)≠0} be the support set of a function f:Rd→R . Denote by diam(A)=sup{‖x−y‖∞:x,y∈A} the diameter of a bounded set A⊂Rd .

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=(a1,…,ad)∈Rd be a vector with positive components. In the sequel, we shortly write a>0 0$]]> in this situation. Moreover, let Π0(a)={x∈Rd,0<xi≤ai,i=1,…,d} and define for any j∈Zd the shifted block Πj(a) by Πj(a)=Π0(a)+ja={x∈Rd,jiai<xi≤ji(ai+1),i=1,…,d}. Clearly {Πj,j∈Zd} forms a partition of Rd . For any U⊂Zd , introduce the sets J−(U,a)={j∈Zd,Πj(a)⊂U},J+(U,a)={j∈Zd,Πj(a)∩U≠∅}U−(a)=⋃j∈J−(U,a)Πj(a),U+(a)=⋃j∈J+(U,a)Πj(a). A sequence of sets Un⊂Rd ( n∈N ) tends to infinity in Van Hove sense or shortly is VH-growing, if for any a>0 0$]]> νd(Un−)→∞andνd(Un−)νd(Un+)→1asn→∞. For a finite set A⊂Zd , define by ∂A={j∈Zd∖A,dist(j,A)=1} its boundary, where dist(j,A)=inf{‖j−x‖∞,x∈A} .

A sequence of finite sets An∈Zd ( n∈N ) is called regularly growing (to infinity), if |An|→∞,and|∂An||An|→0,asn→∞.

The following result that connects regularly and VH-growing sequences can be found in [7, p.174]. Lemma 2.2.

1.

Let Un⊂Rd ( n∈N ) be VH-growing. Then Vn=Un∩Zd ( n∈N ) is regularly growing to infinity.

In what follows, denote by E0(Rd) the collection of all bounded Borel sets in Rd .

Suppose that Λ={Λ(A);A∈E0(Rd)} is an infinitely divisible random measure on some probability space (Ω,A,P) , i.e. a random measure with the following properties:

For every A∈E0(Rd) , let φΛ(A) denote the characteristic function of the random variable Λ(A) . Due to the infinite divisibility of the random variable Λ(A) , the characteristic function φΛ(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 Λ(A) is of a special form, namely φΛ(A)(t)=expνd(A)K(t),A∈E0(Rd), with (2.1) K(t)=ita0−12t2b0+∫Reitx−1−itx1[−1,1](x)v0(x)dx, where νd denotes the Lebesgue measure on Rd , a0 and b0 are real numbers with 0≤b0<∞ and v0:R→R is a Lévy density, i.e. a measurable function which fulfills ∫Rmin{1,x2}v0(x)dx<∞ . The triplet (a0,b0,v0) will be referred to as the Lévy characteristic of Λ. It uniquely determines the distribution of Λ. This particular structure of the characteristic functions φΛ(A) means that the random measure Λ is stationary with the control measure λ:B(R)→[0,∞) given by λ(A)=νd(A)|a0|+b0+∫Rmin{1,x2}v0(x)dxfor allA∈E0(Rd).

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:Rd→R be Λ-integrable; then the function f(t−·) is Λ-integrable for every t∈Rd as well. We define the moving average random field X={X(t),t∈Rd} by (2.2) X(t)=∫Rdf(t−x)Λ(dx),t∈Rd. 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 φX(0) of X(0) is given by φX(0)(u)=exp∫RdK(uf(s))ds, where K is the function from (2.1). The argument ∫RdK(uf(s))ds in the above exponential function can be shown to have a similar structure as K(t) ; more precisely, we have (2.3) ∫RdK(uf(s))ds=iua1−12u2b1+∫Reiux−1−iux1[−1,1](x)v1(x)dx where a1 and b1 are real numbers with b0≥0 and the function v1 is the Lévy density of X(0) . The triplet (a1,b1,v1) 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 (a1,b1,v1) is given by the formulas (2.4) a1=∫RdU(f(s))ds,b1=b0∫Rdf2(s)ds,v1(x)=∫supp(f)1|f(s)|v0xf(s)ds, where supp(f):={s∈Rd:f(s)≠0} denotes the support of f and the function U is defined via U(u)=ua0+∫Rx1[−1,1](ux)−1[−1,1](x)v0(x)dx. Note that the Λ-integrability of f immediately implies that f∈L1(Rd)∩L2(Rd) . 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),t∈Rd} be given as in Section 2.3 and define the function u:R→R by u(x)=x . Suppose further that an estimator uv1ˆ for uv1 is given. In our recent preprint [13], we provided an estimation approach for uv0 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 (2.5) ∫supp(f(s))|f(s)|1/2ds<∞, and define the operator G:L2(R)→L2(R) by Gv=∫supp(f)sgn(f(s))v(·f(s))ds,v∈L2(R). Moreover, define the isometry M:L2(R)→L2(R×,dx|x|) by (Mv)(x)=|x|1/2v(x),v∈L2(R), and let the functions mf,±:R×→C and μf:R×→C be given by mf,+(x)=∫supp(f)sgn(f(s))|f(s)|1/2e−ixlog|f(s)|ds,mf,−(x)=∫supp(f)|f(s)|1/2e−ixlog|f(s)|ds,μf(y)=mf,+(log|y|)ify>0,mf,−(log|y|)ify<0. 0,\\ {} {m_{f,-}}(\log |y|)\hspace{1em}& \text{if}\hspace{2.5pt}y<0.\end{array}\right.\end{aligned}\]]]> Multiplying both sides in (2.4) by u leads to the equivalent relation (2.6) uv1=Guv0. Suppose uv1∈L2(R) and assume that for some β≥0 , (U_β) |mf,±(x)|≳11+|x|β,for allx∈R. Then the unique solution uv0∈L2(R) to equation (2.6) is given by uv0=G−1uv1=M−1F×−1(1μfF×Muv1), cf. [13, Theorem 3.1]. Based on this relation, the paper [13] provides the estimator (2.7) uv0ˆ=M−1F×−1(1μf,nF×Muv1ˆ)=:Gn−1uv1ˆ for uv0 , where (an)n∈N⊆(0,∞) is an arbitrary sequence, depending on the sample size n, that tends to 0 as n→∞ , and the mapping 1μf,n:R→C is defined by 1μf,n:=1μf1{|μf|>an} {a_{n}}\}}}$]]>. Here F×:L2(R×,dx|x|)→L2(R×,dx|x|) denotes the Fourier transform on the multiplicative group R× which is defined by (F×u)(y)=∫R×u(x)e−ilog|x|·log|y|·eiπδ(x)δ(y)dx|x|, for all u∈L1(R×,dx|x|)∩L2(R×,dx|x|) , with δ:R×→R given by δ(x)=1(−∞,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]. Remark 2.3.

The linear operator Gn−1:L2(R)→L2(R) defined in (2.7) is bounded in the operator norm ‖Gn−1‖≤1an , whereas G−1 is unbounded in general.

A random field X={X(t),t∈T} , T⊆Rd , defined on some probability space (Ω,A,P) is called m-dependent if for some m∈N and any finite subsets U and V of T the random vectors (X(u))u∈U and (X(v))v∈V are independent whenever ‖u−v‖∞=max{|ui−vi|,i=1,…,d}>m, m,\]]]> for all u=(u1,…,ud)⊤∈U and v=(v1,…,vd)⊤∈V . Lemma 2.4.

Let the random field X be given in (2.2) and suppose that f has a compact support. Then X is m-dependent with m>diam(supp(f)) \text{diam}(\operatorname{supp}(f))$]]>.

Let Λ={Λ(A),A∈E0(Rd)} be a stationary infinitely divisible random measure defined on some probability space (Ω,A,P) with characteristic triplet (a0,0,v0) , i.e. Λ is purely non-Gaussian. For a known Λ-integrable function f:Rd→R let X={X(t)=∫Rdf(t−x)Λ(dx),t∈Rd} be the infinitely divisible moving average random field defined in Section 2.3.

Fix Δ>0 0$]]> and suppose X is observed on a regular grid ΔZd={jΔ,j∈Zd} with the mesh size Δ, i.e. consider the random field Y given by (3.1) Y=(Yj)j∈Zd,Yj=X(jΔ),j∈Zd. For a finite subset W⊂Zd let (Yj)j∈W be a sample drawn from Y and denote by n the cardinality of W.

Throughout this paper, for any numbers a, b≥0 , we use the notation a≲b if a≤cb for some constant c>0 0$]]>.

Suppose that uv1ˆ is an estimator for uv1 (which we will precisely define in the next section) based on the sample (Yj)j∈W . Then, using the notation in Section 2.4, we introduce the linear functional LˆW:L2(R)→R,LˆWv:=v,uv0ˆL2(R)=v,Gn−1uv1ˆL2(R). The purpose of this paper is to investigate asymptotic properties of LˆW as the sample size |W|=n tends to infinity.

In this section we introduce an estimator for the function uv1 . 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 (3.3) ψ(t)=EeitY0=exp(iγt+∫R(eitx−1)v1(x)dx),t∈R, for some γ∈R and the Lévy density v1 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 −iψ′(t)ψ(t)=γ+F+[uv1](t),t∈R. Neglecting γ for the moment, this relation suggests that a natural estimator F+[uv1]ˆ for F+[uv1] is given by F+[uv1]ˆ(t)=θˆ(t)ψ˜(t),t∈R, with ψ˜(t)=1ψˆ(t)1{|(ψ)ˆ(t)|>n−1/2},t∈R, {n^{-1/2}}\}}},\hspace{1em}t\in \mathbb{R},\]]]> and ψˆ(t)=∑j∈WeitYj , θˆ(t)=∑j∈WYjeitYj being the empirical counterparts of ψ and θ=−iψ′ .

Now, consider for any b>0 0$]]> a function Kb:R→R with the following properties: (K1)

Kb∈L2(R) ;

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 (Yj)j∈Zd is m-dependent with m>Δ−1diam(supp(f)) {\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∈L2+τ(R) implies that f∈Lq(R) for all 0<q≤2+τ . Consequently, f fulfills the integrability condition (2.5). In contrast, if f does not have compact support, the Λ-integrability only ensures f∈L2(R) .

Assumption 3.1, (3) is a moment assumption on Λ. More precisely, it is satisfied if and only if E|Λ(A)|2+τ<∞ for all A∈E0(Rd) , cf. [22]. By Proposition 3.3, (b), this assumption also implies E|X(0)|2+τ<∞ in our setting.

As a consequence of Proposition 3.3, (a) and (c), Assumption 3.1, (4) ensures regularity of uv1 whereas (5) yields the polynomial decay of ψ. It was shown in [13, Theorem 3.10] that ψ and uv1 are connected via the relation |ψ(x)|=exp(−∫0xIm(F+[uv1](y))dy),x∈R; hence, more regularity of uv1 ensures slower decay rates for |ψ(x)| as x→±∞ . 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). Example 3.4

Fix b>0 0$]]> and let for any x∈R , v0(x)=x−1e−bx1(0,∞)(x) . Clearly, Assumption 3.1, (2) and (3) are satisfied for any τ>0 0$]]>. The Fourier transform of uv0 is given by F+[uv0](x)=(b−ix)−1 , x∈R ; hence ∫supp(f)f(s)F+[uv0](f(s)x)ds=∫supp(f)f(s)b−if(s)xds,x∈R. The latter identity shows that Assumption 3.1, (4) holds true for any integrable f with a compact support. Moreover, a simple calculation yields that for any x∈R , Assumption 3.1, (5) becomes (3.6) ∫R(1+x2)−1+εexp(∫supp(f)log(1+x2f2(s)b)ds)dx<∞. This condition is fulfilled for any ε<12−α if α:=∫supp(f)max{1,f2(s)b}ds<12.

In this section, we give an upper bound for the estimation error E|LˆWv−Lv| that allows to derive conditions under which LˆW is consistent for the linear functional L:L2(R)→R given by Lv=v,uv0,v∈L2(R). With the notations from Section 2.4, we have that the adjoint operator G−1∗:Image(G)→L2(R) of G−1 is given by (3.7) G−1∗v=M−1F×−1(1μ¯fF×Mv),v∈Image(G), where μ¯f denotes the complex conjugate function of μf . Moreover, the adjoint Gn−1∗:L2(R)→L2(R) of Gn−1 writes as Gn−1∗v=M−1F×−1(1μ¯f,nF×Mv),v∈L2(R), with 1μ¯f,n=1μ¯f1{|μ¯f|>an} {a_{n}}\}}}$]]>. Notice that Gn−1∗ is a bounded operator whereas G−1∗ is unbounded in general.