Generalized operator-scaling random ball model

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Generalized operator-scaling random ball model

Hermine Biermé, Olivier Durieu, Yizao Wang

To cite this version:

Hermine Biermé, Olivier Durieu, Yizao Wang. Generalized operator-scaling random ball model.

ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada, 2018, �10.30757/ALEA.v15-52�. �hal-01828370v2�

HERMINE BIERMÉ, OLIVIER DURIEU, AND YIZAO WANG

Abstract. This article introduces the operator-scaling random ball model, generalizing the isotropic random ball models investigated recently in the literature to anisotropic setup.

The model is introduced as a generalized random field and results on weak convergence are established in the space of tempered distributions.

1. Introduction

In the past ten years, random ball models have appeared as a simple and yet flexible class of random fields that characterize various types of spatial dependence structures [6, 7, 11–

13, 21, 22, 25, 36]. In particular, in several regimes, their scaling limits are self-similar and with long-range dependence [1, 37, 39]. Such properties are desirable when modeling various real world phenomena and thus such results have a broad range of applications.

In words, a random ball model consists in a collection of random balls in R^d with loca- tions following a homogeneous Poisson point process and with independent and identically distributed random radius and weights. Thus, each realization of random balls on the space can be naturally viewed as a linear functional on an appropriate space of test functions. As- ymptotic behaviors are then of interest, when all the balls are simultaneously rescaled by a parameterρ, and at the same time the intensity of balls also changes with respect toρ. Under mild assumption on the distribution of the radius, limit theorems can be established forρÑ0 or ρ Ñ 8, corresponding to the zoom-out or zoom-in cases respectively. In both cases, the qualitative behavior of the limit random fields, whether exhibiting spatial dependence or not, depends on whether the random balls are dense or sparse in the limit, in certain sense to be specified below.

The random ball models can be viewed as generalizations of certain one-dimensional models based on Poisson point processes that appeared in the study of Internet traffics, see for example [26, 35] and references therein. However, the extension to high dimensions presents new technical challenges, and should not be viewed as simple generalization of the one-dimensional results. In particular, the developments until now have two main limitations. First, results so far in the literature focus on isotropic random ball models (except for [36]). That is, the random fields have the same distribution in each different direction. This feature, from the application point of view, makes the model much less attractive. Second, the tightness of the scaled random fields is difficult to establish. Usually random ball models are defined as a random field tXpµquµPM indexed by a family of measuresM onR^d. The tightness of such random fields, after appropriate normalizations, is only established for very restricted classes of M[12, 13].

Date: October 3, 2018.

2010Mathematics Subject Classification. Primary, 60F05, 60G60; secondary, 60G52.

Key words and phrases. Operator-scaling, stable random field, limit theorem, anisotropic random field, random ball model, generalized random field.

1

The goal of this paper is to establish limit theorems for a general class of random ball models, and to remove the aforementioned two limitations.

First, we provide a general framework of random ball models exhibiting anisotropic features and hence include all previously considered ones as special cases. It is now well understood that a natural generalization of notion of self-similarity, widely used in stochastic processes and time series, is the so-called operator-scaling property for random fields introduced in Biermé et al. [9]. A random fieldtZ_tu_tPR^d is said to bepE, Hq-operator-scaling, if

(1.1) tZ_cEtu_tPRd

“d c^HtZ_tu_tPRd, for allcą0, whereE is an appropriatedˆdmatrix,c^E :“ř₈

k“0pElogcq^k{k!is also a matrix, andHą0.

Taking E to be the identity matrix, the above says that the random field Z is self-similar.

The motivation of allowing general matrixEis to generalize this notion to anisotropic random fields. Such random fields are often of practical importance in various applications, and they also present theoretical challenges. Families of anisotropic random fields are known, and path properties have been investigated. See for example [8, 31, 34, 46]. At the same time, the development of limit theorems for anisotropic random fields is still at an early stage. For some recent results, see for example [5, 15, 30, 32, 38, 41, 45]. In this article, we also consider more general random sets than balls, precisely sets of finite perimeter.

Second, we view the random ball models asdistribution-valued random elements, also known asgeneralized random fields, and establish weak convergence in the space of tempered distribu- tions. A complete description of self-similar generalized Gaussian random fields was obtained in [14] and allows to obtain essentially all Gaussian, translation- and rotation-invariant, H- self-similar generalized random field as scaling limits of a random balls model in [7]. Beyond the Gaussian framework, generalized Lévy random field, including stable generalized random field have been investigated in [43], where they are named as sparse stochastic processes.

Distribution-valued random variables and stochastic processes are already widely used to describe fluctuations of empirical measures of complex particle systems, including notably in- teracting particle systems [29] and branching particle systems [10, 23, 29, 32], just to mention a few.

The paper is organized as follows. Section 2 presents background on generalized random fields, the precise definition of the random ball model, and the four regimes of convergence that we investigate. The limit theorems are stated in Section 3, while their proofs are postponed in Section 6. In Section 4, we study statistical properties of the limit random fields. To conclude, a pointwise representation is obtained in Section 5 and some illustrations are given in the appendix.

Throughout, C stands for real constants that may change values from line to line. Without ambiguity, for x P R^d, |x| denotes its Euclidean norm. We write a_b “ maxpa, bq and a^b“minpa, bqfor a, bPR.

2. Background and definitions

2.1. Generalized random fields. The standard references for generalized random fields include notably [14, 16, 19, 20, 28]. In words, these fields are defined as random variables with values in a space of distributions (or generalized functions). To this end we consider the Schwartz space SpR^dq of all real-valued infinitely differentiable rapidly decreasing functions on R^d, andS¹pR^dqits topological dual, the space of tempered distribution. As usual SpR^dqis equipped with the topology that corresponds to the following notion of convergence: f_nÑf

if and only if for allN PN:“ t0,1,2, . . .u and j“ pj₁, . . . , j_dq PN^d }f_n´f}_N,j :“ sup

zPR^d

p1` |z|q^Nˇ

ˇD^jpf_n´fq pzqˇ

ˇÑ0, asnÑ 8, whereD^jfpzq “ ^Bj1¨¨¨Bjd

Bz₁^j1¨¨¨Bz_d^jdfpzq denotes the partial derivative of orderj.

We will actually also consider the space S₁pR^dq:“

"

f PSpR^dq;

ż

R^d

fpzqdz“0

* . Note thatS₁pR^dq “span D^jf; f PSpR^dq, j P t0,1u^d, j₁` ¨ ¨ ¨ `j_d“1(

. For convenience, we also writeS₀pR^dq “SpR^dq and thus we will be able to useS_npR^dq fornP t0,1uin the sequel.

We denote by S_n¹pR^dq the topological dual of S_npR^dq and by p ¨,¨ q the duality bracket. We usually consider two distinct topologies on S_n¹pR^dq. The strong topology is induced by the family of semi-norms

qBp¨q “sup

fPB

|p ¨, fq|, B bounded inS_npR^dq.

The weak topology on S_n¹pR^dq is the topology induced by the family of semi-norms |p ¨, fq|, f PS_npR^dq. A first remark is that both topologies generate the same Borelσ-field denoted by BpS_n¹pR^dqq, see [4].

A generalized random field is an S_n¹pR^dq-valued random variable, that is a measurable mapping X from a probability spacepΩ,A,Pqto pS_n¹pR^dq,BpS_n¹pR^dqqq. For such a generalized random field X, we let its evaluation at f P S_npR^dq be denoted by Xpfq, which is a real random variable on the same probability space.

The law of a generalized random field X is uniquely determined by its characteristic func- tional

L_Xpfq:“

ż

Ω

e^iXpfqdP, f PS_npR^dq.

Further, Xinduces a family of random variablesXpfq onpΩ,Aqindexed byf PS_npR^dq, with characteristic functions given by

E

´ e^itXpfq

¯

“ ż

Ω

e^itXpfqdP“L_Xptfq, tPR.

By linearity, the finite-dimensional distributions of X are simply obtained with L_Xpa₁f₁` ¨ ¨ ¨ `a_kf_kq “E

´

e^{ira1Xpf1q`¨¨¨`akXpfkqs}

¯ , for all kě1,a₁, . . . , a_kPRand f₁, . . . , f_k PS_npR^dq.

In practice, however, given a family of real random variablestXpfqu_fPSnpRdqon a probability space pΩ,A,Pq satisfying

(2.1) Xpaf`bgq “aXpfq `bXpgq a.s. for all a, bPR, f, gPS_npR^dq,

a priori it is not clear whether a corresponding S_n¹pR^dq-valued random variable exists. When this can be achieved, namely if there exists an S_n¹pR^dq-valued random variable X, possibly˜ defined on another probability space pΩ,˜ A,˜ Pq, such that for all˜ k ě 1, f1, . . . , fk PS_npR^dq, A₁, . . . , A_kPBpRq,

PpXpf₁q PA₁, . . . , Xpf_kq PA_kq “P˜

´Xpf˜ ₁q PA₁, . . . ,Xpf˜ _kq PA_k

¯ ,

we say thatX˜ is a version ofX“ tXpfqu_fPSnpR^d_q[40, Definition 9.1.1]. Let us quote that this notion is weaker than the notion of regularization in [24]. Actually, a regularization X˜ of X should be defined on the same probability spacepΩ,A,Pq thanX and satisfies Xpfq “˜ Xpfq a.s. for all f PS_npR^dq. However, when we deal with convergence in law for most of the part of the paper, the notion of version is enough for our purpose: once the existence of a version is proved, it suffices to work with the characteristic functionals of the original individual random variables. At only a few occasions we shall establish results in the stronger notion of regularization.

We recall below two fundamental theorems when working with limit theorems of generalized random fields, both based on characteristic functionals. The following theorem is a direct consequence of Minlos–Bochner’s theorem, see [4, Corollary 2.2].

Theorem 2.1. Let X “ tXpfqu_fPSnpR^d_q be a collection of real random variables on pΩ,A,Pq satisfying (2.1). If L_X : S_npR^dq Ñ C is continuous then X admits a version that is an S_n¹pR^dq-valued random variable.

Recall that a sequence of generalized random fields tX_mumě1 converges in distribution to X, denoted by X_m ñ X, in S_n¹pR^dq given the strong topology if for all ϕ : S_n¹pR^dq Ñ R continuous for the strong topology and bounded,

ż

S_n¹pR^dq

ϕpuqdPXmpuq ÝÑ

mÑ8

ż

S_n¹pR^dq

ϕpuqdPXpuq.

Similarly,X_mñXinS_n¹pR^dqgiven the weak topology, if the above holds for allϕ:S_n¹pR^dq Ñ R that is bounded and continuous with respect to the weak topology. As a consequence of Lévy’s continuity theorem ([4, Theorem 2.3]), we can state the following result, see [4, Corollary 2.4].

Theorem 2.2. LettXmumě1,XbeS_n¹pR^dq-valued random variables. The following conditions are equivalent:

‚ Xm ñX in S_n¹pR^dq given the strong topology,

‚ Xm ñX in S_n¹pR^dq given the weak topology,

‚ L_Xmpfq ÑL_Xpfq for all f PS_npR^dq.

Since both notions of convergence are equivalent, we shall just writeXmñX inS_n¹pR^dqin the sequel.

Proofs of Theorems 2.1 and 2.2. We refer to [16] for the stated results in the more general framework in terms of nuclear spaces. For the special case S¹pR^dq ”S₀¹pR^dq, we refer to [4]

where self-contained and simplified proofs can be found. Results in [4] can then be extended forS₁¹pR^dqby the following idea from Dobrushin [14, Proposition 2.1]. Let us quote that fixing a function ψ P SpR^dqzS₁pR^dq, one can define the continuous map U : S₁¹pR^dq Ñ S¹pR^dq by UpLqpfq “Lpπpfqq, where for f PSpR^dq,

πpfq “f´cpfqψPS₁pR^dq, withcpfq “ ş

R^dfpxqdx{ş

R^dψpxqdx. Hence any S₁¹pR^dq-valued random variableX coincides with the restriction of an S¹pR^dq-valued random variable Y, defined by Ypfq “ Xpπpfqq, f PSpR^dq. By using the so-defined mapU and applying results on S¹pR^dq, the desired results

forS₁¹pR^dq follow.

2.2. A generalized random ball model. Now we define the random ball model on R^d. Throughout, the operator-scaling is associated to adˆdreal matrixE, of which all eigenvalues have strictly positive real parts, denoted by a₁ ě ¨ ¨ ¨ ě a_d ą 0. Let q “ trpEq ą 0 be the trace of the matrixE.

We consider the kernel operator defined for px, rq PR^dˆ p0,8qand f PSpR^dq, by (2.2) T_r^Efpxq:“

ż

R^d

K_r^Epx, yqfpyqdy withK_r^Epx, yq:“1_BEpx,rqpyq.

Here and throughout, B_Epx, rq is the shifted and scaled “ball” given by BEpx, rq “x`r^EB, xPR^d, rą0,

based on a fixed bounded measurable set B Ă R^d with 0 PB, v_B :“ Leb_dpBq P p0,8q and Leb_dpBBq “0, whereLeb_dis the Lebesgue measure onR^d. Thusv_r:“Leb_dpB_Epx, rqq “r^qv_B. Note that we keep the name “random ball” from the original model but here the setB can be a much more general set than a ball. We only assume that B is a set of finite perimeter in the sense that

(2.3) PerpBq:“sup

"ż

B

divϕpxqdx : ϕPC_c¹pR^d,R^dq,}ϕ}₈ď1

* ă 8,

whereC_c¹pR^d,R^dqis the set of continuously differentiable functions with compact support (e.g.

B can be any bounded convex set). According to [18, Theorem 14], (2.3) is equivalent to the fact that the covariogram gB :R^dQxÞÑLeb_dpBX px`Bqqof the setB is Lipschitz, and thus there exists Cą0such that

(2.4) Leb_dpB∆px`Bqq “2pg_Bp0q ´g_Bpxqq ďC|x|, for all xPR^d.

We first define the model as a collection of random variables indexed byf PSpR^dq, and then prove the existence of regularizations afterwards. The rescaled random ball field is defined as (2.5) X_ρ^Epfq:“

ż

R^dˆR`ˆR

mT_r^EfpxqN_ρpdx, dr, dmq, f PSpR^dq,

whereN_ρis a Poisson random measure onR^dˆR`ˆRwith intensityλpρqdxFpdr{ρqGpdmq. Intuitively, the origins of random balls are distributed as a homogeneous Poisson process with intensity λpρq, and each random ball is scaled with a random radius with distribution Fρpdrq:“Fpdr{ρq, and is associated with a random weightmwith distributionG. Positions, scalings and weights are assumed to be independent. There are a few natural assumptions on F andG. First, the expected volume of a random ball is assumed to be finite. That is,

(2.6) v_B

ż

R`

r^qFpdrq ă 8.

Moreover, we assume that, for some Cβ ą0,

(2.7) Fpdrq “pprqdr with pprq „C_βr^´1´β asr Ñ0^q´β,

with the convention, 0^δ “ 0 if δ ą 0 and 0^δ “ 8 if δ ă 0. This condition is introduced in a compact form for both zoom-in/out scalings to be explained in Section 2.3. It reads as pprq is regularly varying at 0 with index ´1´β, only when β ă q; otherwise (2.6) will be violated. Similarly, pprq is regularly varying at infinity with index´1´β when βąq. Next, for the random weights, their distributionGis assumed to be integrable and in the domain of

attraction of certain stable distributionS_αpσ, b,0q withαP p1,2s,σą0,bP r´1,1s. That is, for independent random variablesMi with common distribution G,

(2.8) M1` ¨ ¨ ¨ `Mn

n^1{α ñSαpσ, b,0q withαP p1,2s.

A standard reference for stable distributions and processes is [40]. Under (2.6) and (2.8) withαą1, the random field (2.5) is well-defined and integrable. This follows from the fact

E`

|X_ρ^Epfq|˘ ď

ż

R^dˆR`ˆR

|m|T_r^E|f|pxqλpρqdxFpdr{ρqGpdmq ďλpρqρ^qEp|M|qvB}f}_L1

ż

R^`

r^qFpdrq, where M is a real random variable of distribution G and }f}

L1 :“ ş

R^d|fpyq|dy. Hence, a centered rescaled random ball field can be defined by

Y_ρ^Epfq:“X_ρ^Epfq ´E`

X_ρ^Epfq˘

, f PSpR^dq.

We come to the generalized random field interpretation ofX_ρ^E and Y_ρ^E.

Proposition 2.3. Under assumption (2.6),X_ρ^E andY_ρ^E are almost surely elements ofS¹pR^dq and therefore ofS₁¹pR^dq. As a consequence, they admit regularizations inS¹pR^dq and therefore inS₁¹pR^dq.

Proof. Let us quote thatf ÞÑT_r^Efpxq PS¹pR^dq, and moreover for allkě0,

|T_r^Efpxq| ď

˜ż

BEpx,rq

p1` |y|q^´kdy

¸ sup

zPR^d

p1` |z|q^k|fpzq|.

It follows that,

|X_ρ^Epfq| ďC_ρ,k^E sup

zPR^d

p1` |z|q^k|fpzq|, with

C_ρ,k^E :“

ż

R^dˆR`ˆR

|m|

ż

BEpx,rq

p1` |y|q^´kdyN_ρpdx, dr, dmq.

Note that E`

C_ρ,k^E ˘

“λpρq ż

R^dˆR`ˆR

|m|

ż

BEpx,rq

p1` |y|q^´kdydxF_ρpdrqGpdmq

“λpρqρ^qEp|M|qv_B ż

R^`

r^qFpdrq ˆż

R^d

p1` |y|q^´kdy

˙ ,

which is finite under assumption (2.6) as soon as ką d. Hence, C_ρ,k^E ă 8 a.s. for k ąd, so thatX_ρ^E PS¹pR^dq a.s. Since we also have f ÞÑ EpX_ρ^Epfqq PS¹pR^dq by taking expectation in the previous computations, it follows that the centered fieldY_ρ^E is also inS¹pR^dqa.s. The last part of the proposition is easy since to obtain a regularization inS¹pR^dqof a processX which is almost surely element ofS¹pR^dq, it suffices to modify it by settingXpωq ”0 for theω PΩ

such thatXpωq RS¹pR^dq, see [16, p.40].

The limit theorems will be based on the characteristic functionals of the centered rescaled random fields

(2.9) L_YE

ρ pfq “Eexp`

iY_ρ^Epfq˘

“exp

˜ż

R^dˆR`

φ_GpT_r^EfpxqqλpρqdxFρpdrq

¸

, f PS_npR^dq, with

(2.10) φGptq:“

ż

pe^imt´1´imtqGpdmq “L_Mptq ´1´itEpMq, tPR, whereM is a real random variable of distributionGsatisfying (2.8).

2.3. Zoom-in/out scalings and four regimes. There are two scalings to be considered in the limit theorems. Recall F_ρpdrq “ Fpdr{ρq. The caseρ Ñ 8 corresponds to enlarging the size of each ball, and ρ Ñ 0 corresponds to shrinking the size of each ball. We refer to the two scalings as the zoom-in and zoom-out scalings, respectively.

Next, for each type of scaling, there are four qualitatively different regimes. Since the spatial dependence of the random field is essentially determined by overlaps of random balls, heuristically we compute the expected weight of rescaled balls covering a fixed pointy, denoted bympρq, independent from y by stationarity. It is natural to expect mpρq Ñc P r0,8s, and we distinguish 8,p0,8q and 0 as three different cases. Take the zoom-in scaling case first.

Clearly only small balls, say with radius less than1 (before the ρ-scaling and the constant1 is irrelevant) should matter, and we compute

m_inpρq:“E

˜ż

R^dˆR`ˆR

m1_tyPBEpx,rqu1_trď1uN_ρpdx, dr, dmq

¸

“EpMqλpρqv_B ż₁

0

r^qF_ρpdrq, with

λpρq ż₁

0

r^qFρpdrq „ ˆ

Cβ

ż₁

0

r^q´β´1dr

˙

λpρqρ^β asρÑ 8.

Similarly for the zoom-out case, we compute for number of balls with radius larger than 1, moutpρq:“E

˜ż

R^dˆR`ˆR

m1_tyPBEpx,rqu1_trą1uN_ρpdx, dr, dmq

¸

“EpMqλpρqvB

ż₈

1

r^qFρpdrq, with

λpρq ż₈

1

r^qFρpdrq „ ˆ

Cβ

ż₈

1

r^q´β´1dr

˙

λpρqρ^β asρÑ0.

The calculations above made use of (2.7), and also explain why it is a reasonable assumption.

Notice that the constant is qualitatively irrelevant, only the common term λpρqρ^β matters, and both cases of scaling can be summarized in the compact form ofρÑ0^β´q.

In summary, there are naturally three regimes of interest, characterized by λpρqρ^β Ñ

$

&

%

8 (dense regime),

cP p0,8q (intermediate regime), 0 ((very-)sparse regime),

asρÑ0^β´q,

where within the caseλpρqρ^β Ñ0 we shall further identify two sub-regimes, named as sparse and very-sparse regimes in the sequel. We shall establish limit theorems for different regimes separately, and in each regime our limit theorem and the proof unify both zoom-in and zoom- out scalings (only zoom-out scaling in the very-sparse regime). Furthermore, in each regime we specify two parameters,β on the tails of the radius of random balls, andnindicating the zoom-in (n“1) and zoom-out (n“0) scalings.

3. Scaling limits

We will treat the four regimes separately. In each regime, we first introduce the limit field as stochastic integral, then show the existence of its generalized random field version by Minlos–Bochner’s theorem and then prove the weak convergence by Lévy’s continuity theorem.

For easy reading, all the proofs of this section are postponed to Section 6. The limit fields appearing here are further investigated in the next sections.

3.1. Dense regime. In the dense regime, we consider λpρqρ^β Ñ 8 asρÑ0^β´q, and the admissible range of parameters β andnis

(3.1) β P pq, αqq n“0 zoom-out scaling,

β P pq´a_d, qq n“1 zoom-in scaling.

The following field appears in the limit. Let αP p1,2s,σ ą0andbP r´1,1sbe given by (2.8) and C_β ą 0 be given by (2.7). Let M_α,β be an α-stable random measure onR^dˆR` with control measure σ^αC_βr^´1´βdrdx, and constant skewness function b. For f P S_npR^dq, let us define the stochastic integral

(3.2) Z_α,β^E pfq:“

ż

R^dˆR`

T_r^EfpxqM_α,βpdr, dxq.

See [40] for more background on stochastic integrals with respect toα-stable random measures.

Proposition 3.1. Let α P p1,2s. For β, n as in (3.1), the processZ_α,β^E :“ tZ_α,β^E pfqu_fPSnpR^d_q in (3.2) is well-defined, has characteristic functional

L_ZE

α,βpfq “exp

#

´Cβσ^α ż

R^dˆR`

|T_r^Efpxq|^α

´

1´ib`

T_r^Efpxq˘

tanαπ 2

¯

r^´1´βdrdx +

, (3.3)

where psq “signpsq, and admits a version with values inS_n¹pR^dq.

Then, we can consider weak convergence inS_n¹pR^dqand state the limit theorem in the dense regime.

Theorem 3.2. Suppose that the assumptions (2.7) and (2.8)on F andGhold. Under (3.1), ifn₁pρq:“ρ^βλpρq Ñ 8 as ρÑ0^β´q, then

1

n₁pρq^1{αY_ρ^E ñZ_α,β^E as ρÑ0^β´q inS_n¹pR^dq.

Remark 3.3. We lettZ_α,β^E pfqu_fPS_npR^dq denote the stochastic process indexed byf via (3.2), and the same notation Z_α,β^E in Theorem 3.2 for the corresponding version taking values in S_n¹pR^dq. Similar notations are used for the other regimes.

3.2. Intermediate regime. In the intermediate regime, we consider (3.4) λpρqρ^β Ña^q´β asρÑ0^β´q with aP p0,8q.

The admissible range of parametersβ and nis the same (3.1) as in the dense regime. In this case, the limit field is represented by a Poisson integral. For a P p0,8q and f P SpR^dq, we first define

(3.5) T_r,a^Efpxq:“

ż

R^d

1_a´EBEpx,rqpyqfpyqdy“T_r{a^E fpa^´Exq and we consider the Poisson integral J_a,α,β^E defined, for f PS_npR^dq, by (3.6) J_a,α,β^E pfq:“

ż

R^dˆR`ˆR`

mT_r,a^EfpxqN˜_βpdr, dx, dmq,

where N˜_β is the compensated Poisson random measure on R^dˆR` ˆR` with intensity C_βr^´1´βdxdrGpdmq, withC_β ą0 given in (2.7). For more background on Poisson integrals, see for example [27].

Proposition 3.4. Let a P p0,8q. For β, n as in (3.1), the process J_a,α,β^E in (3.6) is well- defined on S_npR^dq, has characteristic functional

(3.7) L_JE

a,α,βpfq “exp

#ż

R^dˆR`

φ_GpT_r,a^EfpxqqC_βr^´1´βdrdx +

,

whereφ_G is defined by (2.10)and admits a version with values in S_n¹pR^dq. The limit theorem in the intermediate regime is the following.

Theorem 3.5. Suppose that the assumptions (2.7) and (2.8) on F and G hold. Under (3.1) and (3.4),

Y_ρ^E ñJ_a,α,β^E as ρÑ0^β´q inS_n¹pR^dq.

3.3. Sparse regime. The sparse regime correspond to

(3.8) λpρqρ^β Ñ0asρÑ0^β´q with λpρq Ñ0^q´β. The admissible range of parameters ofβ andn is

(3.9) β P pq, αqq n“0 zoom-out scaling,

β P pq²{pq`adq, qq n“1 zoom-in scaling.

Setγ “β{qP pq{pq`a_dq,1q Y p1, αq. LetMγ^p1q be a γ-stable random measure having control measureσ_1,γdxwith

σ1,γ :“vB

˜ Cβq^´1

ż

R`

p1´cosprqqr^´1´γdr ż

R

|m|^γGpdmq

¸_1{γ ,

and constant skewness function

bγ :“ ´ ş

Rpmq|m|^γGpdmq ş

R|m|^γGpdmq .

We define, forf PSpR^dq,

Z_γ^p1qpfq:“

ż

R^d

fpxqM_γ^p1qpdxq.

Note thatZγ^p1qpfq is well-defined since f PSpR^dq ĂL^γpR^dq and its characteristic functional is given by

(3.10) L

Z^p1qγ pfq “exp ˆ

´σ_1,γ^γ ż

R^d

|fpxq|^γ

´

1´ibγpfpxqqtanγπ 2

¯ dx

˙ .

Proposition 3.6. For α P p1,2s and γ P pq{pq`a_dq,1q Y p1, αq, the process Z_γ^p1q admits a version with values inS₀¹pR^dq ĂS₁¹pR^dq.

Theorem 3.7. Suppose that the assumptions (2.7) and (2.8) on F and G hold. Under (3.8) and (3.9) with n2pρq:“ pλpρq^1{βρq^q and γ “β{q, we have

1

n₂pρqY_ρ^E ñZ_γ^p1q asρÑ0^β´q, in S_n¹pR^dq.

Remark 3.8. Note that the result in the caseβ P pq²{pq`a_dq, qqis also new for the isotropic case when E “I_d (the identity matrix).

3.4. Very-sparse regime. In this regime, consider

(3.11) λpρqρ^β Ñ0, λpρq Ñ 8asρÑ0.

The admissible range of parameters for the very-sparse regime is

(3.12) β P pαq,8q n“0 zoom-out scaling.

Let M_α^p2q be aα-stable random measure having control measureσ_2,αdxwith σ2,α:“σvB

˜ż

R`

r^αqFpdrq

¸_1{α

and constant skewness functionb. For f PSpR^dq, we set Z_α^p2qpfq:“

ż

R^d

fpxqM_α^p2qpdxq.

Proposition 3.9. For αP p1,2s, the process Zα^p2q admits a version with values in S₀¹pR^dq.

Theorem 3.10. Suppose that the assumptions (2.7)and (2.8)onF andGhold. Under (3.11) and (3.12), with n3pρq:“λpρq^1{αρ^q,

1

n₃pρqY_ρ^E ñZ_α^p2q as ρÑ0 in S₀¹pR^dq.

3.5. Summary. For comparison, we summarize in a single statement the limit theorems of the different regimes.

Theorem 3.11. Suppose that the assumptions (2.7) and (2.8) on F and G hold. We have the following weak convergence inS_n¹pR^dq:

pdenseq 1

pρ^βλpρqq^1{αY_ρ^E ñZ_α,β^E if λpρqρ^β Ñ 8, β, n as in (3.1),

pintermediateq Y_ρ^E ñJ_a,α,β^E if λpρqρ^β Ña^q´β P p0,8q, β, n as in (3.1), psparseq 1

pρ^βλpρqq^q{βY_ρ^E ñZ_β{q^p1q if λpρqρ^β Ñ0, λpρq Ñ0^q´β, β, n as in (3.9), pvery sparseq 1

ρ^qλpρq^1{αY_ρ^E ñZ_α^p2q if λpρqρ^β Ñ0, λpρq Ñ 8, β, n as in (3.12), where in all cases the limit is considered as ρÑ0^β´q.

4. Properties of the limit fields

In this section, we provide some properties of the limit generalized random fields. In the dense and intermediate regimes, the limit generalized random fields explicitly depend on E, and in particular so are their anisotropic properties. For the sparse and very-sparse regimes, all the dependence structures in the discrete models are not observable in the limit, and thus the limit generalized random fields have no specific anisotropic properties. Following Dobrushin in [14], using duality, we can define the following groups of transformations on S_npR^dq:

‚ the group of shift transformationsT “ tτhu_hPR^d:

τ_hfptq “fpt´hq, f PS_npR^dq, hPR^d, tPR^d;

‚ the group ofE-operator-scaling transformations∆^E “ tδ_c^Eu_cPp0,8q: δ_c^Efptq “c^´qfpc^´Etq, f PS_npR^dq, cP p0,8q, q“trpEq, tPR^d. Their analogousT,∆^E onS_n¹pR^dq are then defined by

τhLpfq:“Lpτhfq, and δ_c^ELpfq:“Lpδ_c^Efq,

for L P S_n¹pR^dq. Let us note that when the tempered distribution L is given by a function g, one recovers that τ_hL is given by the function gp¨ `hq and δ_c^EL is given by the function gpc^E¨q, thanks to the normalization term.

Proposition 4.1. Let α P p1,2s. For β, n as in (3.1), the generalized random field Z_α,β^E in (3.2) is

‚ shift-invariant: @hPR^d,

τ_hZ_α,β^E “^d Z_α,β^E ,

‚ pE, Hq-operator-scaling for H“ ^q´β_α P p´qp1´1{αq,0q Y p0, a_d{αq: @cą0, δ_c^EZ_α,β^E “^d c^HZ_α,β^E .

Let us remark that in [14] the first property is called the stationary n-th increments while the second one with E “I_d the self-similarity property.

Proof. It suffices to compute the characteristic functional. Observe that for f P S_npRq, one has for allhPR^d,

Z_α,β^E pτ_hfq “ ż

R^dˆR`

T_r^Efpx´hqM_α,βpdx, drq“^d Z_α,β^E pfq, by a change of variable, while for allcą0,

Z_α,β^E pδ^E_cfq “ ż

R^dˆR`

T_r^Eδ_c^EfpxqM_α,βpdx, drq

“ ż

R^dˆR

T_r{c^E fpc^´ExqMα,βpdx, drq

“d c^pq´βq{α ż

R^dˆR

T_r^EfpxqM_α,βpdx, drq “c^pq´βq{αZ_α,β^E pfq,

where the third step also followed from a change of variable argument.

For the intermediate case, the limit random field J_a,α,β^E in (3.6) is not E-operator-scaling but it has aggregate E-operator-scaling property as described below, generalizing aggregate similarity property introduced in [7].

Proposition 4.2. Under the assumption of Theorem 3.5,

δ_k^E1{pq´βqJ_a,α,β^E “^d

k

ÿ

i“1

J_a,α,β^E,piq, for all kPN, where tJ_a,α,β^E,piqui“1,...,k are i.i.d. copies of J_a,α,β^E . Furthermore,

1

a^pq´βq{αJ_a,α,β^E ñZ_α,β^E asaÑ0^β´q.

Proof. The first part of the proof follows from straightforward calculation of characteristic functionals, with a similar change of variable argument as above. The second part of the proof follows from convergence of characteristic functionals for random variables in the domain of

attractions ofS_αpσ, b,0q. The details are omitted.

At last, remark that in the sparse and very-sparse regimes, the limit random fields have essentially no dependence structure, as the limit random fields are stochastic integrals with respect to stable random measures withconstant control measure on R^d. Thus they inherit no specific anisotropic properties. Nevertheless, for anyE¹ satisfying the same assumption as E with possibly different eigenvalues, writing q¹ “trpE¹q, it can be shown that

δ^E_c¹Z_θ^piq “^d c^{1´θθ q1}Z_θ^piq for i“1,2 with legitimate parameter θ.

5. Comments on pointwise representation

Given a tempered distribution LPS¹pR^dq, it is a natural question to wonder if it may be represented by a Borel measurable function g, that is

@f PSpR^dq, Lpfq “ ż

R^d

fptqgptqdt.

We say that a generalized random fieldXadmits a pointwise reprensentation if there exists a measurable random fieldtXptqup _tPRd, meaning as in Definition 9.4.1 of [40] thatXp : ΩˆR^dÑR is a jointly measurable function, such that

Xpfq “ ż

R^d

Xptqfp ptqdt, f PSpR^dq.

Conversely, we have the following property.

Proposition 5.1. Let tXptqup _tPRd be a measurable random field. If there exists k P N such

that ż

R^d

p1` |t|q^´kEp|Xptq|qdtp ă 8, then the random fieldX, defined onS_npR^dqby Xpfq “ş

R^dXptqfp ptqdt, admits a regularization that is a generalized random field. Moreover, if Xp is pE, Hq-operator-scaling for some H ą0 in the sense of (1.1), thenX is pE, Hq-operator-scaling in the sense of Proposition 4.1.

Proof. Under the assumption, one checks that for allf PS_npR^dq, ż

R^d

|Xptqfptq|dtp ďC_ksup

zPR^d

p1` |z|q^k|fpzq|, where the random constant C_k “ ş

R^dp1` |t|q^´k|Xptq|dtp is a.s. finite. This implies that the linear random fieldX is well-defined and a.s. continuous. Hence there exists a regularization ofX on S_n¹pR^dq, see [16, p.40]. The last property of the proposition is straightforward.

Our centered rescaled random ball fieldY_ρ^E defined in Section 2 clearly admits a pointwise representation whereYp_ρ^E “Xp_ρ^E ´EXp_ρ^E and

Xp_ρ^Eptq “ ż

R^dˆR`ˆR

mK_r^Epx, tqN_ρpdx, dr, dmq, tPR^d,

with the same Poisson random measureN_ρthan in (2.5). Let us consider the limit generalized random field Z_α,β^E of the dense regime in the case of symmetric weights (b “ 0). Actually, there are two situations that we treated separately in the following sub-sections.

5.1. The case β P pq´a_d, qq and H “ ^q´β_α P p0, a_d{αq. In this case, as proved in Proposi- tion 5.2 below,Z_α,β^E admits a pointwise representation with

Zp_α,β^E ptq “ ż

R^dˆR`

p1_BEpx,rqptq ´1_BEpx,rqp0qqM_α,βpdr, dxq, tPR^d,

satisfying (1.1) and M_α,β is the same as in the representation of Z_α,β^E . Let us introduce C_Eptq “ tpx, rq;r^´Epx´tq PBuand note that

Zp_α,β^E ptq “Mα,βpC_Eptq XC_Ep0q^cq ´Mα,βpC_Eptq^cXC_Ep0qq, tPR^d. Until here we do not need to assume thatM_α,β has skewness functionb“0.

With the assumption that M_α,β is symmetric, one can check that (5.1)

! Zp_α,β^E ptq

)

tPR^d f.d.d.

“ tM_α,βpV_tqu_tP

R^d,

with V_t“C_Eptq∆C_Ep0q. That is, the random field Zp_α,β^E has a Chentsov’s type representation [40, Chapter 8]. In particular, for H “ ^q´β_α P p0, a_d{αq the random field Zp_α,β^E generalizes

isotropic self-similarpα, Hq-Takenaka random fields (see [40, p.405]), defined by choosing the Euclidean unit ball forB and E“I_d, witha_d“1.

The representation (5.1) allows us to provide several simulations of our operator-scaling random ball model with symmetric α-stable (SαS) weights, following similar ideas as in [3].

See Figures 1–3 in the appendix.

Proposition 5.2. For βP pq´a_d, qq, there exists a measurable version ofZp_α,β^E , also denoted by Zp_α,β^E , such that Z_α,β^E coincides in S₁¹pR^dq with the generalized random field

(5.2) f PSpR^dq ÞÑ

ż

R^d

Zp_α,β^E ptqfptqdt.

Proof. First note that ż

R^d

|1_BEpx,rqptq ´1_BEpx,rqp0q|^αdx“r^qhpr^´Etq,

with hpzq “ Leb_dpB∆pz`Bqq. According to (2.4), h satisfies hpzq ď Cp|z| ^1q for some constantC ą0. It follows that

ż

R^dˆR^`

|1_BEpx,rqptq ´1_BEpx,rqp0q|^ασ^αCβr^´1´βdrdx ďCσ^αC_β

ż

R^`

r^qp|r^´Et| ^1qr^´1´βdrďCσ^αC_β ż

R`

r^qp}r^´E} ^1qr^´1´βdrp1` |t|q

“C_α,β^E p1` |t|q, with C_α,β^E “ Cσ^αCβ

ş

R^`p}r^´E} ^1qr^q´β´1dr ă 8 and } ¨ } the subordinated norm, since β P pq ´a_d, qq. Hence Zp_α,β^E ptq is well-defined and is a SαS random variable with scale parameter bounded by

´

C_α,β^E p1` |t|q

¯1{α

, for everytPR^d. According to [40, Theorem 11.1.1]

there exists a measurable version ofZp_α,β^E since

(1) pt, x, rq PR^dˆR^dˆR^`ÞÑ p1_BEpx,rqptq ´1_BEpx,rqp0qq PR is measurable;

(2) the control measureσ^αC_βr^´1´βdrdx isσ-finite.

Noting that by [40, Property 1.2.17], we have

(5.3) E

´

|Zp_α,β^E ptq|

¯

ďEp|Sα|q`

C_α,β^E p1` |t|q˘1{α

,

withSαaSαSrandom variable of scale parameter1, we may definef PSpR^dq ÞÑş

R^dZp_α,β^E ptqfptqdt that is a.s. inS¹pR^dq, thanks to Proposition 5.1.

Now it remains to show that the right-hand side of (5.2) has the same stable law asZ_α,β^E pfq “ ş

R^dˆR`T_r^EfpxqMα,βpdx, drq. For this we recall that (5.4)

ż

Zp_α,βptqfptqdt“^d ż

R^dˆR`

ˆż

R^d

`1_BEpx,rqptq ´1_BEpx,rqp0q˘ fptqdt

˙

M_α,βpdx, drq, provided that

ż

R^d

|Zp_α,β^E ptq|fptqdtă 8a.s.,

see [40, Theorem 11.4.1]. Sincef decays rapidly, the above follows from (5.3) and hence (5.4) holds. To complete the proof, it remains to remark that for f PS₁pR^dq, one has

ż

R^d

p1_BEpx,rqptq ´1_BEpx,rqp0qqfptqdt“T_r^Efpxq.

5.2. The case β P pq, αqq and H “ ^q´β_α P p´qp1´1{αq,0q. In this case, H ă 0 and we do not have direct pointwise representation, but the limit fieldZ_α,β^E can be obtained as the derivative (in the sense of distributions) of a pointwise process. For all t PR^d, following the same idea as for the definition of Z_α,β^E pfq forf PSpR^dq, we can define the random variable

Zq_α,β^E ptq “pt₁q ¨ ¨ ¨pt_dq ż

R^dˆR`

T_r^E1_r0,tspxqM_α,βpdr, dxq, where the random measure M_α,β is the same as in (3.2), r0, ts “ś_d

i“1r0, tis, andptiq is the sign of t_i. The family Zq_α,β^E “ tZq_α,β^E ptqu_tPRd is a measurable random field and, by successive integrations by parts, we can show that Z_α,β^E “D^p1,...,1qZq_α,β^E , that is for allf PSpR^dq,

Z_α,β^E pfq “ p´1q^d ż

R^d

Zq_α,β^E ptqD^p1,...,1qfptqdt.

This consideration is analogous to [12, Theorem 2.6 and Lemma 3.7] forE“I_dandβ ąq“d inD¹pR^dqthe space of distribution instead ofS¹pR^dq. We thus refer to [12] for technical details.

6. Proofs of the main results

6.1. Preliminary results. The proofs of our limit theorems follow the same scheme as in [7]

or [11] to establish the convergence of the characteristic functions. They use the two following lemmas concerning conditions (2.7) and (2.8).

Lemma 6.1 (Lemma 2.4 in [7], Lemma 3.2 in [11]). Under the assumption (2.7), if tgρuρą0, g are continuous functions on R` such that

(6.1) lim

ρÑ0^β´q|gprq ´gρprq| “0,

and for some 0ăβ_´ăβ ăβ_` there exists a constantC ą0 such that

|gprq| ďCpr^β´ ^r^β`q, (6.2)

|g_ρprq| ďCpr^β´^r^β`q, (6.3)

for all rą0, then, forC_β as in (2.7), ż

R`

g_ρprqF_ρpdrq „C_βρ^β ż

R`

gprqr^´1´βdr, as ρÑ0^β´q.

Lemma 6.2 (Lemma 3.1 in [11]). Suppose thatM is in the domain of attraction ofSαpσ, b,0q for some αą1, σ ą0 and bPR. Then

φGptq “L_Mptq ´1´itEpMq „ ´|t|^αφα,b,σptq, as tÑ0, with

(6.4) φ_α,b,σptq “σ^αp1´ibptqtanpαπ{2qq,