Hermine Biermé

Olivier Durieu
email: olivier.durieu@univ-tours.fr

Yizao Wang
email: yizao.wang@uc.edu

GENERALIZED OPERATOR-SCALING RANDOM BALL MODEL

Keywords: October 3, 2018. 2010 Mathematics Subject Classification. Primary, 60F05, 60G60; secondary, 60G52 Operator-scaling, stable random field, limit theorem, anisotropic random field, random ball model, generalized random field

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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 [START_REF] Beran | Long-memory processes[END_REF][START_REF] Pipiras | Long-range dependence and self-similarity[END_REF][START_REF] Samorodnitsky | Stochastic processes and long range dependence[END_REF]. 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 locations 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. Asymptotic behaviors are then of interest, when all the balls are simultaneously rescaled by a parameter ρ, and at the same time the intensity of balls also changes with respect to ρ. Under mild assumption on the distribution of the radius, limit theorems can be established for ρ Ñ 0 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 [START_REF] Kaj | Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach[END_REF][START_REF] Mikosch | Is network traffic approximated by stable Lévy motion or fractional Brownian motion?[END_REF] 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 [START_REF] Pilipauskaitė | Anisotropic scaling of the random grain model with application to network traffic[END_REF]). 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 tXpµqu µPM indexed by a family of measures M on R d . The tightness of such random fields, after appropriate normalizations, is only established for very restricted classes of M [START_REF] Breton | Functional macroscopic behavior of weighted random ball model[END_REF][START_REF] Breton | Infinite dimensional functional convergences in random balls model[END_REF].

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. [START_REF] Biermé | Operator scaling stable random fields[END_REF]. A random field tZ t u tPR d is said to be pE, Hq-operator-scaling, if (1.1) tZ c E t u tPR d d " c H tZ t u tPR d , for all c ą 0, where E is an appropriate d ˆd matrix, c E :" ř 8 k"0 pE log cq k {k! is also a matrix, and H ą 0. Taking E to be the identity matrix, the above says that the random field Z is self-similar. The motivation of allowing general matrix E is 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 [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF][START_REF] Li | Exact moduli of continuity for operator-scaling Gaussian random fields[END_REF][START_REF] Meerschaert | Fernique-type inequalities and moduli of continuity for anisotropic Gaussian random fields[END_REF][START_REF] Xiao | Sample path properties of anisotropic Gaussian random fields[END_REF]. 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 [START_REF] Biermé | Invariance principles for operator-scaling Gaussian random fields[END_REF][START_REF] Durieu | From random partitions to fractional Brownian sheets[END_REF][START_REF] Lavancier | Invariance principles for non-isotropic long memory random fields[END_REF][START_REF] Li | Occupation time fluctuations of weakly degenerate branching systems[END_REF][START_REF] Puplinskaitė | Scaling transition for long-range dependent Gaussian random fields[END_REF][START_REF] Shen | Operator-scaling Gaussian random fields via aggregation[END_REF][START_REF] Wang | An invariance principle for fractional Brownian sheets[END_REF]. In this article, we also consider more general random sets than balls, precisely sets of finite perimeter.

Second, we view the random ball models as distribution-valued random elements, also known as generalized random fields, and establish weak convergence in the space of tempered distributions. A complete description of self-similar generalized Gaussian random fields was obtained in [START_REF] Dobrushin | Gaussian and their subordinated self-similar random generalized fields[END_REF] and allows to obtain essentially all Gaussian, translation-and rotation-invariant, Hself-similar generalized random field as scaling limits of a random balls model in [START_REF] Biermé | Self-similar random fields and rescaled random balls models[END_REF]. Beyond the Gaussian framework, generalized Lévy random field, including stable generalized random field have been investigated in [START_REF] Unser | An introduction to sparse stochastic processes[END_REF], 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 interacting particle systems [START_REF] Kipnis | of Grundlehren der Mathematischen Wissenschaften[END_REF] and branching particle systems [START_REF] Bojdecki | A long range dependence stable process and an infinite variance branching system[END_REF][START_REF] Holley | Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motions[END_REF][START_REF] Kipnis | of Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Li | Occupation time fluctuations of weakly degenerate branching systems[END_REF], 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, bq for a, b P R.

Background and definitions

2.1. Generalized random fields. The standard references for generalized random fields include notably [START_REF] Dobrushin | Gaussian and their subordinated self-similar random generalized fields[END_REF][START_REF] Fernique | Processus linéaires, processus généralisés[END_REF]19,[START_REF][END_REF][START_REF] Kallianpur | Stochastic differential equations in infinitedimensional spaces[END_REF]. 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 d q of all real-valued infinitely differentiable rapidly decreasing functions on R d , and S 1 pR d q its topological dual, the space of tempered distribution. As usual SpR d q is equipped with the topology that corresponds to the following notion of convergence: f n Ñ f if and only if for all N P N :" t0, 1, 2, . . . u and j " pj 1 , . . . , j d q P N d }f n ´f } N,j :" sup zPR d p1 `|z|q N ˇˇD j pf n ´f q pzq ˇˇÑ 0, as n Ñ 8,

where D j f pzq " B j 1 ¨¨¨B j d Bz j 1 1 ¨¨¨Bz j d d
f pzq denotes the partial derivative of order j.

We will actually also consider the space

S 1 pR d q :" " f P SpR d q; ż R d f pzqdz " 0 * .
Note that S 1 pR d q " span D j f ; f P SpR d q, j P t0, 1u d , j 1 `¨¨¨`j d " 1 ( . For convenience, we also write S 0 pR d q " SpR d q and thus we will be able to use S n pR d q for n P t0, 1u in the sequel. We denote by S 1 n pR d q the topological dual of S n pR d q and by p ¨, ¨q the duality bracket. We usually consider two distinct topologies on S 1 n pR d q. The strong topology is induced by the family of semi-norms

q B p¨q " sup f PB |p ¨, f q|, B bounded in S n pR d q.
The weak topology on S 1 n pR d q is the topology induced by the family of semi-norms |p ¨, f q|, f P S n pR d q. A first remark is that both topologies generate the same Borel σ-field denoted by BpS 1 n pR d qq, see [START_REF] Biermé | Generalized random fields and Lévy's continuity Theorem on the space of tempered distributions[END_REF].

A generalized random field is an S 1 n pR d q-valued random variable, that is a measurable mapping X from a probability space pΩ, A, Pq to pS 1 n pR d q, BpS 1 n pR d qqq. For such a generalized random field X, we let its evaluation at f P S n pR d q be denoted by Xpf q, 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 functional L X pf q :" ż Ω e iXpf q dP, f P S n pR d q.

Further, X induces a family of random variables Xpf q on pΩ, Aq indexed by f P S n pR d q, with characteristic functions given by E ´eitXpfq ¯" ż Ω e itXpf q dP " L X ptf q, t P R.

By linearity, the finite-dimensional distributions of X are simply obtained with

L X pa 1 f 1 `¨¨¨`a k f k q " E ´eira 1 Xpf 1 q`¨¨¨`a k Xpf k qs ¯, for all k ě 1, a 1 , . . . , a k P R and f 1 , . . . , f k P S n pR d q.
In practice, however, given a family of real random variables tXpf qu f PSnpR d q on a probability space pΩ, A, Pq satisfying (2.1)

Xpaf `bgq " aXpf q `bXpgq a.s. for all a, b P R, f, g P S n pR d q, a priori it is not clear whether a corresponding S 1 n pR d q-valued random variable exists. When this can be achieved, namely if there exists an S 1 n pR d q-valued random variable X, possibly defined on another probability space p Ω, Ã, Pq, such that for all k ě 1, f 1 , . . . , f k P S n pR d q, A 1 , . . . , A k P BpRq, PpXpf 1 q P A 1 , . . . , Xpf k q P A k q " P ´Xpf

1 q P A 1 , . . . , Xpf k q P A k ¯,
we say that X is a version of X " tXpf qu f PSnpR d q [40, Definition 9.1.1]. Let us quote that this notion is weaker than the notion of regularization in [START_REF] Itô | Distribution-valued processes arising from independent Brownian motions[END_REF]. Actually, a regularization X of X should be defined on the same probability space pΩ, A, Pq than X and satisfies Xpf q " Xpf q a.s. for all f P S n pR d q. 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 [START_REF] Biermé | Generalized random fields and Lévy's continuity Theorem on the space of tempered distributions[END_REF]Corollary 2.2].

Theorem 2.1. Let X " tXpf qu f PSnpR d q be a collection of real random variables on pΩ, A, Pq satisfying (2.1). If L X : S n pR d q Ñ C is continuous then X admits a version that is an S 1 n pR d q-valued random variable.

Recall that a sequence of generalized random fields tX m u mě1 converges in distribution to X, denoted by X m ñ X, in S 1 n pR d q given the strong topology if for all ϕ : S 1 n pR d q Ñ R continuous for the strong topology and bounded, ż

S 1 n pR d q ϕpuqdP Xm puq ÝÑ mÑ8 ż S 1 n pR d q ϕpuqdP X puq.
Similarly, X m ñ X in S 1 n pR d q given the weak topology, if the above holds for all ϕ : S 1 n pR d q 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 [START_REF] Biermé | Generalized random fields and Lévy's continuity Theorem on the space of tempered distributions[END_REF]Corollary 2.4].

Theorem 2.2. Let tX m u mě1 , X be S 1 n pR d q-valued random variables. The following conditions are equivalent:

' X m ñ X in S 1
n pR d q given the strong topology, ' X m ñ X in S 1 n pR d q given the weak topology, ' L Xm pf q Ñ L X pf q for all f P S n pR d q.

Since both notions of convergence are equivalent, we shall just write X m ñ X in S 1 n pR d q in the sequel.

Proofs of Theorems 2.1 and 2.2. We refer to [START_REF] Fernique | Processus linéaires, processus généralisés[END_REF] for the stated results in the more general framework in terms of nuclear spaces. For the special case S 1 pR d q " S 1 0 pR d q, we refer to [START_REF] Biermé | Generalized random fields and Lévy's continuity Theorem on the space of tempered distributions[END_REF] where self-contained and simplified proofs can be found. Results in [START_REF] Biermé | Generalized random fields and Lévy's continuity Theorem on the space of tempered distributions[END_REF] can then be extended for S 1 1 pR d q by the following idea from Dobrushin [14, Proposition 2.1]. Let us quote that fixing a function ψ P SpR d qzS 1 pR d q, one can define the continuous map U : S 1 1 pR d q Ñ S 1 pR d q by U pLqpf q " Lpπpf qq, where for f P SpR d q, πpf q " f ´cpf qψ P S 1 pR d q,

with cpf q " ş R d f pxqdx{ ş R d ψpxqdx.
Hence any S 1 1 pR d q-valued random variable X coincides with the restriction of an S 1 pR d q-valued random variable Y , defined by Y pf q " Xpπpf qq, f P SpR d q. By using the so-defined map U and applying results on S 1 pR d q, the desired results for S 1 1 pR d q 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 a dˆd real matrix E, of which all eigenvalues have strictly positive real parts, denoted by a 1 ě ¨¨¨ě a d ą 0. Let q " trpEq ą 0 be the trace of the matrix E. We consider the kernel operator defined for px, rq P R d ˆp0, 8q and f P SpR d q, by (2.2)

T E r f pxq :" ż R d K E r px, yqf pyqdy with K E r px, yq :" 1 B E px,rq pyq.
Here and throughout, B E px, rq is the shifted and scaled "ball" given by We first define the model as a collection of random variables indexed by f P SpR d q, and then prove the existence of regularizations afterwards. The rescaled random ball field is defined as (2.5)

B E px, rq " x `rE B, x P R d ,
X E ρ pf q :" ż R d ˆR`ˆR mT E r f pxqN ρ pdx, dr, dmq, f P SpR d q,
where N ρ is a Poisson random measure on R d ˆR`ˆR with intensity λpρqdxF pdr{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 F ρ pdrq :" F pdr{ρq, and is associated with a random weight m with distribution G. Positions, scalings and weights are assumed to be independent. There are a few natural assumptions on F and G. First, the expected volume of a random ball is assumed to be finite. That is,

(2.6) v B ż R
`rq F pdrq ă 8. Moreover, we assume that, for some C β ą 0, (2.7)

F pdrq " pprqdr with pprq " C β r ´1´β as r Ñ 0 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 β ą q. Next, for the random weights, their distribution G is assumed to be integrable and in the domain of attraction of certain stable distribution S α pσ, b, 0q with α P p1, 2s, σ ą 0, b P r´1, 1s. That is, for independent random variables M i with common distribution G,

(2.8) M 1 `¨¨¨`M n n 1{α
ñ S α pσ, b, 0q with α P p1, 2s.

A standard reference for stable distributions and processes is [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF]. Under (2.6) and (2.8) with α ą 1, the random field (2.5) is well-defined and integrable. This follows from the fact

E `|X E ρ pf q| ˘ď ż R d ˆR`ˆR |m|T E r |f |pxqλpρqdxF pdr{ρqGpdmq ď λpρqρ q Ep|M |qv B }f } L 1 ż R `rq F pdrq, where M is a real random variable of distribution G and }f } L 1 :" ş R d |f pyq|dy.
Hence, a centered rescaled random ball field can be defined by

Y E ρ pf q :" X E ρ pf q ´E `XE ρ pf q ˘, f P SpR d q.
We come to the generalized random field interpretation of X E ρ and Y E ρ .

Proposition 2.3. Under assumption (2.6), X E ρ and Y E ρ are almost surely elements of S 1 pR d q and therefore of S 1 1 pR d q. As a consequence, they admit regularizations in S 1 pR d q and therefore in S 1 1 pR d q.

Proof. Let us quote that f Þ Ñ T E r f pxq P S 1 pR d q, and moreover for all k ě 0,

|T E r f pxq| ď ˜żB E px,rq p1 `|y|q ´kdy ¸sup zPR d p1 `|z|q k |f pzq|.
It follows that,

|X E ρ pf q| ď C E ρ,k sup zPR d p1 `|z|q k |f pzq|, with C E ρ,k :" ż R d ˆR`ˆR |m| ż B E px,rq
p1 `|y|q ´kdyN ρ pdx, dr, dmq.

Note that

E `CE ρ,k ˘" λpρq ż R d ˆR`ˆR |m| ż B E px,rq p1 `|y|q ´kdydxF ρ pdrqGpdmq " λpρqρ q Ep|M |qv B ż R `rq F pdrq ˆżR d p1 `|y|q ´kdy ˙,
which is finite under assumption (2.6) as soon as k ą d. Hence, C E ρ,k ă 8 a.s. for k ą d, so that X E ρ P S 1 pR d q a.s. Since we also have f Þ Ñ EpX E ρ pf qq P S 1 pR d q by taking expectation in the previous computations, it follows that the centered field Y E ρ is also in S 1 pR d q a.s. The last part of the proposition is easy since to obtain a regularization in S 1 pR d q of a process X which is almost surely element of S 1 pR d q, it suffices to modify it by setting Xpωq " 0 for the ω P Ω such that Xpωq R S 1 pR d q, see [16, p.40].

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

(2.9) L Y E ρ pf q " E exp `iY E ρ pf q ˘" exp ˜żR d ˆR`φ G pT E r f pxqqλpρqdxF ρ pdrq ¸, f P S n pR d q, with (2.10) φ G ptq :" ż pe imt ´1 ´imtqGpdmq " L M ptq ´1 ´itEpM q, t P R,
where M is a real random variable of distribution G satisfying (2.8).

2.3. Zoom-in/out scalings and four regimes. There are two scalings to be considered in the limit theorems. Recall F ρ pdrq " F pdr{ρq. The case ρ 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 point y, denoted by mpρq, independent from y by stationarity. It is natural to expect mpρq Ñ c 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 than 1 (before the ρ-scaling and the constant 1 is irrelevant) should matter, and we compute

m in pρq :" E ˜żR d ˆR`ˆR m1 tyPB E px,rqu 1 trď1u N ρ pdx, dr, dmq " EpM qλpρqv B ż 1 0 r q F ρ pdrq, with λpρq ż 1 0 r q F ρ pdrq " ˆCβ ż 1 0 r q´β´1 dr ˙λpρqρ β as ρ Ñ 8.
Similarly for the zoom-out case, we compute for number of balls with radius larger than 1,

m out pρq :" E ˜żR d ˆR`ˆR m1 tyPB E px,rqu 1 trą1u N ρ pdx, dr, dmq " EpM qλpρqv B ż 8 1 r q F ρ pdrq, with λpρq ż 8 1 r q F ρ pdrq " ˆCβ ż 8 1 r q´β´1 dr ˙λpρqρ β as ρ Ñ 0.
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 of ρ Ñ 0 β´q . In summary, there are naturally three regimes of interest, characterized by

λpρqρ β Ñ $ & % 8
(dense regime), c P p0, 8q (intermediate regime), 0 ((very-)sparse regime), as ρ Ñ 0 β´q , where within the case λpρqρ β Ñ 0 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 zoomout 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, and n indicating the zoom-in (n " 1) and zoom-out (n " 0) scalings.

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. β P pq, αqq n " 0 zoom-out scaling, β P pq ´ad , qq n " 1 zoom-in scaling.

The following field appears in the limit. Let α P p1, 2s, σ ą 0 and b P r´1, 1s be given by (2.8) and C β ą 0 be given by (2.7). Let M α,β be an α-stable random measure on R d ˆR`w ith control measure σ α C β r ´1´β drdx, and constant skewness function b. For f P S n pR d q, let us define the stochastic integral

(3.2) Z E α,β pf q :" ż R d ˆR`T E r f pxqM α,β pdr, dxq.
See [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF] for more background on stochastic integrals with respect to α-stable random measures.

Proposition 3.1. Let α P p1, 2s. For β, n as in (3.1), the process

Z E α,β :" tZ E α,β pf qu f PSnpR d q in (3.2) is well-defined, has characteristic functional L Z E α,β pf q " exp # ´Cβ σ α ż R d ˆR`| T E r f pxq| α ´1 ´ib `T E r f pxq ˘tan απ 2 ¯r´1´β drdx + , (3.3)
where psq " signpsq, and admits a version with values in S 1 n pR d q. Then, we can consider weak convergence in S 1 n pR d q and state the limit theorem in the dense regime.

Theorem 3.2. Suppose that the assumptions (2.7) and (2.8) on F and G hold. Under (3.1), if n 1 pρq :" ρ β λpρq Ñ 8 as ρ Ñ 0 β´q , then

1 n 1 pρq 1{α Y E ρ ñ Z E α,β
as ρ Ñ 0 β´q in S 1 n pR d q. Remark 3.3. We let tZ E α,β pf qu f PSnpR d q denote the stochastic process indexed by f via (3.2), and the same notation Z E α,β in Theorem 3.2 for the corresponding version taking values in S 1 n pR d q. Similar notations are used for the other regimes.

3.2. Intermediate regime. In the intermediate regime, we consider (3.4) λpρqρ β Ñ a q´β as ρ Ñ 0 β´q with a P p0, 8q.

The admissible range of parameters β and n is 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 d q, we first define

(3.5) T E r,a f pxq :" ż R d 1 a ´E B E px,rq pyqf pyqdy " T E r{a f pa ´E xq
and we consider the Poisson integral J E a,α,β defined, for f P S n pR d q, by

(3.6) J E a,α,β pf q :" ż R d ˆR`ˆR`m T E r,a f pxq Ñβ pdr, dx, dmq,
where Ñβ is the compensated Poisson random measure on R d ˆR`ˆR`w ith intensity C β r ´1´β dxdrGpdmq, with C β ą 0 given in (2.7). For more background on Poisson integrals, see for example [START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF].

Proposition 3.4. Let a P p0, 8q. For β, n as in (3.1), the process J E a,α,β in (3.6) is welldefined on S n pR d q, has characteristic functional

(3.7) L J E a,α,β pf q " exp # ż R d ˆR`φ G pT E r,a f pxqqC β r ´1´β drdx + ,
where φ G is defined by (2.10) and admits a version with values in S 1 n pR d q. The limit theorem in the intermediate regime is the following. The admissible range of parameters of β and n is (3.9) β P pq, αqq n " 0 zoom-out scaling, β P pq 2 {pq `ad q, qq n " 1 zoom-in scaling.

Set γ " β{q P pq{pq `ad q, 1q Y p1, αq. Let M p1q γ be a γ-stable random measure having control measure σ 1,γ dx with

σ 1,γ :" v B ˜Cβ q ´1 ż R `p1 ´cosprqqr ´1´γ dr ż R |m| γ Gpdmq ¸1{γ ,
and constant skewness function

b γ :" ´şR pmq|m| γ Gpdmq ş R |m| γ Gpdmq .
We define, for f P SpR d q,

Z p1q γ pf q :" ż R d f pxqM p1q γ pdxq.
Note that Z p1q γ pf q is well-defined since f P SpR d q Ă L γ pR d q and its characteristic functional is given by

(3.10) L Z p1q γ pf q " exp ˆ´σ γ 1,γ ż R d |f pxq| γ ´1 ´ib γ pf pxqq tan γπ 2 ¯dx ˙.
Proposition 3.6. For α P p1, 2s and γ P pq{pq `ad q, 1q Y p1, αq, the process Z p1q γ admits a version with values in S 1 0 pR d q Ă S 1 1 pR d q.

Theorem 3.7. Suppose that the assumptions (2.7) and (2.8) on F and G hold. Under (3.8) and (3.9) with n 2 pρq :" pλpρq 1{β ρq q and γ " β{q, we have

1 n 2 pρq Y E ρ ñ Z p1q γ as ρ Ñ 0 β´q , in S 1 n pR d q.
Remark 3.8. Note that the result in the case β P pq 2 {pq `ad q, qq is 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 Ñ 0, λpρq Ñ 8 as ρ Ñ 0.

The admissible range of parameters for the very-sparse regime is (3.12) β P pαq, 8q n " 0 zoom-out scaling.

Let M p2q α be a α-stable random measure having control measure σ 2,α dx with σ 2,α :" σv B ˜żR `rαq F pdrq ¸1{α and constant skewness function b. For f P SpR d q, we set

Z p2q α pf q :" ż R d f pxqM p2q α pdxq.
Proposition 3.9. For α P p1, 2s, the process Z p2q α admits a version with values in S 1 0 pR d q.

Theorem 3.10. Suppose that the assumptions (2.7) and (2.8) on F and G hold. Under (3.11) and (3.12), with n 3 pρq :" λpρq 1{α ρ q ,

1 n 3 pρq Y E ρ ñ Z p2q α as ρ Ñ 0 in S 1 0 pR d q.
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 in S 1 n pR d q:

pdenseq 1

pρ β λpρqq 1{α Y E ρ ñ Z E α,β if λpρqρ β Ñ 8, β, n as in (3.1), pintermediateq Y E ρ ñ J E a,α,β if λpρqρ β Ñ a q´β P p0, 8q, β, n as in (3.1), psparseq 1
pρ β λpρqq q{β Y E ρ ñ Z p1q β{q if λpρqρ β Ñ 0, λpρq Ñ 0 q´β , β, n as in (3.9), pvery sparseq 1 ρ q λpρq 1{α Y E ρ ñ Z p2q α if λpρqρ β Ñ 0, λpρq Ñ 8, β, n as in (3.12),
where in all cases the limit is considered as ρ Ñ 0 β´q .

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 [START_REF] Dobrushin | Gaussian and their subordinated self-similar random generalized fields[END_REF], using duality, we can define the following groups of transformations on S n pR d q:

' the group of shift transformations T " tτ h u hPR d :

τ h f ptq " f pt ´hq, f P S n pR d q, h P R d , t P R d ;
' the group of E-operator-scaling transformations ∆ E " tδ E c u cPp0,8q : δ E c f ptq " c ´qf pc ´E tq, f P S n pR d q, c P p0, 8q, q " trpEq, t P R d . Their analogous T , ∆ E on S 1 n pR d q are then defined by τ h Lpf q :" Lpτ h f q, and δ E c Lpf q :" Lpδ E c f q, for L P S 1 n pR d q. Let us note that when the tempered distribution L is given by a function g, one recovers that τ h L is given by the function gp¨`hq and δ E c L 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 E α,β in (3.2) is ' shift-invariant: @h P R d , τ h Z E α,β d " Z E α,β , ' pE, Hq-operator-scaling for H " q´β α P p´qp1 ´1{αq, 0q Y p0, a d {αq: @c ą 0, δ E c Z E α,β d " c H Z E α,β .
Let us remark that in [START_REF] Dobrushin | Gaussian and their subordinated self-similar random generalized fields[END_REF] 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 n pRq, one has for all h P R d ,

Z E α,β pτ h f q " ż R d ˆR`T E r f px ´hqM α,β pdx, drq d " Z E α,β pf q,
by a change of variable, while for all c ą 0,

Z E α,β pδ E c f q " ż R d ˆR`T E r δ E c f pxqM α,β pdx, drq " ż R d ˆR T E r{c f pc ´E xqM α,β pdx, drq d " c pq´βq{α ż R d ˆR T E r f pxqM α,β pdx, drq " c pq´βq{α Z E α,β pf q,
where the third step also followed from a change of variable argument.

For the intermediate case, the limit random field J E a,α,β 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 [START_REF] Biermé | Self-similar random fields and rescaled random balls models[END_REF]. Proposition 4.2. Under the assumption of Theorem 3.5,

δ E k 1{pq´βq J E a,α,β d " k ÿ i"1 J E,piq a,α,β , for all k P N,
where tJ E,piq a,α,β u i"1,...,k are i.i.d. copies of J E a,α,β . Furthermore,

1 a pq´βq{α J E a,α,β ñ Z E α,β
as a Ñ 0 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 of S α pσ, 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 with constant control measure on R d . Thus they inherit no specific anisotropic properties. Nevertheless, for any E 1 satisfying the same assumption as E with possibly different eigenvalues, writing q 1 " trpE 1 q, it can be shown that

δ E 1 c Z piq θ d " c 1´θ θ q 1 Z piq θ
for i " 1, 2 with legitimate parameter θ.

Comments on pointwise representation

Given a tempered distribution L P S 1 pR d q, it is a natural question to wonder if it may be represented by a Borel measurable function g, that is

@f P SpR d q, Lpf q " ż R d f ptqgptqdt.
We say that a generalized random field X admits a pointwise reprensentation if there exists a measurable random field t p Xptqu tPR d , meaning as in Definition 9.4.1 of [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF] that p X : ΩˆR d Ñ R is a jointly measurable function, such that

Xpf q " ż R d p Xptqf ptqdt, f P SpR d q.
Conversely, we have the following property. then the random field X, defined on S n pR d q by Xpf q " ş R d p Xptqf ptqdt, admits a regularization that is a generalized random field. Moreover, if p X is pE, Hq-operator-scaling for some H ą 0 in the sense of (1.1), then X is pE, Hq-operator-scaling in the sense of Proposition 4.1.

Proof. Under the assumption, one checks that for all f P S n pR d q, ż

R d | p Xptqf ptq|dt ď C k sup zPR d p1 `|z|q k |f pzq|,
where the random constant

C k " ş R d p1 `|t|q ´k| p
Xptq|dt is a.s. finite. This implies that the linear random field X is well-defined and a.s. continuous. Hence there exists a regularization of X on S 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 [START_REF] Biermé | Fractional Poisson field and fractional Brownian field: why are they resembling but different?[END_REF]. See Figures 123in the appendix. Proposition 5.2. For β P pq ´ad , qq, there exists a measurable version of p Z E α,β , also denoted by p Z E α,β , such that Z E α,β coincides in S 1 1 pR d q with the generalized random field

(5.2) f P SpR d q Þ Ñ ż R d p Z E α,β ptqf ptqdt. Proof. First note that ż R d
|1 B E px,rq ptq ´1B E px,rq p0q| α dx " r q hpr ´E tq, with hpzq " Leb d pB∆pz `Bqq. According to (2.4), h satisfies hpzq ď Cp|z| ^1q for some constant C ą 0. It follows that ż

R d ˆR`| 1 B E px,rq ptq ´1B E px,rq p0q| α σ α C β r ´1´β drdx ď Cσ α C β ż R `rq p|r ´E t| ^1qr ´1´β dr ď Cσ α C β ż R `rq p}r ´E } ^1qr ´1´β drp1 `|t|q " C E α,β p1 `|t|q, with C E α,β " Cσ α C β ş R `
P R d ˆRd ˆR`Þ Ñ p1 B E px,rq ptq ´1B E px,rq p0qq P R is measurable; (2) the control measure σ α C β r ´1´β drdx is σ-finite.
Noting that by [40, Property 1.2.17], we have

(5.3) E ´| p Z E α,β ptq| ¯ď Ep|S α |q `CE α,β p1 `|t|q ˘1{α ,
with S α a SαS random variable of scale parameter 1, we may define

f P SpR d q Þ Ñ ş R d p Z E α,β ptqf ptqdt that is a.s. in S 1 pR d q, thanks to Proposition 5.1.
Now it remains to show that the right-hand side of (5.2) has the same stable law as Z E α,β pf q " ş R d ˆR`T E r f pxqM α,β pdx, drq. For this we recall that (5.4)

ż p Z α,β ptqf ptqdt d " ż R d ˆR`ˆż R d `1B E px,rq ptq ´1B E px,rq p0q ˘f ptqdt ˙Mα,β pdx, drq, provided that ż R d | p Z E α,β ptq|f ptqdt ă 8 a.s.,
see [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF]Theorem 11.4.1]. Since f decays rapidly, the above follows from (5.3) and hence (5.4) holds. To complete the proof, it remains to remark that for f P S 1 pR d q, one has ż

R d
p1 B E px,rq ptq ´1B E px,rq p0qqf ptqdt " T E r f pxq.

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 field Z E α,β can be obtained as the derivative (in the sense of distributions) of a pointwise process. For all t P R d , following the same idea as for the definition of Z E α,β pf q for f P SpR d q, we can define the random variable

q Z E α,β ptq " pt 1 q ¨¨¨ pt d q ż R d ˆR`T E r 1 r0,ts pxqM α,β pdr, dxq,
where the random measure M α,β is the same as in (3.2), r0, ts " ś d i"1 r0, t i s, and pt i q is the sign of t i . The family q Z E α,β " t q Z E α,β ptqu tPR d is a measurable random field and, by successive integrations by parts, we can show that Z E α,β " D p1,...,1q q Z E α,β , that is for all f P SpR d q,

Z E α,β pf q " p´1q d ż R d q Z E α,β ptqD p1,...,1q f ptqdt.
This consideration is analogous to [12, Theorem 2.6 and Lemma 3.7] for E " I d and β ą q " d in D 1 pR d q the space of distribution instead of S 1 pR d q. We thus refer to [START_REF] Breton | Functional macroscopic behavior of weighted random ball model[END_REF] for technical details.

Proofs of the main results

6.1. Preliminary results. The proofs of our limit theorems follow the same scheme as in [START_REF] Biermé | Self-similar random fields and rescaled random balls models[END_REF] or [START_REF] Breton | Rescaled weighted random ball models and stable self-similar random fields[END_REF] 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 [START_REF] Biermé | Self-similar random fields and rescaled random balls models[END_REF], Lemma 3.2 in [START_REF] Breton | Rescaled weighted random ball models and stable self-similar random fields[END_REF]). Under the assumption (2.7), if tg ρ u ρą0 , g are continuous functions on R `such that (6.1) lim ρÑ0 β´q |gprq ´gρ prq| " 0, and for some 0 ă β ´ă β ă β `there exists a constant C ą 0 such that |gprq| ď Cpr β ´^r β `q, (6.2)

|g ρ prq| ď Cpr β ´^r β `q, (6.3) for all r ą 0, then, for C β as in (2.7), ż R `gρ prqF ρ pdrq " C β ρ β ż R `gprqr
´1´β dr, as ρ Ñ 0 β´q . Lemma 6.2 (Lemma 3.1 in [START_REF] Breton | Rescaled weighted random ball models and stable self-similar random fields[END_REF]). Suppose that M is in the domain of attraction of S α pσ, b, 0q for some α ą 1, σ ą 0 and b P R. Then φ G ptq " L M ptq ´1 ´itEpM q " ´|t| α φ α,b,σ ptq, as t Ñ 0, with (6.4) φ α,b,σ ptq " σ α p1 ´ib ptq tanpαπ{2qq, where ptq " signptq. Furthermore, there exists C ą 0 such that for all t P R,

(6.5) |φ G ptq| ď C|t| α .
The key ingredients for our generalized random ball model are the precise continuity properties of the operators T E r stated in the following proposition. Recall that we write v r " Leb d pB E p0, rqq " r q v B , r ą 0, and for γ ą 0, }f } γ L γ " ş R d |f pxq| γ dx. Proposition 6.3. (i) For all γ P r1, 2s, r ą 0, and f P SpR d q, (6. 6)

}T E r f } L γ ď v r }f } L γ , and
(6.7) }T E r f } L γ ď v 1{γ r }f } L 1 .
As a consequence, for γ P p1, 2s and β P pq, γqq, there exists some constant C ą 0 such that

(6.8) ż R `}T E r f } γ L γ r ´1´β dr ď C}f } γ L 1 XL γ , f P SpR d q, with }f } L 1 XL γ :" }f } L 1 _ }f } L γ .
(ii) For all γ P r1, 2s, r ą 1, and f P S 1 pR d q, (6.9)

}T E r f } γ L γ ď Cr q´a d p| log r| _ 1q d ´1}f } γ´1 L 1 ż R d |y||f pyq|dy,
where d ď d is the number of eigenvalues of E having the minimal real part a d (counted with multiplicities). As a consequence, for β P pq ´ad , qq there exists a constant C such that

(6.10) ż R `}T E r f } γ L γ r ´1´β dr ď C}f } γ´1 L 1 ż R d p1 `|y|q|f pyq|dy, f P S 1 pR d q.
Proof. (i) Note that

}T E r f } L 1 :" ż R d |T E r f pxq|dx ď ż R d ż R d K E r px, yq|f pyq|dydx,
with K E r px, yq " 1 B E px,rq pyq by (2.2). Hence, by Fubini's theorem, (6.11)

}T E r f } L 1 ď v r }f } L 1 . Moreover, }T E r f } 2 L 2 " ż R d |T E r f pxq| 2 dx ď ż R d v r ż R d K E r px, yq|f pyq| 2 dydx " v 2 r }f } 2 L 2
, where we first applied the Cauchy-Schwarz inequality, and Fubini's theorem at the end. According to the Riesz-Thorin interpolation theorem (see [START_REF] Bergh | Interpolation spaces. An introduction[END_REF]), combining this with (6.11), we get (6.6). Moreover, since by the Cauchy-Schwarz inequality we also have

}T E r f } 2 L 2 ď ż R d ż R d K E r px, yq|f pyq|dy}f } L 1 dx " v r }f } 2 L 1 ,
it follows by Hölder's inequality that, for p ą 1 such that γ " 1{p `2p1 ´1{pq,

}T E r f } γ L γ ď }T E r f } 1{p L 1 }T E r f } 2p1´1{pq L 2 ď v 1{p r }f } 1{p L 1 v 1´1{p r }f } 2p1´1{pq L 1 " v r }f } γ L 1 .
Since v r " r q v B with q " trpEq we can conclude that for β P pq, γqq, by (6.6) and (6.7), ż

R `}T E r f } γ L γ r ´β´1 dr ď ´pv B }f } γ L 1 q _ pv γ B }f } γ L γ q
¯żR `rq´β´1 ^rγq´β´1 dr. Therefore we have proved (6.8).

(ii) The assumption that f P S 1 pR d q implies that ş R d f pzqdz " 0 so that

T E r f pxq " ż R d KE r px, yqf pyqdy,
with KE r px, yq " 1 B E px,rq pyq ´1B E px,rq p0q. Then, by Hölder's inequality, one has

}T E r f } γ L γ " ż R d ˇˇˇż R d `1B E py,rq pxq ´1B E p0,rq pxq ˘f pyqdy ˇˇˇγ dx ď }f } γ´1 L 1 ż R d ˆżR d ˇˇ1 B E py,rq pxq ´1B E p0,rq pxq ˇˇγ |f pyq|dy ˙dx. Also, ż R d
ˇˇ1 B E py,rq pxq ´1B E p0,rq pxq ˇˇγ dx " Leb d pB E py, rq B E p0, rqq " r q hpr ´E yq with hpzq " Leb d pB E p0, 1q B E pz, 1qq " Leb d pB pz `Bqq, that does not depend on E. By (2.4), hpyq ď C|y| for all y P R d and it follows that, (

)

}T E r f } γ L γ ď C}f } γ´1 L 1 ż R d r q |r ´E y||f pyq|dy.
Recall that according to the Jordan decomposition theorem, given E, there exists an invertible matrix P such that D " P ´1EP has the real canonical form ¨J1 0 . . .

0 J p ‹ ',
where p corresponds to the number of distinct real parts of eigenvalues and each block matrix J is either (i) a Jordan cell matrix of size

¨a 0 0 1 a . . . . . . . . . 0 0 1 a ‹ ‹ ‹ ' , with a a real eigenvalue of E, or (ii) a 2 ˆ2 matrix in form of ¨Λ 0 I 2 Λ . . . . . . 0 I 2 Λ ‹ ‹ ‹ ' with Λ " ˆa b b a ˙and I 2 " ˆ1 0 0 1 ˙,
with a ˘ib (b ‰ 0) being complex conjugated eigenvalues of E.

In either case, for the subordinated norm }¨} of the Euclidean norm on R d , for each block J with the corresponding real part of eigenvalue denoted by a, it is shown in [8, Lemma 3.2] that r a ď › › r J › › ď ? 2 er a p| log r| _ 1q ´1, for all r ą 0. (This is slightly different from [8, Lemma 3.2], but can be easily established by following the proof carefully.) Recall that it is assumed that the real parts of eigenvalues of E satisfy a 1 ě ¨¨¨ě a d ą 0. Let d be the size of the Jordan block associated with a d and note that the other Jordan blocks, if they exist, are associated with a strictly greater real part. Then, there exists a constant C ą 0, such that › › r E › › ď Cr a d p| log r| _ 1q d ´1, for all r P p0, 1q. Now, it follows from (6.12) that for f P S 1 pR d q one has for r ą 1,

}T E r f } γ L γ ď Cr q´a d p| log r| _ 1q d ´1}f } γ´1 L 1 ż R d |yf pyq|dy.
Hence, for β P pq ´ad , qq, f P S 1 pR d q, combining the above inequality for r ą 1 with (6.7) for r ď 1, we obtain ż

R `}T E r f } γ L γ r ´1´β dr ď C ˆ}f } γ´1 L 1 ż R d p1 `|y|q|f pyq|dy żR `r´1´β`q ^´r ´1´β`q´a d p| log r| _ 1q d ´1¯d r,
which proves (6.10).

Dense regime.

Proof of Proposition 3.1. First, the stochastic integral Z E α,β pf q in (3.2) is well-defined as soon as ż

R d ˆR`| T E r f pxq| α r ´1´β drdx " ż R `}T E r f } α L α r ´1´β
dr ă 8 and this condition follows from Proposition 6.3, with γ " α, β, n as in (3.1). It is well known (see [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF]Chap. 3]) that the characteristic functional L Z E α,β of Z E α,β on S n pR d q is given by (3.3). Now, according to Theorem 2.1, to prove the existence of a generalized-random-field version of Z E α,β , it suffices to prove that L Z E α,β is continuous on S n pR d q, that is, for all tf k u kPN and

f in S n pR d q such that f k Ñ f in S n pR d q, lim kÑ8 L Z E α,β pf k q " L Z E α,β
pf q. This shall follow from the convergence in distribution of the random variables Z E α,β pf k ´f q to 0 as k Ñ 8, or equivalently from

lim kÑ8 ż R d ˆR`} T E
r pf k ´f q} α L α r ´1´β dr " 0. By (6.8) and (6.10) of Proposition 6.3 with γ " α, this is straightforward, since f k ´f Ñ 0 in S n pR d q clearly implies that the upper bounds also tend to 0.

Proof of Theorem 3.2. Note that, by Theorem 2.2, the result follows from the pointwise convergence of the characteristic functional. Further, by (2.9), we clearly have for f P S n pR d q,

L n 1 pρq ´1{α Y E ρ pf q " exp ˜żR d ˆR`φ G ˆT E r f pxq n 1 pρq 1{α ˙λpρqdxF ρ pdrq ¸.
Since n 1 pρq Ñ 8, by Lemma 6.2,

φ G ˆT E r f pxq n 1 pρq 1{α ˙"
´1 n 1 pρq |T E r f pxq| α φ α,b,σ pT E r f pxqq, as ρ Ñ 0 β´q , for φ α,b,σ defined in (6.4). Hence, under (2.7), one can apply Lemma 6.1 to prove that

L n 1 pρq ´1{α Y E ρ pf q Ñ L Z E α,β pf q.
Indeed, recall the uniform bound (6.5) on φ G and, thanks to Proposition 6.3, the fact that for n " 0,

}T E r f } α L α ď C E }f } α L 1
XL α pr q ^rαq q, and for n " 1,

}T E r f } α L α ď C E }f } α´1 L 1
ˆżR d p1 `|y|q|f pyq|dy ˙pr q ^rq´ap | logprq| d´1 q.

We can then apply Lemma 6.1 with g ρ prq " n 1 pρq ş R d φ G pn 1 pρq ´1{α T E r f pxqqdx to both cases β P pq, αqq and β P pq ´ad , qq.

Intermediate regime.

Proof of Proposition 3.4. Recall that the Poisson integral J E a,α,β pf q in (3.6) is well-defined as soon as ż

R d ˆR`ˆR``| mT E r,a f pxq| ^|mT E r,a f pxq| 2 ˘r´1´β dxdrGpdmq ă 8.
Let us remark that |mT E r,a f pxq| ^|mT E r,a f pxq| 2 ď |mT E r,a f pxq| γ , for any γ P r1, 2s. Hence, for β P pq ´ad , qq Y pq, αqq, choosing γ P r1, αq such that β P pq ´ad , γqq, one has ż

R d ˆR`ˆR`| mT E r,a f pxq| γ r ´1´β dxdrGpdmq ď Ep|M | γ q ż R `}T E r,a f } γ L γ r ´1´β dr ă 8,
in view of Proposition 6.3, since }T E r,a f } γ L γ " a q }T E r{a f } γ L γ (see (3.5)). It follows that the Poisson integral J E a,α,β pf q is well-defined for all f P S n pR d q and the characteristic functional

L J E a,α,β
of J E a,α,β is given by (3.7). Again, to show the existence of a version of J E a,α,β with values in S 1 n pR d q, using Theorem 2.1, it is sufficient to prove that the characteristic functional L J E a,α,β is continuous on S n pR d q. Let β P pq ´ad , qq Y pq, αqq and assume that f k Ñ 0 in S n pR d q. We will show that J E a,α,β pf k q converges in L γ to 0, which is sufficient to prove the continuity of L J E a,α,β . Actually, following the proof of Proposition 3.1 in [START_REF] Breton | Functional macroscopic behavior of weighted random ball model[END_REF], we can bound γ-moments of the real random variable J E a,α,β pf q for f P S n pR d q. Since J E a,α,β pf q is centered, for γ P r1, αq, following [17, p.461] and using Lemma 2 and Lemma 4 of [START_REF] Von Bahr | Inequalities for the rth absolute moment of a sum of random variables, 1 ď r ď 2[END_REF],

E `|J E a,α,β pf q| γ ˘ď Apγq ż 8 0 ˆ1 ´ˇˇL J E a,α,β pθf q ˇˇ2 ˙θ´1´γ dθ, with Apγq :" p ş 8 0 p1 ´cos xqx ´1´γ dxq ´1 ă 8. But ˇˇL J E a,α,β pθf q ˇˇě exp ˆ´C|θ| α ż R d ˆR`| T E r,a f pxq| α C β r ´1´β drdx ˙,
using the upper bound on |φ G | given (6.5). It follows that for γ P r1, αq one has

E `|J E a,α,β pf q| γ ˘ď Apγq ż 8 0 p1 ´exp ˆ´2C|θ| α ż R `}T E r,a f } α L α C β r ´1´β dr ˙θ´1´γ dθ ď ApγqApα, γq ˆC ż R `}T E r,a f } α L α C β r ´1´β dr ˙γ{α ,
with Apα, γq :" ş 8 0 p1 ´expp´s α qqs ´1´γ ds ă 8. Hence the result follows from Proposition 6.3 since }T E r,a f } α L α " a q }T E r{a f } α L α . Proof of Theorem 3.5. Again, by Theorem 2.2, the result follows from the convergence of the characteristic functionals. Observe that,

L J E a,α,β pf q " exp # ż R d ˆR`φ G pT E r,a f pxqqC β r ´1´β drdx + " exp # C β ż R d ˆR`φ G pT E s f pyqqa q´β s ´1´β dsdy
+ by the changes of variables y " a ´E x and s " r{a. The rest of the proof can be done similarly as for Theorem 3.2, starting from (2.9) and applying Lemma 6.1 with gprq " g ρ prq " ş R d φ G pT E r f pxqqdx and the help of Proposition 6.3. 6.4. Sparse regime.

Proof of Proposition 3.6. Using Theorem 2.1, it is sufficient to prove that Z p1q γ pf k q converges in distribution to 0 when f k Ñ 0 in SpR d q. This last assertion is obvious since convergence in SpR d q implies convergence in L γ pR d q.

To prove Theorem 3.7, we consider the maximal function f ˚associated to a function f of SpR d q,

f ˚pxq :" sup rą0 1 r q v B ż 1 B E px,rq pyq|f pyq|dy, x P R d ,
and we shall need the following lemma.

Lemma 6.4. For all f P SpR d q and all α ą 1, f ˚P L α pR d q.

Proof. By Lemma 6.1.5 in Meerschaert and Scheffler [START_REF] Meerschaert | Limit theorems for continuous-time random walks with infinite mean waiting times[END_REF], there exists a norm } ¨}0 on R d such that the mapping p0, 8q ˆtx

P R d | }x} 0 " 1u Ñ R d zt0u, pt, θq Þ Ñ t E θ, is a homeomorphism.
Further, the function t Þ Ñ }t E x} 0 is increasing for all x P R d . Thus, any x P R d zt0u can be uniquely written as x " τ pxq E θpxq with τ pxq ą 0 and }θpxq} 0 " 1. The function τ is a continuous function that can be extended to R d by setting τ p0q " 0. By Lemma 2.2 in Biermé et al. [START_REF] Biermé | Operator scaling stable random fields[END_REF], one can find κ ě 1 such that (6.13) τ px `yq ď κ pτ pxq `τ pyqq .

Therefore we can introduce the function δpx, yq " τ py ´xq, x, y P R d , which is a quasi-distance on R d . We also introduce the sets (6.14) C E px, rq " ty P R d | δpx, yq ă ru, r ą 0.

Since B is a bounded subset of R d , we can find a real r 0 ą 0 such that B Ă C E p0, r 0 q. With no loss of generality we assume that r 0 " 1 and we denote C :" C E p0, 1q. Thus C E px, rq " x `rE C for all x P R d and r ą 0, and B E px, rq Ă C E px, rq. We infer that for all x P R d ,

f ˚pxq ď v C v B sup rą0 1 r q v C ż 1 C E px,rq pyq|f pyq|dy.
The desired result is now a consequence of Theorem 1 and Example 2.4 in Stein [START_REF] Stein | Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[END_REF].

Proof of Theorem 3.7. By Theorem 2.2, it is sufficient to prove the convergence of the characteristic functionals. The characteristic functional of n 2 pρq ´1Y E ρ is given by, recalling that F ρ pdrq " F pdr{ρq,

L n 2 pρq ´1Y E ρ pf q " exp # ż R d ˆR`λ pρqφ G ˆT E r f pxq n 2 pρq ˙dxF ρ pdrq + " exp # ż R d ˆR`λ pρqφ G ˜T E n 2 pρq 1{q r f pxq n 2 pρq ¸dxF ρn 2 pρq ´1{q pdrq + .
We shall show that (6.15)

ż R d ˆR`λ pρqφ G ˜T E n 2 pρq 1{q r f pxq n 2 pρq ¸dxF ρn 2 pρq ´1{q pdrq Ñ C β ż R d ˆR`φ
G pf pxqv B r q qr ´1´β drdx as ρ Ñ 0 β´q and λpρq Ñ 0 q´β .

From this, we infer that

L n 2 pρq ´1Y E ρ pf q Ñ exp # C β ż R d ˆR`φ G pf pxqv B r q qr ´1´β drdx + " L Z p1q γ pf q,
for L Z p1q γ pf q given in (3.10), which completes the proof. The last equality above is obtained by following the same lines as in [11, pages 3650-3651].

To prove (6.15), recalling that λpρqρ β n 2 pρq ´β{q " 1, it suffices to check the conditions of Lemma 6.1 for

g ρ prq :" ż R d φ G ˜T E n 2 pρq 1{q r f pxq n 2 pρq ¸dx and gprq :" ż R d φ G pf pxqv B r q qdx.
First, remark that for f P S n pR d q,

T E n 2 pρq 1{q r f pxq n 2 pρq ÝÑ ρÑ0 β´q v B r q f pxq for dx-almost all x, so that φ G ˜T E n 2 pρq 1{q r f pxq n 2 pρq ¸ÝÑ ρÑ0 β´q φ G pv B r q f pxqq for dx-almost all x by continuity of φ G . But, by Lemma 6.2, ˇˇˇˇφ G ˜T E n 2 pρq 1{q r f pxq n 2 pρq ¸ˇˇˇˇď C ˇˇˇˇT E n 2 pρq 1{q r f pxq n 2 pρq ˇˇˇˇα ď Cpv B r q q α f ˚pxq α .
Since f ˚belongs to L α pR d q by Lemma 6.4, Condition (6.1) follows by Lebesgue's theorem.

Next, for Condition (6.2), we deal with the cases n " 0 and n " 1 separately. Now, since |φ G puq| ď Cp|u| ^|u| α q and f P L 1 pR d q X L α pR d q,

|gprq| ď C ż R d |f pxqv B r q | ^|f pxqv B r q | α dx ď Cp}f } L 1 v B _ }f } α L α v α B qpr q ^rαq q.
This establishes Condition (6.2) for β P pq, αqq and n " 0 with β ´" q and β `" αq. Next, when f P S 1 pRq, remark that

gprq " ż R d φ G pf pxqv B r q qdx " ż R d
φG pf pxqv B r q qdx, with φG puq " ş pe imu ´1qGpdmq so that now | φG puq| ď Cp1 ^|u| δ q for any δ P p0, 1s. Hence |gprq| ď Cv δ B }f } δ L δ r qδ . Choosing δ " q{pq `ad q P p0, 1q and δ " 1 respectively, we infer that for n " 1, Condition (6.2) holds for β P pq 2 {pq `ad q, qq with β ´" q 2 {pq `ad q and β `" q, respectively.

It remains to prove that (6.3) holds. We first consider β P pq, αqq. Using |φ G puq| ď C|u| and (6.7), (6.16)

|g ρ prq| ď C 1 n 2 pρq }T E n 2 pρq 1{q r f } L 1 ď C}f } L 1 r q .
Then, using |φ G puq| ď C|u| α , we can write

|g ρ prq| ď Cr αq ż R d ˇˇˇˇT E
n 2 pρq 1{q r f pxq n 2 pρqr q ˇˇˇˇα dx ď C}f ˚}α L α r αq , that finishes to prove (6.3) when β P pq, αqq. Finally, when β P pq 2 {pq `ad q, qq and f P S 1 pR d q, we write g ρ " g where κ ě 1 comes from the quasi-triangular inequality given in (6.13). With this choice we may write for any z P pn 2 pρq 1{q rq E B, τ pxq ď κ pτ px `zq `τ pzqq ď κ ´τ px `zq `n2 pρq 1{q r ¯,

where, with no loss of generality, we have again assumed that B Ă C E p0, 1q (recall (6.14)). It follows that τ px `zq ą 1 2κ τ pxq for any z P pn 2 pρq 1{q rq E B and x such that τ pxq ą 2κn 2 pρq 1{q r. Since f is rapidly decreasing, we get for N ě 1, ˇˇˇ1 n 2 pρq T E n 2 pρq 1{q r f pxq ˇˇˇď where here and below, the constant C " Cpf q does not depend on r and ρ. Using that | φG puq| ď C|u| δ for δ P p0, 1s, choosing N " N pδ, qq such that N δ ą q `1, it follows that (6.17 ˇˇT E n 2 pρq 1{q r f pxq ˇˇδ dx ď Cn 2 pρq ´δ }T E n 2 pρq 1{q r f } δ L pδ pn 2 pρqr q q 1´1{p , by Hölder's inequality for p ą 1. When n 2 pρq 1{q r ď 1, we use (6.7) with pδ P r1, 2s. It follows that (6.18) ˇˇg p1q ρ prq ˇˇď Cn 2 pρq ´δ pn 2 pρqr q q 1{p ˆpn 2 pρqr q q 1´1{p ď Cn 2 pρq 1´δ r q ď Cr qδ , since n 2 pρq ď r ´q. When n 2 pρq 1{q r ą 1, we use (6.9) for pδ P r1, 2s. By the assumption that β ą q 2 {pq `ad q, we can choose b P p0, a d q such that β ą q 2 {pq `bq and }T E n 2 pρq 1{q r f } δ L pδ ď C ´n2 pρq 1{q r ¯pq´bq{p , by (6.9) since b ă a d . Hence, ˇˇg p1q ρ prq ˇˇď Cn 2 pρq ´δ`1´b{qp r q´b{p . Now we can choose δ " q{pq `bq P p0, 1q and p " p1 `b{qq ą 1 such that δp " 1 and ˇˇg p1q ρ prq ˇˇď Cr q´b{p1`b{qq " Cr q 2 {pq`bq .

Combining with the previous bounds (6.17) and (6.18) for the same δ " q{pq `bq, we get |g ρ prq| ď Cr q 2 {pq`bq , and we have that (6.3) holds with β ´" q 2 {pq `bq and β `" q (which we have shown in (6.16) when considering the case β P pq, αqq). We have thus proved (6.15) and the theorem.

6.5. Very-sparse regime. Proposition 3.9 can be obtained as before using Theorem 2.1. The proof of Theorem 3.10 is similar to the one of Theorem 3.7 (see also the proof of [START_REF] Breton | Rescaled weighted random ball models and stable self-similar random fields[END_REF]Theorem 2.19]). The details of this part are thus omitted.

α " 1.7 α " 1.9 α " 2

3. 1 .

 1 Dense regime. In the dense regime, we consider λpρqρ β Ñ 8 as ρ Ñ 0 β´q , and the admissible range of parameters β and n is(3.1)

Theorem 3 . 5 .

 35 Suppose that the assumptions (2.7) and (2.8) on F and G hold. Under (3.1) and (3.4), Y E ρ ñ J E a,α,β as ρ Ñ 0 β´q in S 1 n pR d q. 3.3. Sparse regime. The sparse regime correspond to (3.8) λpρqρ β Ñ 0 as ρ Ñ 0 β´q with λpρq Ñ 0 q´β .

Proposition 5 . 1 .

 51 Let t p Xptqu tPR d be a measurable random field. If there exists k P N such that ż R d p1 `|t|q ´kEp| p Xptq|qdt ă 8,

τ pxqď2κn 2 pρq 1{q r φG ˜T E n 2 pρq 1{q r f pxq n 2 pρq ¸dx and g p2q ρ :" ż τ pxqą2κn 2

 2 pρq 1{q r φG ˜T E n 2 pρq 1{q r f pxq n 2 pρq ¸dx,

ż R d 1 r

 1 E B pzq ˇˇf px `n2 pρq E{q zq ˇˇdz ď C ż R d 1 r E B pzq ´1 `τ px `n2 pρq E{q zq ¯´N dz ď Cv B r q p1 `τ pxqq ´N ,

  ) ˇˇg p2q ρ prq ˇˇď Cr qδ ż R d p1 `τ pxqq ´N δ dx ď Cr qδ . Moreover, ˇˇg p1q ρ prq ˇˇď Cn 2 pρq ´δ ż τ pxqďCn 2 pρq 1{q r

Figure 1 ." 2 Figure 2 .

 122 Figure 1. Operator-scaling random ball with a1 " 1.2 and β " 1.6: the set B is an Euclidean ball, the weights vary according to a SαSpσq distribution with σ " 0.1.

8 Figure 3 .

 83 Figure 3. Operator-scaling random ball in high intensity with H " 1`a 1 ´β α " 0.3, a1 " 1.3, weights following a SαSpσq distribution with σ " 0.1, and different balls: B1 " tx P R 2 : |x1| `|x2| ď 1u, B 1{2 " tx P R 2 : |x1| 1{2 `|x2| 1{2 ď 1u and B8 " tx P R 2 : maxp|x1|, |x2|q ď 1u. Top: α " 2 (Gaussian case). Bottom: α " 1.9.

  r ą 0, based on a fixed bounded measurable set B Ă R d with 0 P B, v B :" Leb d pBq P p0, 8q and Leb d pBBq " 0, where Leb d is the Lebesgue measure on R d . Thus v r :" Leb d pB E px, rqq " r q v B . , R d q is 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 g B : R d Q x Þ Ñ Leb d pB X px `Bqq of the set B is Lipschitz, and thus there exists C ą 0 such that (2.4) Leb d pB∆px `Bqq " 2pg B p0q ´gB pxqq ď C|x|, for all x P R d .

	Note that we keep the name "random ball" from the original model but here the set B 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
		"ż	*
	(2.3)	PerpBq :" sup	divϕpxq dx : ϕ P C 1 c pR d , R d q, }ϕ} 8 ď 1	ă 8,
		B
	where C 1 c pR d

  Actually, there are two situations that we treated separately in the following sub-sections. B E px,rq ptq ´1B E px,rq p0qqM α,β pdr, dxq, t P R d , satisfying (1.1) and M α,β is the same as in the representation of Z E α,β . Let us introduce C E ptq " tpx, rq; r ´E px ´tq P Bu and note that p Z E α,β ptq " M α,β pC E ptq X C E p0q c q ´Mα,β pC E ptq c X C E p0qq , t P R d . Until here we do not need to assume that M α,β has skewness function b " 0. Hq-Takenaka random fields (see[40, p.405]), defined by choosing the Euclidean unit ball for B and E " I d , with a d " 1.

	Our centered rescaled random ball field Y E ρ defined in Section 2 clearly admits a pointwise
	representation where p Y E ρ " p X E ρ	´E p X E ρ and
		ż
	p X E ρ ptq "	R d ˆR`ˆR	mK E
					Proposi-
	tion 5.2 below, Z E α,β admits a pointwise representation with
	ż
	p Z E α,β ptq " 1 With the assumption that M α,β is symmetric, one can check that R d ˆR`p
	(5.1)	! Z E p α,β ptq	tPR d )	f.d.d.

1

n pR d q, see

[16, p.40]

. The last property of the proposition is straightforward. r px, tqN ρ pdx, dr, dmq, t P R d , with the same Poisson random measure N ρ than in (2.5). Let us consider the limit generalized random field Z E α,β of the dense regime in the case of symmetric weights (b " 0). 5.1. The case β P pq ´ad , qq and H " q´β α P p0, a d {αq. In this case, as proved in " tM α,β pV t qu tPR d , with V t " C E ptq∆C E p0q. That is, the random field p Z E α,β has a Chentsov's type representation [40, Chapter 8]. In particular, for H " q´β α P p0, a d {αq the random field p Z E α,β generalizes isotropic self-similar pα,

  p}r ´E } ^1qr q´β´1 dr ă 8 and } ¨} the subordinated norm, since β P pq ´ad , qq.

	Hence p Z E α,β ptq is well-defined and is a SαS random variable with scale
	parameter bounded by ´CE α,β p1 `|t|q	¯1{α	, for every t P R d . According to [40, Theorem 11.1.1]
	there exists a measurable version of p Z E α,β since
	(1) pt, x, rq

Acknowledgments. YW's research was supported in part by NSA grant H98230-16-1-0322, Army Research Laboratory grant W911NF-17-1-0006, and Charles Phelps Taft Research Center at University of Cincinnati.

Appendix A. Illustrations

We provide several simulations of our operator-scaling random ball model, obtained by following similar ideas as in [START_REF] Biermé | Fractional Poisson field and fractional Brownian field: why are they resembling but different?[END_REF]. For the sake of simplicity we choose E " diagpa 1 , a 2 q with a 1 ě a 2 :" 1 and β P pq ´ad , qq " pa 1 , a 1 `1q.


Étienne Le Quentrec
email: elequentrec@unistra.fr

Loïc Mazo

Étienne Baudrier

Mohamed Tajine

Local turn-boundedness: a curvature control for a good digitization

This paper focuses on the classical problem of the control of information loss during the digitization step. The properties proposed in the literature rely on smoothness hypotheses that are not verified by the curves including angular points. The notion of turn introduced by Milnor in the article On the Total Curvature of Knots generalizes the notion of integral curvature to continuous curves. Thanks to the turn, we are able to define the locally turn-boundedness. This promising property of curves do not require smoothness hypotheses and shares several properties with the par(r)-regularity, in particular well-composed digitizations. Besides, the locally turn-boundedness enables to constraint spatially the continuous curve in function of its digitization.

Introduction

The loss of information caused by a digitization process is inevitable. Therefore a fundamental point concerns the control of this information loss. Indeed, the border of a compact connected shape S can be arbitrarily far from the digitization of S or can oscillate around this latter border. Therefore, hypotheses on the border of the shape S, which is a Jordan curve noted by C, are needed. One of the most used hypothesis, called par(r)-regularity, has been introduced by Pavlidis in [START_REF] Pavlidis | Algorithms for graphics and image processing[END_REF]. It demands that any point c ∈ C has an interior osculating disk entirely included in the interior of C except for the point c and an exterior osculating disk entirely included in the exterior of C except for the point c. It has been used to prove some preserving topology properties [2, [START_REF] Latecki | Preserving topology by a digitization process[END_REF][START_REF] Pavlidis | Algorithms for graphics and image processing[END_REF] or to control the behavior of the projection from the digitized curve to the continuous curve C [3, 4]. However, the notion can be hard to manipulate in geometry and most of the authors using this notion add the assumption that the curve C is of class C 2 . The par(r)-regularity encompasses two ideas: the border of the shape has a bounded curvature from above and the shape has a positive minimal thickness. In this article, relaxing the assumption of a bounded curvature, we develop a new notion called local turn-boundedness that is defined on continuous curves, including polygons. The local turn-boundedness relies on the notion of turn adapted to regular curves and polygons, firstly introduced by Milnor [START_REF] Milnor | On the total curvature of knots[END_REF] to study the geometry of knots. The main properties and definitions around the notion of turn are recalled in Section 2. The local turn-boundedness involves the Euclidean distance between any two points of a curve and the turn of this curve between these two points. The properties of locally turn-bounded curves are given and illustrated in Section 3.

Turn of a simple curve

The definitions and properties given in this section are detailed in [START_REF] Alexandrov | Properties of gauss digitized shapes and digital surfaces integration[END_REF]. As presented in Proposition 2, the turn extends to continuous curves the notion of integral curvature already defined for regular curves.

Terminology and notations

-Let c ∈ R 2 and r ≥ 0, B(c, r) is the open disk of center c and radius r. -A parametrized curve is a continuous application from an interval [a, b] of R (a < b) to R 2 . It is simple if it is injective on [a, b) and closed if γ(b) = γ(a). A (geometric) curve is the image of a parametrized curve. A Jordan curve is a simple closed curve. -A polygonal line with vertices x 0 , ..., x N is noted [x 0 x 1 ...x N ] (if x N = x 0 ,
the polygonal line is a polygon). -Let N a positive integer and x 0 , x 1 , ..., x N points of R 2 . The polygonal line PL = [x 0 x 1 . . . x N ] can be considered as the image of the parametrized curve pl :

[0, N ] → R 2 such that pl(t) = x t (t -t ) + (1 -t + t )x t +1
where for r ∈ R, r in the integer part of the real r. In other words, for any integer i between 0 and N , if t ∈ [i, i + 1), then pl(t) = (t -i) -The angle between two vectors u and v is noted (u, v) ((u, v) ∈ R/2πZ). The geometric angle between two vectors u and v, noted (u,v), or two directed straight lines oriented by u and v, is the absolute value of the reference angle taken in (-π, π] between the two vectors. Thus, (u,v) ∈ [0, π).

x i + (1 -t + i)x i+1
Definition 1 (Turn).

-The turn κ(L) of a polygonal line L = [x i ] N i=0 is defined by:

κ(L) := N -1 i=1 (x i-1 x i , x i x i+1 ).
-The turn κ(P ) of a polygon P = [x i ] N i=0 (where x N = x 0 ) is defined by:

κ(P ) := N i=1 (x i-1 x i , x i x i+1 ).
-The turn κ(C) of a simple curve C is the upper bound of the turn of its inscribed polygonal lines -The turn κ(C) of a Jordan curve C is the upper bound of the turn of its inscribed polygons.

It should be noticed that the turn does not depend of the orientation of the curve. Indeed, it is well-known that (u

, v) = (-u, -v) = -(-v, -u). Thus κ [x i ] N i=0 = κ [x i ] 0
i=N . Furthermore, since the turn of a polygon is equal the upper bound of the turn of the polygonal lines inscribed in it (cf Corollary p. 119 [START_REF] Alexandrov | Properties of gauss digitized shapes and digital surfaces integration[END_REF]), the turn of the polygon seen as a closed curve is equal to the turn of the polygon. Hence the notation κ is well defined. In the same way that we estimate the length of a curve, the following proposition make it possible to calculate the turn thanks to multiscale samplings. Given a curve C, we denote by L(C) the length of C.

Proposition 1 (Convergence of the length and turn of a sequence of polygonal lines [START_REF] Alexandrov | Properties of gauss digitized shapes and digital surfaces integration[END_REF], p. 23, 30, 121, 122). Let C be a simple curve and (L m ) m∈N a sequence of polygonal lines inscribed in C and with same endpoints as C. If lim m→+∞ λ m = 0, where λ m is the maximal Euclidean distance between two consecutive vertices of L m , then

lim m→+∞ L(L m ) = L(C) and lim m→+∞ κ(L m ) = κ(C). Moreover, if κ(C) is finite, then L(C) is also finite, that is C is rectifiable. In Prop. 1, if we assume that the sequence (L m ) is increasing (L m is inscribed in L m+1 ), then the sequences (L(L m )) and (κ(L m )) are both increasing ([1, Lemma 5.1.1]).
Proposition 2 (Turn for regular curves [START_REF] Alexandrov | Properties of gauss digitized shapes and digital surfaces integration[END_REF], p. 133). Let γ : [0, ] → R 2 be a parametrization by arc length of a simple curve C. Assume that γ is of class C 2 and denote by k(s) the curvature at the point γ(s). Then,

κ(γ) = 0 k(s)ds.
For regular curves, therefore, the turn corresponds to the integral of the curvature (with respect to an arc-length parametrization).

The following theorem gives a lower bound of the turn for closed curves.

Theorem 1 (Fenchel's Theorem, [START_REF] Alexandrov | Properties of gauss digitized shapes and digital surfaces integration[END_REF] theorem 5.1.5 p.125). For any Jordan curve C, κ(C) ≥ 2π. Moreover κ(C) = 2π if and only if the interior of C is convex.

Proposition 3. Let C be a simple curve from a to b and γ be its parametrization. Let C x2 a and C b x1 be two arcs of C that overlap with a < x 1 < x 2 < b. Then,

κ(C) ≤ κ(C x2 a ) + κ(C b x1 ). Proof. Let (L m ) be a sequence of polygonal lines [l m,i ] Nm i=0 inscribed in C such that l m,0 = a, l m,Nm = b and : l m,i+1 -l m,i 2 ≤ 1 m .
Let m > 2/(x 2 -x 1 ) and denote by l m,i2 the largest l m,i such that l m,i lies in C x2 a and l m,i1 the smallest l m,i that lies in C b x1 . Then,

l m,i1 < l m,i2 .
Therefore, from Definition 1, we derive that

κ([l m ] Nm i=0 ) ≤ κ([l m ] i2 i=0 ) + κ([l m ] Nm i=i1 ).
Then, by Proposition 1,

κ(C) ≤ κ(C x2 a ) + κ(C b x1 ).
Without the hypothesis of an overlapping non reduced to a singleton, Proposition 3 is false in presence of angular points as it is illustrated in Figure 2.

Remark 1. The turn is stable under homothetic maps. Indeed, obviously, the turn is invariant by any conformal map, in particular by the homotheties.

σ(θ) = inf{ a -b 2 | κ(C b a ) > θ ∧ κ(C a b ) > θ}. ( 1
)
Let θ ≥ 0, δ ≥ 0. A Jordan curve C is locally turn-bounded with parameters (θ, δ) if δ ≤ σ(θ).
The Figure 3 illustrates the definition of the local turn with different curves.

π/4 • a • b σ(θ) θ = π/4 . . . a2 a1 • • • . . . b2 b1 • • • σ(θ) θ = π/3 σ(θ) • a • b θ = π/2
Figure 3: For the chosen value of θ, the corresponding θ-turn-step σ(θ).

On a locally turn-bounded curve with parameters (θ, δ) the smallest turn of the curve C between two points of C a and b such that b -a 2 < δ is less than or equal to θ.

The turn step function θ → σ(θ) is increasing. Indeed, the value of the θ-turn step is an infimum of a set that decreases (for the inclusion order) in function of θ. If the turn of C is finite, it exists a value θ max above which this set is empty and the θ max -turn step is then infinite.

Examples

-The 0-turn step of a Jordan curve is 0.

-The π-turn step of a convex Jordan curve is +∞ (see Theorem 1).

-The θ-turn step of a circle with radius r is 2r sin(θ/2) if θ ≤ π and is infinite for θ > π.

-The θ-turn step of a polygon having vertices with interior angles strictly less than π -θ is zero. Indeed, the turn of an arc joining two points arbitrary close, located before and after such a vertex is greater than θ. -The turn between two points on a polygonal curve is a finite sum of geometric angles, then the turn step function is a step function.

Remark 2. Local turn-boundedness is scale invariant: let C be a locally turnbounded Jordan curve with parameters (θ, δ). Then, the curve k C, k > 0, is locally turn-bounded with parameters (θ, kδ). It is a direct consequence of Remark 1.

The next proposition makes it possible to localize a locally turn-bounded curve from a sufficiently tight sampling. Figure 4 For π/2 < θ < π, in particular for polygons with acute angles, Proposition 5 does not hold (the intersection of the curve with a ball near an acute angle may have two connected components). Nevertheless, a weak version of the proposition could be stated: the intersection of the disk D(a, δ) with the dilation of C by the structuring element B(0, (1/ sin θ -1)δ) is connected. The information on the local turn makes it possible to localize the continuous curve from its digitization as illustrated by Fig. 6.

Proposition 7 (Hausdorff distance between the curve and a pixel).

Assuming an n-regular tiling of the plane with edge length h (n ∈ {3, 4, 6}), let C be a locally turn-bounded Jordan curve with parameters

(θ < 2π/n, δ > h √ n -2).
Let T be a tile crossed by C and a, b be respectively the infimum and the supremum of C ∩ T (C is ordered by some parametrization). Then, the Hausdorff distance between T and one of the segments of C bounded by a and b is less than h(1 -cos(θ))/2 sin(θ).

Proof. Let C 1 , C 2 be the arcs of C bounded by a and b. We assume κ(C 1 ) ≤ κ(C 2 ) (then, κ(C 2 ) ≥ π > θ). As the diameter of T is √ n -2h, by the hypothesis δ ≥ √ n -2h, κ(C 1 ) ≤ θ. If the arc C 1 leaves T , it intersects the border of T on a point c. By the definition of a and b there exists d = c such that d is on the border of T . The point d belongs to the same edge as c. Indeed, if it wasn't the case, there would be a polygonal line included in the border of T containing at least one vertex and separating an arc segment of C 1 from the line segment [c, d]. Then, by Lemma 1, the turn of C 1 would be greater than or equal to 2π/n > θ. Hence, c and d belongs to the same edge. Therefore, by the proposition 4, we derive that C 1 lies in the union of T with the interior of n truncated circles whose Hausdorff distance to T is h(1 -sin θ)/2 cos θ.

The next proposition makes a link between the well-composedness of the digitized object and a local turn-boundedness hypothesis. Let us first define the Gauss digitization and the well-composedness. Let h > 0 be a sampling grid step, the Gauss digitization of a shape K is defined as K ∩ (hZ) 2 . By abuse of language, given a Jordan curve C which is the border of the compact shape K, we define its Gauss digitization -we write Dig h (C)-as the border of the union of the squares p ⊕ [-h/2, h/2] × [-h/2, h/2] where ⊕ denotes the Minkowski sum and p ∈ K ∩ (hZ) 2 . The Gauss digitization of C is well-composed if it is a Jordan curve. Proposition 8. Let C be a locally turn-bounded Jordan curve with parameters (θ ∈ (0, π/2], δ) where δ is less than the diameter of C. Then, the Gauss digitization of C for a grid step h < δ/ √ 2 is almost surely a disjoint union of Jordan curves (the digitization of the shape is well-composed).

• b θ = π 3 , σ > h e = h( √ 3/6) e a b h • • θ = π 3 , σ > √ 2h e = h( √ 3/6) e • a • b θ = π 4 , σ > 2h e = h( √ 2 -
Proof. The proof is made by contradiction. So, let a be a double point on Dig h (C) traveled counterclockwise. Since the interior points of (hZ) 2 are on the left of the discrete curve, there is only one configuration modulo rotations and symmetries depicted in Fig 7 . Furthermore, we can assume almost surely C ∩ (hZ) 2 = ∅. Indeed, the set of translations for which a given point x ∈ (hZ) 2 lies on the translation of C is {x -p | p ∈ C}. In other words, it is a translation of -C (in the translation parameter space). Then, since C is rectifiable, this set is onedimensional. Therefore, the set S of translations for which there exists a point of (hZ) 2 on the translation of C is a countable union of one-dimensional sets. Thus, the Lebesgue measure of S is zero.

With the notation of the figure, we claim that on each edge of the square We began the proof of Prop.8 without any restriction on the localization of the curve (omitting "almost surely"). But there are many special cases to consider and it can not be given as part of a short article.

T = [I 1 , E 1 , I 2 , E 2 , I 1 ]

Conclusion

In this article, the notion of local turn, adapted to both regular curves and polygons having large enough interior angles, has been developed to have control on the curve without smoothness assumption. Indeed, it makes it possible to bound the domain where an arc of such a curve is lying, to guarantee (almost surely) the well-composedness of its Gauss digitization and to bound the Hausdorff distance between one pixel and an arc crossing it. The local turn needs to be related to others notions used in discrete geometry for similar purposes. To this end, we will prove in an upcoming article that a par(r)-regular curve of class C 1 is locally turn-bounded with parameters (θ, 2r sin(θ/2)) where θ is any angle in [0, π). Besides, we expect that these results will make it possible to correctly associate the vertices of a Gauss digitization to points on the continuous curve without smoothness assumption.

  , and thus pl([i, i + 1]) is the segment [x i , x i+1 ] of R 2 . A polygonal line is simple if it is simple for the previous parametrization and thus a simple polygon is a Jordan curve. -Given a curve C and two points a, b on C (a = b), we write C b a for the arc ending at a and b if C is not closed and C b a and C a b for the two arcs of C ending at a and b if C is closed.

Figure 1 :

 1 Figure 1: The turn of the polygon is the sum of the green angles.

Figure 2 :

 2 Figure 2: Counterexample: the turn of the arcs C a2 a1 and C a3 a2 are zero but the turn of the arc C a3a1 is nonzero. This is why the arcs are required to be overlapping in Proposition 3.

Figure 4 :

 4 Figure 4: Illustration of Prop. 4 for three values of the parameter θ: π/3, π/2, 2π/3. Given two points a, b ∈ C such that a -b 2 < σ(θ), then one of the arc of C between a and b belongs to the grey area.

Proposition 6 . 7 . 1 .Figure 5 :

 6715 Figure 5: Thick, blue: the curve C and the line segment [a, b]. Black: the polygonal line P = [ap 1 p 2 b]. Black, dashed: the projection of p 1 and p 2 on C yields the points q 1 and q 2 . Red: the polygonal line Q = [aq 1 q 2 b].

1 )/ 2 eFigure 6 :

 126 Figure 6: Gray: a tilel T with edge length h. Blue, thick: a locally turn-bounded curve arc passing through T . Red: the union of n circle arcs from which the edges of T are viewed from an angle 2π -θ. The Hausdorff distance between the circle arcs and the pixel T is e.

Figure 7 :

 7 Figure 7: Blue: the curve C. Red: the line segments [a 0 a], [aa 1 ], [a 2 a], [aa 3 ] are edges of Dig h (C). Black: the points E 1 , E 2 , I 1 , I 2 are in (hZ) 2 , E 1 , E 2 are exterior to C while I 1 , I 2 are interior to C.