On a linear functional for infinitely divisible moving average random fields

Given a low-frequency sample of the infinitely divisible moving average random field $\{\int_{\mathbb{R}^d}f(t-x)\Lambda (dx), t\in \mathbb{R}^d\}$, in [13] we proposed an estimator $\hat{uv_0}$ for the function $\mathbb{R}\ni x\mapsto u(x)v_0(x)=(uv_0)(x)$, with $u(x)=x$ and $v_0$ being the L\'{e}vy density of the integrator random measure $\Lambda$. In this paper, we study asymptotic properties of the linear functional $L^2(\mathbb{R})\ni v\mapsto \left \langle v,\hat{uv_0}\right \rangle_{L^2(\mathbb{R})}$, if the (known) kernel function $f$ has a compact support. We provide conditions that ensure consistency (in mean) and prove a central limit theorem for it.


Introduction
Consider a stationary infinitely divisible indepently scattered random measure Λ whose Lévy density is denoted by v 0 . For some (known) Λ-integrable function f : R d → R with a compact support, let A wide class of spatio-temporal processes with the spectral representation (1.1) is provided by the so-called ambit random fields, where a space-time Lévy process serves as integrator. Such processes are, e.g., used to model the growth rate of tumours, where the spatial component describes the angle between the center of the tumour cell and the nearest point at its boundary (cf. [3,16]). Ambit fields cover quite a number of different processes and fields including Ornstein-Uhlenbeck type and mixed moving average random fields (cf. [1,2]). A further interesting application of (1.1) is given in [17], where the author uses infinitely divisible moving average random fields in order to model claims of natural disaster insurance within different postal code areas.
We point out that there is a large number of literature concerning estimation of the Lévy density v 1 (its Lévy measure, respectively) in the case when X is a Lévy process (cf. [5,9,10,14,19]). Moreover, in the recent paper [4] the authors provide an estimator for the Lévy density v 0 of the integrator Lévy process {L s } of a moving average process X(t) = R f (t − s)dL s , t ∈ R, which covers the case d = 1 in (1.1). For a discussion on the differences between our approach and the method provided in [4], we refer to [13] and [18].
In this paper, we investigate asymptotic properties of the linear functional L 2 (R) ∋ v →L W v = v, uv 0 L 2 (R) as the sample size |W | tends to infinity. It is motivated by the paper of Nickl and Reiss [20], where the authors provide a Donsker type theorem for the Lévy measure of pure jump Lévy processes. Since our observations are m-dependent, the classical i.i.d. theory does not apply here. Instead, we combine results of Chen and Shao [8] for m-dependent random fields and ideas of Bulinski and Shashkin [7] with exponential inequalities for weakly dependent random fields (see e.g. [15,11]) in order to prove our limit theorems.
It turns out that under certain regularity assumptions on uv 0 ,L W v is a mean consistent estimator for Lv = v, uv 0 L 2 (R) with a rate of convergence given by O(|W | −1/2 ), for any v that belongs to a subspace U of L 1 (R) ∩ L 2 (R). Moreover, we give conditions such that finite dimensional distributions of the process {|W | 1/2 (L W −L)v; v ∈ U} are asymptotically Gaussian as |W | is regularly growing to infinity.
From a practical point of view, a naturally arising question is wether a proposed model for v 0 (or equivalently uv 0 ) is suitable. Knowledge of the asymptotic distribution of |W | 1/2 (L W − L) can be used in order to construct tests for different hypotheses, e.g., on regularity assumptions of the model for v 0 . Indeed, the scalar product · , · L 2 (R) naturally induces that the class U of test functions is growing, when uv 0 becomes more regular. This paper is organized as follows. In Section 2, we give a brief overview of regularly growing sets and infinitely divisible moving average random fields. We further recall some notation and the most frequently used results from [13]. Section 3 is devoted to asymptotic properties ofL W . Here we discuss our regularity assumptions and state the main results of this paper (Theorems 3.7 and 3.12). Section 4 is dedicated to the proofs of our limit theorems. Some of the shorter proofs as well as external results that will frequently be used in Section 3 are moved to Appendix.

Notation
Throughout this paper, we use the following notation.
By B(R d ) we denote the Borel σ-field on the Euclidean space R d . The Lebesgue measure on R d is denoted by ν d and we shortly write ν d (dx) = dx when we integrate w.r.t. ν d . For any measurable space (M, M, µ) we denote by L α (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 || L α (M) = M |f | α (x)µ(dx) 1 the Sobolev space of order δ > 0 equipped with the Sobolev norm ||f || H δ (R) = ||F + f (·)(1 + · 2 ) δ/2 || L 2 (R) , where F + is the Fourier transform on L 2 (R). For f ∈ L 1 (R), F + f is defined by F + f (x) = R e itx f (t)dt, x ∈ R. Throughout the rest of this paper (Ω, A, P) denotes a probability space. Note that in this case L α (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 ∈ R d : f (x) = 0} be the support set of a function f : R d → R. Denote by diam(A) = sup{ x − y ∞ : x, y ∈ A} the diameter of a bounded set A ⊂ R d .

Regularly growing sets
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 , . . . , a d ) ∈ R d be a vector with positive components. In the sequel, we shortly write a > 0 in this situation. Moreover, let A sequence of sets U n ⊂ R d (n ∈ N) tends to infinity in Van Hove sense or shortly is VH-growing, if for any a > 0 A sequence of finite sets A n ∈ Z d (n ∈ N) is called regularly growing (to infinity), if |A n | → ∞, and |∂A n | |A n | → 0, as n → ∞.
Remark 2.1. Regular growth of a family A n ⊂ Z d means that the number of points in the boundary of A n grows significantly slower than the number of its interior points.
The following result that connects regularly and VH-growing sequences can be found in [7, p.174].
is regularly growing to infinity.
2. If (U n ) n∈N is a sequence of finite subsets of Z d , regularly growing to infinity,

Infinitely divisible random measures
In what follows, denote by E 0 (R d ) the collection of all bounded Borel sets in R d .
is an infinitely divisible random measure on some probability space (Ω, A, P ), i.e. a random measure with the following properties: (a) Let (E m ) m∈N be a sequence of disjoint sets in E 0 (R d ). Then the sequence (Λ(E m )) m∈N consists of independent random variables; if, in addition, (b) The random variable Λ(A) has an infinitely divisible distribution for any choice of A ∈ E 0 (R d ).
For every A ∈ E 0 (R d ), let ϕ Λ(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 ϕ Λ(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 where ν d denotes the Lebesgue measure on R d , a 0 and b 0 are real numbers with 0 ≤ b 0 < ∞ and v 0 : R → R is a Lévy density, i.e. a measurable function which fulfills R min{1, x 2 }v 0 (x)dx < ∞. 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 ϕ Λ(A) means that the random measure Λ is stationary with the control measure λ : B(R) → [0, ∞) 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 : R d → R be Λ-integrable; then the function f (t − ·) is Λ-integrable for every t ∈ R d as well. We define the moving average random field X = {X(t), t ∈ 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 ϕ X(0) of X(0) is given by where K is the function from (2.1). The argument R d K(uf (s)) ds 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 ≥ 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 supp(f ) := {s ∈ R d : f (s) = 0} denotes the support of f and the function U is defined via Hence, all integrals above are finite. For details on the theory of infinitely divisible measures and fields we refer the interested reader to [21].

A plug-in estimation approach
for v 0 Let the random field X = {X(t), t ∈ R d } be given as in Section 2.3 and define the function u : R → R by u(x) = x. Suppose further that an estimator uv 1 for uv 1 is given. In our recent preprint [13], we provided an estimation approach for uv 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 supp(f (s)) |f (s)| 1/2 ds < ∞, (2.5) and define the operator G : Moreover, define the isometry M : and let the functions m f,± : R × → C and µ f : R × → C be given by Suppose uv 1 ∈ L 2 (R) and assume that for some β ≥ 0, Then the unique solution uv 0 ∈ L 2 (R) to equation (2.6) is given by cf. [13,Theorem 3.1]. Based on this relation, the paper [13] provides the estimator for uv 0 , where (a n ) n∈N ⊆ (0, ∞) is an arbitrary sequence, depending on the sample size n, that tends to 0 as n → ∞, and the mapping 1 µ f,n : R → C is defined by denotes the Fourier transform on the multiplicative group R × which is defined by for all u ∈ L 1 (R × , dx |x| )∩L 2 (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]. an , whereas G −1 is unbounded in general.

m-dependent random fields
A random field X = {X(t), t ∈ T }, T ⊆ R d , 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 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 )).

The setting
Let Λ = {Λ(A), A ∈ E 0 (R d )} be a stationary infinitely divisible random measure defined on some probability space (Ω, A, P) with characteristic triplet (a 0 , 0, v 0 ), i.e. Λ is purely non-Gaussian. For a known Λ-integrable function f : , t ∈ R d } be the infinitely divisible moving average random field defined in Section 2.3. Fix ∆ > 0 and suppose X is observed on a regular grid ∆Z d = {j∆, j ∈ Z d } with the mesh size ∆, i.e. consider the random field Y given by For a finite subset W ⊂ Z d let (Y j ) 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.
Assumption 3.1. Let the function u : R → R be given by u(x) = x. We make the following assumptions: for some τ > 0 Suppose that uv 1 is an estimator for uv 1 (which we will precisely define in the next section) based on the sample (Y j ) j∈W . Then, using the notation in Section 2.4, we introduce the linear functional The purpose of this paper is to investigate asymptotic properties ofL W as the sample size |W | = n tends to infinity.

An estimator for uv 1
In this section we introduce an estimator for the function uv 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 γ ∈ 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.
Neglecting γ for the moment, this relation suggests that a natural estimator F + [uv 1 ] for F + [uv 1 ] is given by andψ(t) = j∈W e itYj ,θ(t) = j∈W Y j e itYj being the empirical counterparts of ψ and θ = −iψ ′ . Now, consider for any b > 0 a function K b : R → R with the following properties: Then, for any b > 0, we define the estimator uv 1 for uv 1 by If uv 1 is a consistent estimator for uv 1 , it is reasonable to assume that γ = 0 (cf. [18]). Indeed, for the asymptotic results below, the value of γ is irrelevant. Even if γ = 0, the functionalL W estimates the intended quantity with uv 1 given in (3.4) (cf. Section 4.3).
yields the estimator uv 1 that we introduced in [18] and [13], originally designed by Comte and Genon-Catalot [10] in the case when X is a pure jump Lévy process.

Discussion and examples
In order to explain Assumption 3.1, we prepend the following proposition whose proof can be found in Appendix. Proposition 3.3. Let the infinitely divisible moving average random field X = {X(t), t ∈ R d } be given as above and suppose u(x) = x.
Example 3.4 (Gamma random measure). Fix b > 0 and let for any . Clearly, Assumption 3.1, (2) and (3) are satisfied for any τ > 0. The Fourier transform of uv 0 is given by 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, This condition is fulfilled for any

Consistency ofL W
In this section, we give an upper bound for the estimation error E|L W v − Lv| that allows to derive conditions under whichL W is consistent for the linear functional L : L 2 (R) → R given by With the notations from Section 2.4, we have that the adjoint operator G −1 * : whereμ f denotes the complex conjugate function of µ f . Moreover, the adjoint G −1 * n : n is a bounded operator whereas G −1 * is unbounded in general.
Remark 3.5. Notice that G −1 * n = G −1 * if a n = 0 for any n ∈ N. Hence, G −1 * n uv 1 in this case only is well-defined if uv 1 is an element of Image(G * ) what is indeed a very restrictive assumption. For a detailed discussion we refer to [13].
With the previous notations we now derive an upper bound for E|L W v − Lv|. Therefore, recall condition (U β ) from Section 2.4. Lemma 3.6. Let γ = 0 and suppose Assumption 3.1, (1)-(3) hold true for some τ ≥ 0. Moreover, let condition (U β ) be satisfied for some β ≥ 0 and assume that A proof of Lemma 3.6 as well as of Theorem 3.7 below can be found in Appendix.
for some β 2 > β 1 . Moreover, let a = a n and b = b n be sequences with the properties a n → 0, b n → 0 and a n = o n b n , as n → ∞, and assume that conditions (K1)-(K3) are fulfilled. Then, under Assumption 3.1, (1)-(4), E|L W v − Lv| → 0 as n → ∞ with the order of convergence given by as a solution of Guv 0 = uv 1 . In Lemma 3.6, it can be replaced by the more (and most) general assumption m f,± = 0 almost everywhere on R. Moreover, condition (K3) can be replaced by sup b>0, x∈R |F (b) In order to deduce the convergence rate in Theorem 3.7 explicitely, condition (3.9) is essential. Moreover, it ensures that the function v belongs to the range of G (cf. [13, Theorem 3.1]); hence the expression G −1 * v is well-defined.
(d) Under the conditions of Theorem 3.7, the convergence rate of 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 G −1 * .

A central limit theorem forL W
Provided the assumptions of Theorem 3.7 are satisfied, is bounded in mean. In this section, we give conditions under which err W (v) is asymptotically Gaussian. For this purpose, introduce the following notation.
Definition 3.10. Let Assumption 3.1 be satisfied and suppose that condition The linear subspace of all admissible functions of index (ξ, β 2 ) is denoted by U(ξ, β 2 ).  For any j ∈ W and any admissible function v ∈ U(ξ, β 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∈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. the sequences a n and b n be given by Then, as W is regularly growing to infinity, for any admissible function v ∈ U(ξ, β 2 ), where N v is a Gaussian random variable with zero expectation and variance given by (3.10) A proof of Theorem 3.12 can be found in Section 4.
Remark 3.13. Unfortunately, we could not provide a rate for the convergence err W (v) d → N v in Theorem 3.12. Therefore, it would be sufficient to provide, e.g., L 1 (Ω, P)-rates for the convergence sup x |ψ(x) − ψ(x)|, sup x |θ(x) − θ(x)| → 0 (as |W | → ∞), that seems to be a hard problem in the dependent observations setting.

Proof of Theorem 3.12
In order to prove Theorem 4, we adopt the strategy of the proof of [20,Theorem 2]. Nevertheless, the main difficulty in our setting is that the observations (Y j ) j∈W are not independent; hence the classical theory cannot be applied here. Instead, we use asymptotic results for partial sums of m-dependent random fields (cf. [8]) in combination with the theory developed by Bulinski and Shashkin in [7] for weakly dependent random fields.
We start with the following lemma.
Lemma 4.1. Let γ = 0 and suppose that v ∈ U(ξ, β 2 ) is an admissible function. Then Assumption 3.1 implies: 1. xP has a bounded Lebesgue density on R, where P denotes the distribution of X(0). Proof.
1. Let µ(dx) = (uv 1 )(x)dx. By Proposition 3.3, (a), uv 1 ∈ L 1 (R); hence, µ defines a finite signed measure on R. Since θ = ψF + [uv 1 ], we conclude that i.e. xP (dx) = (µ * P )(dx); thus, xP has the density given by Further, we have for any x ∈ R, The last expression is bounded and square integrable, hence 1 immediately follows from Definition 3.10, (i) (cf. Remark 3.11, (b)). Moreover, by Proposition 3.3, (a), we find that where the latter is finite due to Definition 3.10, (i). The bound in part (2) finally yields In order to prove Theorem 3.12, consider the following decomposition that can be obtained by the isometry property of F + : with E 1 , . . . , E 5 given by , and R n = 1 −ψ ψ θ ψ − θ ψ . We call the expression E 1 main stochastic term and the expression E 2 remainder term.
Subsequently, we give a step by step proof for Theorem 3.12 by considering each of the above terms E 1 , . . . , E 5 seperately.
We first show that the deterministic term E 3 tends to zero as the sample size n tends to infinity.
Next, we observe that E 5 is asymptotically negligible in mean.
Proof. From the proofs of Theorem 3.6 and Corollary 3.7 we conclude that as n → ∞, for any admissible function v ∈ U(ξ, β 2 ).
Proof. Since θ ψ 2 = i 1 ψ ′ , we obtain by conditions (K2) and (K3), with S := sup x∈R, b>0 |F + [K b ](x)|. In order to bound the summands on the righthand side of the latter inequality, we start with the following observation: ∃ n 0 ∈ N such that for all n ≥ n 0 , This shows (4.1). In the sequel, we assume that n ≥ n 0 and consider each summand in the above inequality seperately: 1. Using the m-dependence of (Y j ) j∈Z d , we conclude as in the first part of the proof of [18, Lemma 8.3] that , for all n ≥ n 0 , where the last inequality uses the fact that |ψ(x)| ≤ 1. Hence, the first integral tends to zero as n → ∞.
2. For the second integral, by the triangle inequality we observe that for any n ≥ n 0 , Applying Lemma A.2 with q = 1/2 (cf. Appendix A.4), we find that hence, by Lemma 4.1, (3) and the finite (2 + τ )-moment condition, I 1 is majorized by an integrable function. Moreover, applying the Cauchy-Schwarz inequality, (4.2) and again Lemma A.2 (with q = 1) yields All in all, this shows the assertion of the lemma.

Main stochastic term
In this section we show the asymptotic normality of the main stochastic term. For this purpose, let P n : B(R) → [0, 1] be the empirical measure given by where δ x : B(R) → {0, 1} denotes the Dirac measure concentrated in x ∈ R. Further, for any v ∈ U(ξ, β 2 ), define the random fields (Z and The following theorem is the main result of this section.
Theorem 4.5. Let the assumptions of Theorem 3.12 be satisfied. Then, as W is regularly growing to infinity, for any v ∈ U(ξ, β 2 ), where N v is a Gaussian random variable with zero expectation and variance σ 2 given in (3.10).
In order to prove Theorem 4.5, we first show some auxiliary statements. We begin with the following representation for the main stochastic term. Lemma 4.6. Let v ∈ U(ξ, β 2 ). Then, under the assumptions of Theorem 3.12, the main stochastic term can be represented by j,v,n , k = 1, 2, given in (4.3) and (4.4).
Now, taking into account thatψ(x) = R e itx P n (dt) andθ(x) = R e itx tP n (dt), Fubini's theorem yields the desired result.
The following lemma justifies the asymptotic variance σ 2 in Theorem 3.12.
as |W | → ∞. Hence, Lemma A.4 yields the assertion of the lemma.
We now can give a proof of Theorem 4.5.
Proof of Theorem 4.5. If σ 2 v = 0, then Lemma 4.7 yields as W is regularly growing to infinity; hence, n −1/2 j∈W Z 0,v,n → 0 in probability. Now, assume that σ 2 v > 0 and choose n 0 ∈ N such that σ 2 v,n > 0 for all n ≥ n 0 (which is indeed possible, since σ 2 v,n → σ 2 v > 0 as n → ∞). For any n ≥ n 0 , let and denote by F n the distribution function of j∈W X j,n . In the proof of Lemma 4.7 we have seen that (X j,n ) j∈Z d is a centered m-dependent random field and E|X j,n | 2+τ ≤ cn −1−τ /2 σ −(2+τ ) v,n for any n ∈ N and a constant c > 0. Hence, applying [8, Theorem 2.6] with p = 2 + τ yields as n → ∞. This completes the proof.

Remainder term
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 ∈ R, j ∈ Z d , let the centered random variables ξ (l) j (t), l = 1, 2, be given by Thenψ − ψ andθ − θ can be rewritten aŝ j (t) and In the sequel, we shortly write ξ (l) (t),ξ (l) (t) for the random fields (ξ  j (t)) j∈Z d , l = 1, 2. Moreover, for any K > 0, we define the random fields ξ For any finite subset V ⊂ Z d and any random field Y = (Y j ) j∈Z d , let Lemma 4.9. Let the assumptions of Theorem 3.12 be satisfied and suppose K ≥ 1. Then and for any t ∈ R, x ≥ 0 and l = 1, 2.

Taking into account that
and P sup Proof. We use the same idea as in the proof of [4, Theorem 2]: divide the interval [−T, T ] by 2J equidistant points (t k ) k=1,...,2J = D, where t k = −T + k T J , k = 1, . . . , 2J. Then, for any t ∈ [−T, T ] such that |t − t k | ≤ T J , we have for any j ∈ Z d that |ξ (l) Hence, by Markov's inequality and Lemma 4.9, for any n ∈ N, we obtain that All in all, we have for any n such that ε n < T 2 , Hence, it follows for any n with ε n < min{1, as n → ∞, where O P denotes the probabilistic order of convergence.
(b) For large n in Theorem 4.11 is understood in the following sense: for any fixed m, there exists n 0 = n 0 (m) such that the bound holds for all n ≥ n 0 . Of course, the function m → n 0 (m) is increasing.
The following corollary is an immediate consequence of Theorem 4.11.

Neglecting the drift γ
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 (Ỹ j ) j∈W given byỸ j = Y j − γ, j ∈ W . Moreover, let ψ * (t) = E[e itỸ0 ] be the characteristic function ofỸ 0 and writê ψ * for its empirical counterpart, i.e.ψ * (t) = 1 n j∈W e itỸj . Then, with the notation we have for any t ∈ R, where θ * (t) = E[Ỹ 0 e itỸ0 ] andθ * (t) = 1 n j∈WỸ j e itỸj . For any v ∈ U(ξ, β 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 ψ * 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 → ∞, since b n = o(n −1/2 ). Moreover, Taking into account that |ψ(x)| = |ψ * (x)|, relation (4.2) with p = 1/2 yields Applying again (4.2) with p = 1 implies as n → ∞; thus, by dominated convergence, we have All in all, this shows that Theorem 3.12 holds for any fixed γ ∈ R.
(c) The proof of Theorem 3.10 in [13] yields that |ψ(x)| coincides with the inverse of the right-hand side in (3.2). This shows part (c).
A.2 Proof of Lemma 3.6 Let v ∈ Image(G) be such that R In order to prove the upper bound in Theorem 3.6, decompose E|L W v − Lv| as follows: .
We estimate parts (I) and (II) seperately. Using the isometry property of F + , we obtain Hence, by the Cauchy-Schwarz inequality we obtain that where the last line follows from (K2) and again by applying the isometry property of F + . For the second part, we find that where the identity |F + [uv 1 ]|(x) = θ(x) ψ(x) was used in the last line. Hence, it remains to bound expression (III). Indeed, applying the triangle inequality followed by the Cauchy-Schwarz inequality and the bounds in [18, Lemma 8.1 and 8.3] yields with constants c 1 , c 2 , c 3 > 0. Hence, by integrability of uv 1 it follows for some constant c > 0. This finishes the proof.
A.3 Proof of Theorem 3.7 Using Assumption 3.1, (4), (K3) and F + [G −1 * ] ∈ L 1 (R) we find that Moreover, applying the same arguments as in the proof of [13, Corollary 3.7], we observe that Hence, if γ = 0, the assertions of the theorem immediately follow by the upper bound in Lemma 3.6. Otherwise, if γ = 0, consider the sample (Ỹ j ) j∈W defined in Section 4.3. Following the computations there, one finds that on the right-hand side of (3.8) the additional term arises. Using G −1 * v ∈ H 1 (R), F + [G −1 * ] ∈ L 1 (R) and (K3) yields that the latter expression can be estimated from above by .
This completes the proof.

A.4 Moment inequalities for m-dependent random fields
In this section, we sum up some moment inequalities that are quite helpful for the proofs in Section 3. We start with the following Bernstein type inequality that is due to [15, p. 316].
Theorem A.1. Let (X j ) j∈V , V ⊂ Z d , be a centered m-dependent random field satisfying 0 < EX 2 j < ∞ and, for some H > 0, shows that lim n→∞ z n = t∈Z d : t ∞ ≤m σ t .