# On a linear functional for infinitely divisible moving average random fields

Given a low-frequency sample of the infinitely divisible moving average random field {∫_R^d f(t-x) Λ(dx), t ∈R^d }, in [1], we proposed an estimator uv_0 for the function R∋ x u(x)v_0(x) = (uv_0)(x), with u(x) = x and 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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## 1. Introduction

Consider a stationary infinitely divisible indepently scattered random measure whose Lévy density is denoted by . For some (known) -integrable function with compact support, let be the corresponding moving average random field. In our recent preprint [1], we proposed an estimator for the function , , based on low frequency observations of , with and a finite subset of .
In this paper, we investigate asymptotic properties of the linear functional as the sample size tends to infinity. It is motivated by the paper of Nickl and Reiss [2], where the authors provide a Donsker type theorem for the Lévy measure of pure jump Lévy processes. Since our observations are -dependent, the classical i.i.d. theory does do not apply here. Instead, we combine results of Chen and Shao [3] for -dependent random fields and ideas of Bulinski and Shashkin [4] with exponential inequalities for weakly dependent random fields (see e.g. [5][6]) in order to prove our limit theorems.
It turns out that under certain regularity assumptions on , is a mean consistent estimator for with a rate of convergence given by , for any that belongs to a subspace of . Moreover, we give conditions such that finite dimensional distributions of the process are asymptotically Gaussian as is regularly growing to infinity.
From a practical point of view, a naturally arising question is wether a proposed model for (or equivalently ) is suitable. Knowing the asymptotic distribution of can be used in order to construct tests for different hypotheses e.g. on regularity assumptions of the model for . Indeed, a behaviour which is naturally induced by the scalar product , is that the class of test functions is growing, when 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 [1]. Section 3 is devoted to asymptotic properties of . Here we discuss our regularity assumptions and state the main results of this paper (Theorems 3.6 and 3.11). Sections 4 and LABEL:proof:clt_multivariate are 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.

## 2. Preliminaries

### 2.1. Notation

Throughout this paper, we use the following notation.

By we denote the Borel -field on the Euclidean space . The Lebesgue measure on is denoted by and we shortly write when we integrate w.r.t. . For any measurable space we denote by , , the space of all -mesurable functions with . Equipped with the norm , becomes a Banach space and even in the case a Hilbert space with scalar product , for any . With (i.e. if ) we denote the space of all real valued bounded functions on . In case we denote by

 Hδ(R)={f∈L2(R): ∫R|F+f|2(x)(1+x2)δdx<∞}

the Sobolev space of order equipped with the Sobolev norm , where is the Fourier transform on . For , is defined by , . Throughout the rest of this paper

denotes a probability space. Note that in this case

is the space of all random variables with finite

-th moment. For an arbitrary set

we introduce furthermore the notation or briefly for its cardinality. Let be the support set of a function . Denote by the diameter of a bounded set .

### 2.2. Regularly growing sets

In this secion, we briefly recall some basic facts about regular growing sets. For a more detailed investigation on this topic, see e.g. [4].

Let

be a vector with positive components. In the sequel, we shortly write

in this situation. Moreover, let

 Π0(a)={x∈Rd, 0

and define for any the shifted block by

 Πj(a)=Π0(a)+ja={x∈Rd, jiai

Clearly forms a partition of . For any , 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 () tends to infinity in Van Hove sense or shortly is VH-growing, if for any

 νd(U−n)→∞andνd(U−n)νd(U+n)→1as n→∞.

For a finite set , define by its boundary, where .
A sequence of finite sets () is called regularly growing (to infinity), if

 |An|→∞,and|∂An||An|→0,as n→∞.

The following result, that connects regularly and VH-growing sequences can be found in [4, p.174].

###### Lemma 2.1.
1. Let () be VH-growing. Then () is regularly growing to infinity.

2. If is a sequence of finite subsets of , regularly growing to infinity, then is VH-grwoing, where .

### 2.3. Infinitely divisible random measures

Subsequently, denote by the collection of all bounded Borel sets in .

Suppose to be an infinitely divisible random measure on some probability space , i.e. a random measure with the following properties:

1. Let be a sequence of disjoint sets in . Then the sequence consists of independent random variables; if, in addition, , then we have almost surely.

2. The random variable has an infinitely divisible distribution for any choice of .

For every , let

denote the characteristic function of the random variable

. Due to the infinite divisibility of the random variable , the characteristic function has a Lévy-Khintchin representation which can, in its most general form, be found in [7, p. 456]. Throughout the rest of the paper we make the additional assumption that the Lévy-Khintchin representation of is of a special form, namely

 φΛ(A)(t)=exp{νd(A)K(t)},A∈E0(Rd),

with

 (2.1) K(t)=ita0−12t2b0+∫R(eitx−1−itx1I[−1,1](x))v0(x)dx,

where denotes the Lebesgue measure on , and are real numbers with and is a Lévy density, i.e. a measurable function which fulfils . The triplet will be referred to as Lévy characteristic of . It uniquely determines the distribution of . This particular structure of the characteristic functions means that the random measure is stationary with control measure given by

 λ(A)=νd(A)⎡⎢⎣|a0|+b0+∫Rmin{1,x2}v0(x)dx⎤⎥⎦for all A∈E0(Rd).

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

1. Let be a real simple function on , where are pairwise disjoint. Then for every we define

 ∫Af(x)Λ(dx)=n∑j=1xjΛ(A∩Aj).
2. A measurable function is said to be -integrable if there exists a sequence of simple functions as in (1) such that holds -almost everywhere and such that, for each , the sequence converges in probability as . In this case we set

 ∫Af(x)Λ(dx)=P-limm→∞∫Af(m)(x)Λ(dx).

A useful characterization for -integrability of a function is given in [7, Theorem 2.7]. Now let be -integrable; then the function is -integrable for every as well. We define the moving average random field 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 above is (strictly) stationary and infinitely divisible and the characteristic function of is given by

 φX(0)(u)=exp(∫RdK(uf(s))ds),

where is the function from (2.1). The argument in the above exponential function can be shown to have a similar structure as ; more precisely, we have

 (2.3) ∫RdK(uf(s))ds=iua1−12u2b1+∫R(eiux−1−iux1I[−1,1](x))v1(x)dx

where and are real numbers with and the function is the Lévy density of . The triplet is again referred to as Lévy characteristic (of ) and determines the distribution of uniquely. A simple computation shows that the triplet is given by the formulas

 a1=∫RdU(f(s))ds,b1=b0∫Rdf2(s)ds, (2.4) v1(x)=∫supp(f)1|f(s)|v0(xf(s))ds;

where denotes the support of and where the function is defined via

 U(u)=u(a0+∫Rx[1I[−1,1](ux)−1I[−1,1](x)]v0(x)dx).

Note that -integrability of immediately implies that . Hence, all integrals above are finite.

For details on the theory of infinitely divisible measures and fields we refer the interested reader to [7].

### 2.4. A plug-in estimation approach for v0

Let the random field be given as in Section 2.3 and define the function by . Suppose further, an estimator for is given. In our recent preprint [1], we provided an estimation approach for based on relation (2.4) which we briefly recall in this section. Therefore, quite a number of notation are required.

Assume that satisfies the integrability condition

 (2.5) ∫supp(f(s))|f(s)|1/2ds<∞.

and define the operator by

 Gv=∫supp(f)sgn(f(s))v(⋅f(s))ds,v∈L2(R).

Moreover, define the isometry by

 (Mv)(x)=|x|1/2v(x),v∈L2(R)

and let the functions and be given by

 mf,+(x)=∫supp(f)sgn(f(s))|f(s)|1/2e−ixlog|f(s)|dsmf,−(x)=∫supp(f)|f(s)|1/2e−ixlog|f(s)|ds.μf(y)={mf,+(log|y|)if y>0,mf,−(log|y|)if y<0.

Multiplying both sides in (2.4) by leads to the equivalent relation

 (2.6) uv1=Guv0.

Suppose and assume that for some ,

 (Uβ) |mf,±(x)|≳11+|x|β,for all x∈R.

Then, the unique solution to equation (2.6) is given by

 uv0=G−1uv1=M−1F−1×(1μfF×Muv1),

cf. [1, Theorem 3.1]. Based on this relation, the paper [1] provides the estimator

 (2.7) ˆuv0=M−1F−1×(1μf,nF×Mˆuv1)=:G−1nˆuv1

for , where is an arbitrary sequence – depending on the sample size – that tends to as , and, the mapping is defined by . Here, denotes the Fourier transform on the multiplicative group , that was shortly introduced in [1, Section 2.2]. A more detailed introduction to harmonic analysis on locally compact abelian groups can be found e.g. in [8].

###### Remark 2.2.

The linear operator defined in (2.7) is bounded with the operator norm , whereas is unbounded in general.

### 2.5. m-dependent random fields

A random field , , defined on some probability space is called -dependent if for some and any finite subsets and of the random vectors and are independent whenever

 ∥u−v∥∞=max{|ui−vi|, i=1,…,d}>m,

for all and . Notice that the random field in (2.2) is -dependent with if has compact support.

## 3. A linear functional for infinitely divisible moving averages

### 3.1. The setting

Let be a stationary infinitely divisible random measure defined on some probability space with characteristic triplet given by , i.e. is purely non-Gaussian. For a known -integrable function let be the infinitely divisible moving average random field defined in Section 2.3.
Fix and suppose is observed on a regular grid with mesh size , i.e. consider the random field given by

 (3.1) Y=(Yj)j∈Zd,Yj=X(jΔ), j∈Zd.

For a finite subset let be a sample drawn from and denote by the cardinality of .

Throughout this paper, for any numbers , , we use the notation if for some constant .

###### Assumption 3.1.

Let the function be given by . We make the following assumptions: for some

1. has compact support;

2. is bounded;

3. ;

4. for all ;

5. such that the function

 (3.2) R∋x↦exp(∫supp(f)∫f(s)x0Im(F+[uv0](y))dyds)

is contained in .

Suppose to be an estimator for (which we precisely define in the next section) based on the sample . Then, using the notation in Section 2.4, we introduce the linear functional

It is the purpose of this paper to investigate asymptotic properties of as the sample size tends to infinity.

### 3.2. An estimator for uv1

In this section we introduce an estimator for the function . Therefore, let denote the characteristic function of . Then, by Assumption 3.1, (2), together with formula (2.3), we find that can be rewritten as

 (3.3) ψ(t)=EeitY0=exp(iγt+∫R(eitx−1)v1(x)dx),t∈R,

for some and the Lévy density given in (2.4). We call the drift parameter or shortly drift of .
Taking derivatives in (3.3) leads to the identity

 −iψ′(t)ψ(t)=γ+F+[uv1](t),t∈R.

Neglecting for the moment, this relation suggests that a natural estimator for is given by

 ˆF+[uv1](t)=^θ(t)~ψ(t),t∈R

with

 ~ψ(t)=1^ψ(t)1{|^(ψ)(t)|>n−1/2},t∈R

and , being the empirical counterparts of and .
Now, consider for any a function with the following properties:

(K1)

;

(K2)

;

(K3)

for all .

Then, for any , we define the estimator for by

 (3.4) ˆuv1(t)=F−1+[ˆF+[uv1]F+[Kb]](t)=12π∫Re−itx^θ(x)~ψ(x)F+[Kb](x)dx,t∈R.
###### Remark 3.2.
1. If is supposed to be a consistent estimator for , it is reasonable to assume that (cf. [9]). In contrast, for the asymptotic results below, the value of is irrelevant. Even if , the functional estimates the intended quantity with given in (3.4) (cf Section 4.3).

2. Choosing yields the estimator that we introduced in [9] and [1], originally designed by Comte and Genon-Catalot [10] in case that is a pure jump Lévy-process.

### 3.3. 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 be given as above and suppose .

1. Let Assumption 3.1, (1) and (2) be satisfied. Then, is bounded. Moreover,

 (3.5) F+[uv1](x)=∫supp(f)f(s)F+[uv0](f(s)x)ds,for all x∈R;

that is, the expression in Assumption 3.1, (4).

2. Let Assumption 3.1, (1) and (3) hold true. Then, (also in case that ).

3. Assumption 3.1, (5) is satisfied, if and only if the function – with given in (3.3) – is contained in for some .

The compact support property in Assumption 3.1, (1) ensures that the random field is -dependent with . In particular, increases when the grid size of the sample is decreasing. Moreover, compact support of together with implies that for all . Consequently, fulfills the integrability condition (2.5). In contrast, if does not have compact support, -integrability only ensures .
Assumption 3.1, (3) is a moment assumption on . More precisely, it is satisfied, if and only if

 E|Λ(A)|2+τ<∞

for all , cf. [11]. By Proposition 3.3, (b), this assumption also implies in our setting.
As a consequence of Proposition 3.3, (a) and (c), Assumption 3.1, (4) ensures regularity of whereas (5) yields polynomial decay of . It was shown in [1, Theorem 3.10] that and are connected via the relation

 |ψ(x)|=exp(−∫x0Im(F+[uv1](y))dy),x∈R;

hence, more regularity of ensures slower decay rates for as . Further results on polynomial decay of infinitely divisible characteristic functions as well as sufficient conditions for this property to hold can be found in [12].

Let us give some examples for and satisfiying Assumption 3.1, (1)–(5).

###### Example 3.4 (Gamma random measure).

Fix and let for any , . Clearly, Assumption 3.1, (2) and (3) is satisfied for any . The Fourier transform of is given by , ; 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 with compact support. Moreover, a simple calculation yields that for any , 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 if

 α:=∫supp(f)max{1,f2(s)b}ds<12.

### 3.4. Consistency of ^LW

In this section, we give an upper bound for the estimation error that allows to derive conditions under which is consistent for the linear functional given by

 Lv=⟨v,uv0⟩,v∈L2(R).

With the notations from Section 2.4, we have that the adjoint operator of is given by

 (3.7) G−1∗v=M−1F−1×(1¯μfF×Mv),v∈Image(G),

where denotes the complex conjugate function of . Moreover, the adjoint of writes as

 G−1∗nv=M−1F−1×(1¯μf,nF×Mv),v∈L2(R),

with . Notice that is a bounded operator whereas is unbounded in general.

With the previous notations we now derive an upper bound for . Therefore, recall condition from Section 2.4.

###### Lemma 3.5.

Let and suppose Assumption 3.1, (1) – (3) hold true for some . Moreover, let condition be satisfied for some and assume to be a function with properties (K1)–(K3). Then

 (3.8) E|^LWv−Lv|≤S√πE|Y0|(nb)1/2∥(G−1∗n−G−1∗)v∥L2(R)+12π⟨|F+[G−1∗v]|,|F+[uv1]||1−F+[Kb]|⟩L2(R)+c⋅S2π√n(√E|Y0|2+∥uv1∥L1(R))∫R|F+[G−1∗v]|(x)|ψ(x)|dx

for any such that and for some constant , where .-

A proof of Lemma 3.5 as well as of Theorem 3.6 below can be found in Appendix.

###### Theorem 3.6.

Fix . Suppose that condition is satisfied for some and let such that ,