Self-similar Random Fields and Rescaled Random Balls Models

DOI 10.1007/s10959-009-0259-x

Self-similar Random Fields and Rescaled Random Balls Models

Hermine Biermé·Anne Estrade·Ingemar Kaj

Received: 11 December 2008 / Revised: 23 July 2009 / Published online: 2 December 2009

© Springer Science+Business Media, LLC 2009

Abstract We study generalized random fields which arise as rescaling limits of spa- tial configurations of uniformly scattered random balls as the mean radius of the balls tends to 0 or infinity. Assuming that the radius distribution has a power-law behavior, we prove that the centered and renormalized random balls field admits a limit with self-similarity properties. Our main result states that all self-similar, translation- and rotation-invariant Gaussian fields can be obtained through a unified zooming proce- dure starting from a random balls model. This approach has to be understood as a microscopic description of macroscopic properties. Under specific assumptions, we also get a Poisson-type asymptotic field. In addition to investigating stationarity and self-similarity properties, we giveL²-representations of the asymptotic generalized random fields viewed as continuous random linear functionals.

Keywords Self-similarity·Generalized random field·Poisson point process· Fractional Poisson field·Fractional Brownian field

Mathematics Subject Classification (2000) Primary 60G60·60G78·Secondary 60G20·60D05·60G55·60G10·60F05

This work was supported by ANR grant “mipomodim” ANR-05-BLAN-0017.

H. Biermé (

Laboratory MAP5, Université Paris Descartes, CNRS UMR 8145, 45 rue des Saints-Pères, 75006 Paris, France

e-mail:[email protected] A. Estrade

e-mail:[email protected]

I. Kaj

Department of Mathematics, Uppsala University, P.O. Box 480, 751 06, Uppsala, Sweden e-mail:[email protected]

Introduction

In this work we construct essentially all Gaussian, translation- and rotation-invariant, H-self-similar generalized random fields onR^din a unified manner as scaling limits of a random balls model. The self-similarity indexH ranges over all ofR\Zand the random balls model is of germ-grain type. It arises by aggregation of spherical grains attached to uniformly scattered germs given by a Poisson point process in d-dimensional space. By a similar scaling procedure, we obtain also non-Gaussian random fields with interesting properties, in particular a model of the type “fractional Poisson field.” Its covariance functional coincides with that of the GaussianH-self- similar field, so that it fulfills a second-order self-similarity property. Although not self-similar in law, this Poisson field presents a property of “aggregate similarity”

which takes into account both Poisson structure and self-similarity.

We observe two distinctly separate behaviors depending on whether the self- similarity indexHbelongs to an interval of type(m, m+1/2)or of type[m−1/2, m) for some integerm. In the first case, the scaling limit applies to random balls models with balls of arbitrarily small radii. In the opposite case, the corresponding germ- grain models have arbitrarily large spherical grains.

The scaling procedure which acts on the random balls model is based on the as- sumption that the grains have random radius, independent and identically distributed, with a distribution having a power-law behavior either in zero or at infinity. The re- sulting configuration of mass, obtained by counting the number of balls that cover any given point in space, suitably centered and normalized, exhibits limit distributions un- der scaling. For the case of the random balls radius distribution being heavy-tailed at infinity, the corresponding scaling operation amounts to zooming out over larger areas of space while renormalizing the mass. In the opposite case, when the radius of balls is given by an intensity with prescribed power-law behavior close to zero, the scaling which is applied entails zooming in successively smaller regions of space. Infinites- imally small microballs will emerge and eventually shape the resulting limit fields.

In particular, our results unify and extend in some directions the previous works on similar topics in Kaj et al. [15] (caseH∈(−d/2,0)) and Biermé and Estrade [4]

(caseH∈(0,1/2)). Preliminary and less general versions of some of the results pre- sented here have appeared in Biermé et al. [5] (caseH∈(−d/2,0)∪(0,1/2)). Let us emphasize that the main novelty of this paper is the extension to any noninteger values ofHand the complete description of the asymptotic fields.

Dobrushin [9] characterized the stationary self-similar Gaussian generalized ran- dom fields in their spectral form. In this work we obtain the subclass of such random fields that are isotropic, since the random balls models under consideration are rota- tionally symmetric. In order to obtain the whole range of self-similarity behavior, it is necessary to work not only with stationary random fields but with the wider class of generalized random fields with stationary increments or stationarynth increments.

In this sense our approach also links to the line of work initiated by Matheron [18].

The paper is organized as follows. After having introduced the modeling frame- work and the setting of the investigation, we discuss in Sect.2some principles for scaling limit analysis and state two main results, which cover a Gaussian limit regime and a Poisson limit regime. Section3is devoted to the properties of the limiting ran-

dom fields: stationarity and self- or aggregate-similarity. The main results, in partic- ular Theorem4.7, are presented in Sect.4with the study of all self-similar isotropic stationary generalized random fields. In particular we prove that all such Gaussian fields arise as scaling limits of the random balls model. In Sect.5we give a pointwise representation of the generalized self-similar fields with positive self-similarity index H >0 and discuss a few explicit examples.

1 Setting

We present first a unified framework which includes and extends both of the distinct modeling scenarios studied in [15] and [4], respectively. LetB(x, r)denote the ball inR^dwith center atxand radiusrand consider a family of grainsX_j+B(0, Rj)in R^dgenerated by a Poisson point process(X_j, R_j)_j inR^d×R⁺. Equivalently, we let N (dx,dr)be a Poisson random measure onR^d×R⁺and associate with each random point(x, r)∈R^d×R⁺the random ballB(x, r). We assume that the intensity measure ofN is given byκdx F (dr),whereκ is a positive constant andF is a nonnegative measure onR⁺,σ-finite on(0,+∞). Moreover, we assume throughout the paper that the ball radius intensityF (dr)is such that

R⁺r^dF (dr) <+∞. (1)

Note that if F is a probability measure, this assumption implies that the expected volume of a ball is finite.

For measurable setsA⊂R^d×R⁺, we letN (A)=

AN (dx,dr)denote the num- ber of balls with random location and radius(x, r)contained inAand view the values ofA→N (A)as integer-valued random variables on a probability space(,A,P).

We recall the basic facts (see [17], Chap. 10 for instance) thatN (A)is Poisson distrib- uted with mean

Aκdx F (dr)(if the integral diverges, thenN (A)is countably infi- nite with probability one) and that ifA1, . . . , A_nare disjoint, thenN (A1), . . . , N (A_n) are independent. We also recall that for measurable functionsk:R^d×R⁺→R, the stochastic integral

k(x, r) N (dx,dr)ofkwith respect toN existsP-a.s. if and only

if

R^d×R⁺min(|k(x, r)|,1)dx F (dr) <∞. (2) 1.1 Power-law Assumption

Forβ=d, we introduce the following asymptotic power-law assumption for the be- havior ofF near 0 or at infinity:

A(β): F (dr)=f (r)dr withf (r)∼r^−β−1asr→0^d−β, where by convention 0^α=0 ifα >0 and 0^α= +∞ifα <0.

The range of parameter values under consideration will bed−1< β <2d. Then, according to (1), under assumption A(β), it is natural to consider the asymptotic behavior ofF near 0 ford−1< β < dand at infinity ford < β <2d.

1.2 Random Field

We consider random fields defined on a space of measures, in the same spirit as the random functionals of [15] or the generalized random fields of [3]. LetMdenote the space of signed measuresμonR^dwith finite total variation

μ := |μ| R^d

<∞, (3)

where|μ|is the total variation measure ofμ, and · is a norm onM(see, e.g., [21], p. 161). For anyμ∈M,μ(B(x, r))is a measurable function onR^d×R⁺for which

R^d×R⁺|μ(B(x, r))|dx F (dr)≤v_d|μ| R^d

R⁺r^dF (dr) <+∞, (4) in view of (1), wherevdis the Lebesgue measure of the unit ball inR^d. In particular, (2) applies withk(x, r)=μ(B(x, r)). We may hence introduce a generalized random fieldXdefined onMby

X(μ)=

R^d×R⁺μ(B(x, r))N (dx,dr), μ∈M. (5) Condition (4) is even sufficient and necessary forX(μ)to have finite expected value, and in this case

EX(μ)=

R^d×R⁺μ(B(x, r))κdx F (dr)=κvdμ R^d

R⁺r^dF (dr).

Let us also note that the random fieldXis linear on each vectorial subspace ofMin the sense that for allμ1, . . . , μ_n∈Manda1, . . . , a_n∈R, almost surely,

X(a1μ1+ · · · +a_nμ_n)=a1X(μ1)+ · · · +a_nX(μ_n).

Furthermore the characteristic function ofX(μ)is given by (see [17], Lemma 10.2) E

e^{it X(μ)}

=exp

R^d×R⁺

eit μ(B(x,r))−1

κdx F (dr)

, t∈R. (6) Our first proposition adds to this a simple topological structure.

Proposition 1.1 The random fieldX:(M, · )→(L²(,A,P), · 2)is a con- tinuous random linear functional, where · is given by (3), and · 2is the usual norm onL²(,A,P).

Proof Letμ∈M. The random variableX(μ)is inL²(,A,P), and so Xcan be considered as a linear functionalX:M→L²(,A,P). Moreover, for anyμ∈M, by Fubini’s theorem,

Var(X(μ))=

R^d×R⁺μ(B(x, r))²κdx F (dr)

≤κμ

R^d×R^d|μ(B(x, r))|dx F (dr) (7)

≤κ v_d

R⁺r^dF (dr)

μ²<∞.

Similarly, |E(X(μ))| ≤κ v_d(_+∞

0 r^dF (dr))μ. Therefore, according to (1), one can find a positive constantc_d>0 such that

X(μ)2=

Var(X(μ))+E(X(μ))²≤c_dμ,

which shows the continuity ofX.

The random linear functional X−E(X) is also a continuous linear functional from(M, · )to(L²(,A,P), · 2). The corresponding subordinated norm of X−E(X)is given by

|||X−E(X)||| = sup

μ≤1X(μ)−E(X(μ))2= sup

μ≤1

Var(X(μ)).

For μ = δ0, the Dirac mass at the origin of R^d, we get Var(X(δ0)) = κ v_d(

R⁺r^dF (dr))and may conclude in view of (7) that

|||X−E(X)||| =

κ vd

R⁺r^dF (dr)

. (8)

2 Scaling Limit

2.1 Scaled Random Fields

Let us introduce now the notion of “scaling,” by which we indicate an action: a change of scale acts on the size of the grains. The scaling procedure performed in [15] acts on grains of volumev changed by shrinking into grains of volumeρ v with small parameterρ(“small scaling” behavior). The same is performed in [4] in the context of a homogenization, but the scaling acts in the opposite way: the radiirof grains are changed intor/ε(which is a “large scaling” behavior). To cover both mechanisms we introduce the random field which is obtained by applying the rescaling of measures μ→μ^ρ, whereμ^ρ(B)=μ(ρB)forρ >0 and measurable subsetsBofR^d. Let us denote byF_ρ(dr)the image measure ofF (dr)by the change of scaler→ρr and remark that

X μ^ρ

=

R^d×R⁺μ(B(x, r))N

dρ⁻¹x,dρ⁻¹r

, ∀μ∈M,

where the intensity measure ofN (dρ⁻¹x,dρ⁻¹r) isκ ρ^−ddx Fρ(dr). It is natural from this viewpoint to haveμrepresenting an observation window and interpret lim- itsρ→0 as zoom-out and limitsρ→ ∞as zoom-in of the random configurations of balls in space.

Let us multiply the intensity measure byλ/κ (λ >0) and consider the associated random field onMgiven by

R^d×R⁺μ(B(x, r))Nλ,ρ(dx,dr),

where N_λ,ρ(dx,dr) is the Poisson random measure with intensity measure λdx Fρ(dr)andμ∈M. Choosingλ=κρ^−d, this random field has the same law as {X(μ^ρ);μ∈M}. Results are expected concerning the asymptotic behavior of this scaled random balls model under hypothesis A(β)as ρ→0 orρ→ +∞. We chooseρ as the basic model parameter, considerλ=λ(ρ) as a function ofρ, and define onMthe random field

Xρ(μ)=

R^d×R⁺μ(B(x, r))Nλ(ρ),ρ(dx,dr). (9) Then, we are looking for a normalization termn(ρ)such that the centered field con- verges in distribution,

Xρ(.)−E(Xρ(.)) n(ρ)

→fddW (.), (10)

and we are interested in the nature of the limit fieldW. The convergence (10) holds whenever

E

exp

iXρ(μ)−E(Xρ(μ)) n(ρ)

→E(exp(iW (μ)))

for allμin a convenient subspace ofM. A scaling analysis of power law tails reveals that under A(β)we expect

Var(Xρ(μ))∼λ(ρ)ρ^βVar(X(μ)), ρ→0^β−d,

which suggests the asymptotic relation n(ρ)² ∼λ(ρ) ρ^β to obtain the conver- gence of (10) in(L²(,A,P),·2). However, in view of (8), the norm of(X_ρ− E(X_ρ))/n(ρ) as a continuous linear functional from (M,·) to (L²(,A,P),

·2)is given by

X_ρ−E(X_ρ) n(ρ)

=

vd

R⁺r^dF (dr)

λ(ρ)ρ^d

n(ρ)² . (11)

In particular, (11) is not bounded forn(ρ)²=λ(ρ)ρ^βasρ→0^β−d, and the Banach–

Steinhaus theorem states that there exists a dense subset ofMon which the rescaled process(X_ρ(μ)−E(X_ρ(μ)))/ λ(ρ)ρ^β cannot converge in(L²(,A,P), · 2).

Therefore, we study in the sequel the convergence (10) on strict subspaces ofM.

This will allow us to get in the limit a continuous linear functional taking values in(L²(,A,P), · 2), despite the fact that the convergence holds only for finite- dimensional distributions.

2.2 Gaussian Limit Regime

Forβ=d, let us define the space of measures M^β=

μ∈M: ∃αs.t.α < β < dord < β < α and

R^d×R^d|z−z|^d−α|μ|(dz)|μ|(dz) <+∞

,

where|z|denotes the Euclidean norm ofz∈R^d, and|μ|is the total variation measure ofμ∈M. We remark that the integral assumption is a finite Riesz energy assumption forβ > d and thatM^β = {0}whenβ≥2d. In both casesd−1< β < d andd <

β <2d, ifμ∈Msatisfies

R^d×R^d|z−z|^d−α|μ|(dz)|μ|(dz) <+∞

for someα(α < β < d andd < β < α, respectively), then the same holds for anyγ betweenβ andα. In particular, for anyμ∈M^β,

R^d×R^d|z−z|^d−β|μ|(dz)|μ|(dz) <+∞.

We also introduce the subspace of finite signed measures of vanishing total mass, M1=

μ∈M:

R^dμ(dz)=0

, and consider the subspaces

Mβ=

M^β ford < β <2d,

M^β∩M1 ford−1< β < d. (12) Theorem 2.1 Letd−1< β <2d withβ=d. LetF be a nonnegative measure on R⁺which satisfies A(β). For all positive functionsλsuch thatλ(ρ)ρ^β −→

ρ→0^β−d+∞, the limit

X_ρ(μ)−E(X_ρ(μ)) λ(ρ)ρ^β

−→fdd ρ→0^β−d

Wβ(μ)

holds for allμ∈Mβ, in the sense of finite-dimensional distributions of the random functionals. HereW_βis the centered Gaussian random linear functional onMβwith covariance functional

Cov(Wβ(μ), W_β(ν))=E(W_β(μ)W_β(ν))=c_β

R^d×R^d|z−z|^d−βμ(dz)ν(dz) (13) for a constantcβ only depending onβ.

Remark 2.2 Equation (13) defines a covariance function, called generalized covari- ance function in [18]. The value of the constantc_β is given by (19) below.

Proof We begin with two lemmas. The first lemma describes the covariance function and is based on some technical estimates for the intersection volume of two balls.

The second one, inspired by Lemma 1 of [15], stands for Lebesgue’s theorem with assumptions that are well adapted to the present setting.

Lemma 2.3 Letd−1< β <2d withβ =d. There exists a real constantcβ such that for allμ∈Mβ,

0<

R^d×R⁺μ(B(x, r))²r^−β−1drdx=c_β

R^d×R^d|z−z|^d−βμ(dz)μ(dz) <+∞. Proof Let us introduce the functionγ defined on[0,∞)by

γ (u)=Lebesgue measure ofB(0,1)∩B(ue,1) (14) for any unit vector e∈R^d. The functionγis decreasing, supported on[0,2], bounded byγ (0)=v_d, continuous on[0,2], and smooth on(0,2). Defineγ_βas

γ_β(u)=

γ (u)−γ (0), d−1< β < d, γ (u), d < β <2d.

We notice that ford−1< β < d,|γβ(u)| ≤γ (0)and|γβ(u)| ≤sup_v>0|γ(v)|u.

Hence, for some constantC >0,|γ_β(u)| ≤C u^d−α for any 0≤d−α≤1, that is, anyαin[d−1, d]. Ford < β <2d, one can findC >0 such that|γ_β(u)| ≤C u^d−α for anyα≥β. In particular, we may takeα such thatd−1< α < β for the case d−1< β < d andαsuch thatβ < α <2d ford < β <2d, and for both cases, we have aC >0 with

∀u >0, |γ_β(u)| ≤Cu^d−α. (15) Step 1. Forμ∈Mβ, let us prove that

R^d×R⁺μ(B(x, r))²r^−β−1drdx <+∞. We introduce the functionϕdefined by

ϕ(r)=

R^dμ(B(x, r))²dx, r >0. (16) Using successively Fubini’s theorem, homogeneity, and (14), we get

ϕ(r)=

R^d×R^d

R^d1B(z,r)(x)1B(z,r)(x)dx

μ(dz)μ(dz)

=r^d

R^d×R^dγ (|z−z|/r)μ(dz)μ(dz).

Thereforeϕ(r)≤γ (0)|μ|(R^d)²r^d. Moreover, sinceμ∈Mβ, ϕ(r)=r^d

R^d×R^dγ_β(|z−z|/r)μ(dz)μ(dz), (17)

and we can chooseαsuch that

R^d×R^d|z−z|^d−α|μ|(dz)|μ|(dz) <+∞and (15) holds. Then

ϕ(r)≤Cr^α

R^d×R^d|z−z|^d−α|μ|(dz)|μ|(dz).

Finally, one can findC >0 such that

ϕ(r)≤Cmin r^d, r^α

(18)

and _+∞

0

ϕ(r)r^−β−1dr=

R^d×R⁺μ(B(x, r))²r^−β−1drdx <+∞. Step 2. We prove the equality stated in the lemma, which is

_+∞

0

ϕ(r)r^−β−1dr=c_β

R^d×R^d|z−z|^d−βμ(dz)μ(dz),

using the previous notation. To this end we wish to replaceϕby (17) in the left-hand side integral. Using estimates (15) on|γβ|, one can show that the integral

Iβ(u):=

R⁺γβ(u/r)r^d−β−1dr

is well defined for allu∈R₊. Furthermore,I_β is homogeneous of orderd−β so that

∀u >0, I_β(u)=I_β(1)u^d−β. This proves that

_+∞

0

ϕ(r)r^−β−1dr=I_β(1)

R^d×R^d|z−z|^d−βμ(dz)μ(dz), which completes the proof of the lemma with

c_β=I_β(1)=

R⁺γ_β(1/r)r^d−β−1dr. (19) Now let us state a second lemma, which is the main tool to establish our scaling limit results.

Lemma 2.4 LetF be a nonnegative measure onR⁺satisfying A(β)forβ=d.

(i) Assume thatgis a continuous function onR⁺such that for some 0< p < β < q, there existsC >0 such that

|g(r)| ≤Cmin r^q, r^p

.

Then

R⁺g(r)F_ρ(dr)∼ρ^β

R⁺g(r)r^−β−1dr asρ→0^β−d. (ii) Letg_ρ be a family of continuous functions onR⁺. Assume that

lim

ρ→0^β−d

ρ^βgρ(r)=0 and ρ^β|gρ(r)| ≤Cmin r^p, r^q for some 0< p < β < qandC >0. Then

lim

ρ→0^β−d

R⁺g_ρ(r)F_ρ(dr)=0.

Proof (i) Let us assume, for instance, thatβ < d (the proof of the case β > d is similar and can be found in [15]). Let ε >0. Since F satisfies A(β), there exists δ >0 such that

r < δ ⇒ f (r)−r^−β−1≤εr^−β−1. (20) Let us remark that the assumptions ongensure that

_+∞

0 |g(r)|r^−β−1dr <+∞. On the one hand, since_δρ

0 g(r)F_ρ(dr)=_δρ

0 g(r)f (_ρ^r)^dr_ρ, we get by (20) _δρ

0

g(r)Fρ(dr)−ρ^β _δρ

0

g(r)r^−β−1dr ≤ερ^β

R⁺|g(r)|r^−β−1dr.

On the other hand, forδρ >1, since|g(r)| ≤Cr^p, _∞

δρ

g(r)F_ρ(dr)−ρ^β _∞

δρ

g(r)r^−β−1dr

≤CC₁(δ)ρ^p+ C

β−pδ^p−βρ^p, whereC1(δ)=_+∞

δ r^pF (dr)≤δ^p−d

R⁺r^dF (dr) <∞. Sincep < β, we obtain (i).

(ii) We follow the same lines as for (i) and can assume similarly thatβ < d. Since F satisfies A(β), there existsδ >0 such that

r < δ ⇒ |f (r)| ≤2r^−β−1. (21) The assumptions ong_ρensure that for allρ >0,

_+∞

0

ρ^β|gρ(r)|r^−β−1dr <+∞ with lim

ρ→+∞

_∞

0

ρ^β|gρ(r)|r^−β−1dr=0, by Lebesgue’s theorem. Since_δρ

0 g_ρ(r)F_ρ(dr)=_δρ

0 g_ρ(r)f (_ρ^r)^dr_ρ, we get by (21) _δρ

0

g_ρ(r)F_ρ(dr) ≤2

_∞

0

ρ^β|g_ρ(r)|r^−β−1dr.

Therefore,

ρ→+∞lim _δρ

0

g_ρ(r)F_ρ(dr)=0. (22)

Moreover, forδρ >1, sinceC1(δ)=_+∞

δ r^pF (dr) <+∞and|gρ(r)| ≤Cρ^−βr^p, _∞

δρ

g_ρ(r)F_ρ(dr)

≤Cρ^−β _∞

δρ

r^pF_ρ(dr)≤CC₁(δ)ρ^−(β−p). (23) We conclude the proof using (22) and (23), sincep < β.

We start now with the proof of Theorem2.1. Let us denote n(ρ):= λ(ρ)ρ^β

and define the functionϕρ onR⁺by ϕρ(r)=

R^dΨ

μ(B(x, r)) n(ρ)

dx, where

Ψ (v)=e^iv−1−iv. (24)

According to (6), the characteristic function of the normalized field (X_ρ(.)− E(X_ρ(.)))/n(ρ)is given by

E

exp

iX_ρ(μ)−E(X_ρ(μ)) n(ρ)

=exp

R⁺λ(ρ)ϕ_ρ(r)F_ρ(dr)

.

By assumption,n(ρ)tends to+∞as ρ→0^β−d so thatΨ (^μ(B(x,r))_n(ρ) )behaves like

−¹₂(^μ(B(x,r))_n(ρ) )². Therefore, we write

R⁺λ(ρ)ϕ_ρ(r)F_ρ(dr)= −1 2

R⁺ϕ(r)λ(ρ)n(ρ)⁻²F_ρ(dr)+

R⁺_ρ(r)F_ρ(dr), (25) where the functionϕis introduced in (16), and

ρ(r)=λ(ρ)ϕρ(r)+1

2λ(ρ)n(ρ)⁻²ϕ(r) (26)

=λ(ρ)

R^d

Ψ

μ(B(x, r)) n(ρ)

+1

2

μ(B(x, r) n(ρ)

2 dx.

Since μ∈ Mβ, the function ϕ is continuous on R⁺ and satisfies (18). Thus, by Lemma 2.4(i), the first term of the right-hand side of (25) converges to

−¹₂

R⁺ϕ(r)r^−β−1dr. Moreover, by Lemma2.3, we obtain lim

ρ→0^β−d

R⁺ϕ(r)λ(ρ)n(ρ)⁻²F_ρ(dr)=c_β

R^d×R^d|z−z|^d−βμ(dz)μ(dz).

For the second term, let us verify that_ρ given by (26) satisfies the assumptions of Lemma2.4(ii). First, let us remark that the function_ρis continuous onR⁺since μ∈M. Because|Ψ (v)−(−^v₂²)| ≤^|v₆^|3 and

R^d|μ(B(x, r))|³dx≤ μ²

R^d|μ(B(x, r))|dx≤v_dμ³r^d, we also check that

λ(ρ)⁻¹n(ρ)²ρ(r)≤1

6vdμ³n(ρ)⁻¹r^d. Finally, since|Ψ (v)| ≤^|v₂^|2, by (18) there existsC >0 such that

λ(ρ)⁻¹n(ρ)²_ρ(r)≤Cr^α for someαwith(α−β)(β−d) >0. Therefore,

R⁺ρ(r)Fρ(dr)tends to 0 accord- ing to Lemma2.4(ii), and so

lim

ρ→0^β−dE

exp

iX_ρ(μ)−E(X_ρ(μ)) n(ρ)

=exp

−1 2cβ

R^d×R^d|z−z|^d−βμ(dz)μ(dz)

.

Hence(X_ρ(μ)−E(X_ρ(μ)))/n(ρ)converges in distribution to the centered Gaussian random variableW (μ)whose variance is equal to

E W (μ)²

=c_β

R^d×R^d|z−z|^d−βμ(dz)μ(dz).

By linearity, the covariance ofW satisfies (13).

With similar arguments, we can state a further scaling result leading to a non- Gaussian limit.

2.3 Poisson Limit Regime

In this section we keep the notation introduced in Sect.2.2for the Gaussian limit regime.

Theorem 2.5 Letd −1< β <2d with β=d. Let F be a nonnegative measure onR⁺satisfying A(β). For all positive functionsλsuch thatλ(ρ)ρ^β −→

ρ→0^β−d

a^d−β for somea >0, we have, in the sense of finite-dimensional distributions of random functionals, the scaling limit

X_ρ(μ)−E(X_ρ(μ))^fdd→J_β(μ_a)

for allμ∈Mβ. HereJ_βis the centered random linear functional onMβ defined as J_β(μ)=

R^d×R⁺μ(B(x, r))N_β(dx,dr),

whereN_βis a compensated Poisson random measure with intensity dx r^−β−1dr, and μ_ais defined byμ_a(A)=μ(a⁻¹A).

Proof Let us recall that a compensated Poisson measure N of intensity n is such that N+n is a Poisson measure of intensity n. Therefore, the stochastic integral k(x, r)N (dx, dr) of a measurable functionk:R^d×R⁺→R with respect to a compensated Poisson measureNof intensitynexistsP-a.s. if and only if

R^d×R⁺min

|k(x, r)|, k(x, r)²

n(dx,dr) <∞ (27) (see [17], Theorem 10.15 for instance).

By Lemma2.3, using once again the functionϕ introduced in (16), for allμ∈ Mβ, we have

R^d

R⁺μ(B(x, r))²r^−β−1drdx=

R⁺ϕ(r)r^−β−1dr <+∞.

Hence, in view of (27) withn(dx,dr)=dx r^−β−1drandk(x, r)=μ(B(x, r)), the random fieldJ_β is well defined onMβ, with characteristic function

E(exp(iJβ(μ)))=exp

R⁺×R^dΨ (μ(B(x, r)))dx r^−β−1dr

, (28)

whereΨ is given by (24).

On the other hand, the characteristic function for the centered Poisson random balls model equals

E(exp(i(Xρ(μ)−E(X_ρ(μ))))=exp

R⁺×R^dΨ (μ(B(x, r)))dx λ(ρ)Fρ(dr)

. Define forr >0,

ϕ(r)=

R^dΨ (μ(B(x, r)))dx.

Forμ∈Mβ, using|Ψ (v)| ≤ |v|²/2 and (18), we get that there existsC >0 such that

|ϕ(r)| ≤Cmin r^d, r^α for someαwith(α−β)(β−d) >0. Thus, by Lemma2.4(i),

R⁺λ(ρ)ϕ(r)F_ρ(dr) ∼

ρ→0^β−d

a^d−β _∞

0

ϕ(r)r^−β−1dr,

and hence, lim

ρ→0^β−dE(exp(i(Xρ(μ)−E(Xρ(μ))))=exp

a^d−β

R⁺ϕ(r) r^−β−1dr

.

Finally, it is sufficient to remark that a^d−β

R⁺ϕ(r)r^−β−1dr=a^d

R⁺ϕ(a⁻¹r)r^−β−1dr with

a^dϕ a⁻¹r

=a^d

R^dΨ μ

B

x, a⁻¹r dx=

R^dΨ (μ_a(B(x, r)))dx, to obtain

lim

ρ→0^β−dE(exp(i(Xρ(μ)−E(X_ρ(μ)))))=E(exp(iJβ(μ_a))).

Lemma2.3and (13) yield the following remark.

Remark 2.6 The covariance function ofJ_βis given for allμ, ν∈Mβ by Cov(Jβ(μ), J_β(ν))=

R^d×R⁺μ(B(x, r))ν(B(x, r))dx r^−β−1dr

=c_β

R^d×R^d|z−z|^d−βμ(dz)ν(dz), and soJβ andWβ have the same covariance function onMβ.

3 Properties of the Limiting Random Generalized Fields

In this section we discuss some of the main properties of the fields we obtain as scal- ing limits. The limits inherit from the random balls model a stationarity property and acquire, due to the nature of the performed scaling, certain self-similarity properties.

3.1 Stationarity

Following the same ideas as in [9] or [18], we define a notion of stationarity which characterizes the translation invariance of a random linear functional over a subset of signed measures. We say as usual that a subspaceS⊂M is closed for trans- lations if, for anyμ∈S and anys∈R^d, we haveτsμ∈S, whereτsμ is defined byτ_sμ(A)=μ(A−s), for any Borel set A. To provide a more general framework for stationary random fields, we introduce the following subspaces of measures with

vanishing moments. For anyn∈N {0}, denote byMn the subspace of measures μ∈Msuch that

R^d|z|ⁿ⁻¹|μ|(dz) <+∞which satisfy

R^dz^jμ(dz)=

R^dz₁^j1· · ·z^j_d^dμ(dz)=0 (29) for allj =(j1, . . . , jd)∈N^d with 0≤j1+ · · · +jd< n(see [18], where similar spaces of measures are introduced). Here, the classM1 was already used for the setting of Theorem2.1. For convenience, we also putM0=M. A simple but tedious computation shows that whenμ∈Mnsatisfies

R^d|z|²ⁿ⁻²|μ|(dz) <+∞forn≥1,

then

R^d×R^d|z−z|^2kμ(dz)μ(dz)=0, 0≤k < n.

In particular, the subspacesMndefined by (29) are closed under translations for any n∈N.

Definition 3.1 Letn∈N. LetX be a random field defined on a subspaceS⊂Mn

closed for translations. The fieldXis translation invariant if

∀μ∈S, ∀s∈R^d, X(τsμ)^fdd=X(μ). (30) More precisely, one says that X is stationary when n=0 and has stationary nth increments whenn >0.

It follows that ifXhas stationarynth increments on a subspaceS⊂Mn, then its restriction onS∩Mn+1⊂Mn+1has stationary(n+1)th increments. This termi- nology comes from [9], whereS=S(R^d)is the Schwartz space. In this setting the generalized fieldXhas stationarynth increments if all its partial derivatives of order nare stationary.

By the translation invariance of the Lebesgue measure, for anyρ >0, the random fieldXρ defined by (9) is stationary onM. The fieldsWβ andJβ obtained as limit fields onMβ in Theorem2.1and Theorem2.5are not defined on the full spaceM.

ButMβ is closed for translations. Therefore, when considering the limiting random fields onMβ, one has the following property.

Proposition 3.2 Letd−1< β <2d withβ=d. ThenW_β andJ_β are translation invariant onMβ.

In other words, from (12),W_β andJ_β defined onMβ are both stationary ifd <

β <2d, and they have stationary first increments ifd−1< β < d.

3.2 Self-similarity

Leta >0 and denote byμ_a the dilated measure defined byμ_a(A)=μ(a⁻¹A)for any Borel setA. A subspace S⊂Mis said to be closed for dilations if, for any μ∈Sand anya >0, we haveμ_a∈S. The following definition extends the standard definition of self-similarity for pointwise defined random fields.

Definition 3.3 LetH∈R. A random fieldX, defined on a subspaceS ofMwhich is closed for dilations, is said to be self-similar with indexH if

∀μ∈S, ∀a >0, X(μ_a)^fdd=a^HX(μ).

Once noticed thatMβ is closed for dilations and observing the consequence of dilation on the covariance ofWβ, the following property is straightforward.

Proposition 3.4 The fieldWβ, defined onMβ, is self-similar with indexH=^d−₂^β that runs over(−d/2,1/2)\ {0}.

In contrast to the Gaussian fieldW_β, the Poisson limit fieldJ_β is not self-similar.

A similarity property which applies in great generality to long-range dependent processes is discussed in [14]. The following is a version for spatial random fields.

Definition 3.5 A random fieldXwithEX=0, defined on a subspaceSofMwhich is closed for dilations, is said to be aggregate-similar if there exists a sequence of positive real numbers(am)m≥1such that

∀μ∈S, ∀m≥1, X(μ_am)^fdd= m i=1

Xⁱ(μ),

where(Xⁱ)_i≥1are i.i.d. copies ofX.

Thus, a random field is aggregate-similar if the pathμa_m→X(μa_m), as we trace along the sequence of dilations given byam, passes all aggregates_m

i=1Xⁱ ofX, in the distributional sense. We may write, equivalently,

∀μ∈S, ∀m≥1, X(μ)^fdd= m i=1

Xⁱ(μ_a−1 m ),

which immediately shows that an aggregate-similar random field is infinitely divisi- ble.

Any self-similar zero-mean Gaussian random field is aggregate-similar. Indeed, ifX_H is Gaussian withEX_H =0 and self-similar with indexH, then lettinga_m= m^1/2H, we have

XH(μa_m) ^fdd= m^1/2XH(μ)^fdd= m

i=1

Xⁱ_H(μ), m≥1. (31)

In particular,W_β is aggregate-similar on Mβ with respect to the sequence a_m= m^1/(d−β). Ford−1< β < d, we havea_m⁻¹→0, and henceμ_amrepresents a zoom- in ofW_β asm→ ∞. This is in contrast to the cased < β <2d, for whicha_m⁻¹→ ∞.

Consequently, the succession of aggregates_m

i=1W_βⁱ(μ) of W_β(μ)appears as the sequence of measuresμ_amperforms a zoom-out, in the limitm→ ∞.

Turning next to the non-Gaussian fieldJ_β, by (28),

logE(exp(iJβ(μ_a)))=a^d−β logE(exp(iJβ(μ))).

Thus,J_β is aggregate-similar with respect toa_mgiven bya^d_m^−β=m. This property provides an interpretation of the dilation parameterain Theorem2.5. If we assume in the theorem thatλ(ρ)ρ^β →a^d_m^−β=masρ^β−d→0 for arbitrarym≥1, then

Xρ(μ)−E(Xρ(μ))→^fddJβ(μa_m)^fdd= m

i=1

J_βⁱ(μ).

The guiding asymptotic quantityλρ^β may be interpreted as the expected number of very large (β > d) or very small (β < d) balls which cover a point asymptotically.

Thus, the more of such extreme grains are allowed asymptotically, the larger number of i.i.d. copies of the basic fieldJβ appears in the limit.

We may continue this line of reasoning by providing a limit result forJβ(μa_m)as m→ ∞. In view of Theorems2.5and2.1, this result is not at all surprising.

Proposition 3.6 Asa^d−β→ ∞, for allμinMβ, 1

a^(d−β)/2J_β(μ_a)^fdd→ W_β(μ).

Proof Consider the subsequence a_m = m^1/(d−β). It follows immediately from aggregate-similarity and the central limit theorem that

1 a_m^(d−β)/2

J_β(μ_am)^fdd= 1

√m m i=1

J_βⁱ(μ)→^fddW_β(μ), m→ ∞,

sinceJ_β(μ)andW_β(μ)have the same variance. A standard argument completes the proof of convergence in distribution along an arbitrary sequence.

4 Self-similar Random Fields of Arbitrary Order

We consider in this section an extension of our methods in order to obtain random fields with the self-similarity property for any indexH∈R\Z. To state our main results, Theorems4.7and4.8, a preliminary study of self-similar random fields of arbitrary order is required.

4.1 Dobrushin’s Characterization of Self-similar Random Fields

Dobrushin [9] gives a complete description of Gaussian translation-invariant self- similar generalized random fields onR^d. For this purpose, he considers continuous random linear functionals of S(R^d), where S(R^d) is the topological dual of the