Invariance principles for self-similar set-indexed random fields

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Invariance principles for self-similar set-indexed random fields

Hermine Biermé, Olivier Durieu

To cite this version:

Hermine Biermé, Olivier Durieu. Invariance principles for self-similar set-indexed random fields.

Transactions of the American Mathematical Society, American Mathematical Society, 2014, 366 (11),

http://www.ams.org/journals/tran/2014-366-11/S0002-9947-2014-06135-7/. �hal-00716437v2�

FIELDS

HERMINE BIERM ´E^1,2 & OLIVIER DURIEU²

Abstract. For a stationary random field (Xj)_j∈

Z^dand some measureµonR^d, we consider the set-indexed weighted sum processSn(A) =P

j∈Z^dµ(nA∩Rj)¹²Xj, whereRjis the unit cube with lower cornerj. We establish a general invariance principle under ap-stability assumption on the Xj’s and an entropy condition on the class of setsA. The limit processes are self- similar set-indexed Gaussian processes with continuous sample paths. Using Chentsov’s type representations to choose appropriate measuresµand particular sets A, we show that these limits can be L´evy (fractional) Brownian fields or (fractional) Brownian sheets.

1. Introduction

Let (Xj)_j∈Zd be a centered stationary random field withX0∈L². One can naturally associate to (Xj)_j∈Zd a set-indexed process by considering, forA∈ B(R^d),

S(A) =X

j∈A

Xj.

When theX_j⁰sare independent and identically distributed with Var(X0) = 1, the Central Limit Theorem ensures that, for convenient setsA, the normalized sequence n^−d/2S(nA)

n≥1converges in distribution to a centered Gaussian variable with variance given byλ(A), whereλis the Lebesgue measure on R^d. In order to establish invariance principles Alexander and Pyke [1] consider a

”smoothed” version defined as

(1) S_n(A) = X

j∈Z^d

b_n,j(A)X_j,

wherebn,j(A) =λ(nA∩Rj), withRj the unit cube with lower cornerj, and restrict the process to a sub-classAof Borel sets satisfying a metric entropy condition. The limit process (W(A))_A∈A is a Brownian process indexed byAwhich is a centered Gaussian process with

Cov(W(A), W(B)) =λ(A∩B), A, B∈ A.

Whend= 1 one recovers the classical Donsker Theorem, with Brownian motion at the limit, by considering the class R={[0, t];t∈[0,1]}. For dimensiond >2, it is usual to consider the class R = {[0, t];t ∈ [0,1]^d} with [0, t] = [0, t1]×. . .×[0, td] for t = (t1, . . . , td) ∈ [0,1]^d such that (W([0, t]))_t∈[0,1]d= (B(t))_t∈[0,1]d where (B(t))_t∈Rd is the Brownian sheet characterized by

(2) Cov(B(t),B(s)) = 1

2^d

d

Y

i=1

(|ti|+|si| − |ti−si|), for allt, s∈R^d.

Contrarily to the Brownian motion, the Brownian sheet does not have stationary increments. The independence property is also lost. A second generalization ford-dimensionally indexed Brownian field was introduced by L´evy [22] as a centered Gaussian random field (W(t))_t∈Rd with

Cov(W(t), W(s)) = 1

2(|t|+|s| − |t−s|), for allt, s∈R^d,

Date: March 20, 2013.

2010Mathematics Subject Classification. 60F17; 60G60; 60G18; 60G10; 60D05.

Key words and phrases. Dependent random field; Invariance principle; Set-indexed process; L´evy fractional Brownian field; Chentsov’s type representation; Physical dependence measure; Vapnik-Chervonenkis dimension.

1

where | · | denotes the Euclidean norm. Such a field has stationary increments and is linearly additive [23], which means that (W(a+rs))_r∈R has independent increments for any a, s ∈ R^d. Then, a natural question is to find a classAand weights (b_n,j(A))_A∈A to get the L´evy Brownian fieldW as the limit of an invariance principle. To our knowledge this question was only raised by Ossiander and Pyke [25] but their construction does not fit the setting of (1). The main ingredient is the geometric Chentsov construction of the L´evy Brownian field [7], which allows to identify (W(t))_t∈Rdas (M(At))_t∈Rd, for some convenient Borel setsAtandM a Gaussian random measure with control measureµgiven in a specific way (see Section 8.3 of [27]).

In this paper, we also deal with dependent data (Xj)_j∈Zd. Central limit theorems for stationary random sequences have been extensively studied under several dependence assumptions, see Hall and Heyde [16], Dedecker et al. [10], Bradley [5]. Extensions to stationary random fields are often more difficult due to the lack of order ofR^d. Nevertheless, limit theorems for dependent random fields can be found in the literature. Bolthausen [4] proved a central limit theorem forα-mixing random fields. Goldie and Greenwood [15] gave an invariance principle for the set-indexed process (1) with bn,j(A) = λ(nA∩Rj) in case of uniform φ-mixing random fields (see also Chen [6]).

Later, Dedecker [9] (see also [8]) obtained the invariance principle under a projective criterion.

Here we adopt the setting of physical dependence measure (p-stability) introduced in Wu [33]

in dimension 1 and extended by El Machkouri, Voln´y and Wu [14] to general dimension d ≥1.

Several limit theorems are proved underp-stability, see [34] and references therein. An invariance principle for the set-indexed process (1) withbn,j(A) =λ(nA∩Rj) is obtained in [14].

In Section 2 we recall the definition of p-stability and its main properties. We also give sev- eral examples of random fields satisfying p-stability. In Section 3, we consider Central Limit Theorem for weighted sums X

j∈Z^d

b_n,jX_j of 2-stable random fields, under appropriate assumptions on weights. A similar result was already obtained by Wang [32], but its proof, based on anm- dependent approximation and on a coefficient averaging procedure, requires asymptotic properties for averaged coefficients (named regular property in Definition 1 of [32]). Following [14] we use an m_n-dependent approximation and a result of Heinrich [18] which enable us to state the Central Limit Theorem under conditions that only depend on the coefficients. In Section 4, we prescribe the coefficients to be of the form

bn,j(A) = q

µ(nA∩Rj),

for some measure µ defined on (R^d,B(R^d)) absolutely continuous with respect to the Lebesgue measure and depending on a Borel setAin a particular classA. We provide general assumptions on bothµandAwhich ensure the convergence for finite dimensional distributions of the normal- ized set-indexed sequence (S_n(A))_A∈A defined in (1) to a centered set-indexed Gaussian process (σW(A))_A∈Awithσ >0 and

Cov(W(A), W(B)) =µ(A∩B) = 1

2(µ(A) +µ(B)−µ(A4B)).

An invariance principle is obtained under an additional entropy assumption on the classA, which may be obtained using Vapnik-Cheronenkis dimension ofA. Section 5 illustrates previous results with particular examples of self-similar processes. Our setting allows on the one hand to recover classical results for Brownian sheet consideringµ=λthe Lebesgue measure andA=R. On the other hand, it allows more flexibility and we can use Chentsov random fields representations. The L´evy Brownian field may be obtained as a particular case of field defined for a Chentsov measure on (R^d,B(R^d))

µ(dx) = dx

|x|^d−2H for some H∈(0, d/2]

and considering (D_t)_t∈[0,1]d with D_tthe Euclidean ball of diameter [[0, t]]. The limit random field isH-self-similar but does not have stationary increments except whenH= ¹₂, which corresponds to the L´evy Brownian field. Chentsov’s type construction for self-similar fields with stationary increments has been considered by Takenaka in [29] and generalized toα-stable fields in [30, 28].

This construction involves Borel setsV_tofR^d×R, identified in our setting withR^d+1, indexed by t∈R^d, and a Takenaka measure on (R^d×R,B(R^d×R)) defined by

µ(dx, dr) =r^2H−d−11r>0dxdr forH∈(0,1/2).

The limit field (BH(t))_t∈Rd is the well-known L´evy fractional Brownian field of Hurst parameter H, satisfying

Cov(BH(t), BH(s)) = 1

2(|t|^2H+|s|^2H− |t−s|^2H), for allt, s∈R^d.

Note that such a field is well defined for any H ∈ (0,1). However Chentsov’s construction is only possible when H ≤ ¹₂. The case H = ¹₂ corresponds to the L´evy Brownian field, whose construction was previously considered. Generalizations to others self-similar Brownian sheets are studied using products of L´evy Chentsov or Takenaka measures. Invariance principles are obtained using a tightness criterion which involves entropy conditions on the considered class of sets. These entropy conditions may be obtained using Vapnik-Cheronenkis dimension of the class, which are computed for the different class of sets considered for theses examples in the last section of this paper. Some technical proofs are postponed to Appendix.

2. Measure of dependence

Let (εj)_j∈Zd be a sequence of i.i.d. random variables,g be a measurable function fromR^Z

d to Rand consider the stationary random field (Xj)_j∈Zd defined by

(3) X_j=g(ε_j−k :k∈Z^d), j∈Z^d.

Such a process is called Bernoulli shift in [12]. The following measure of dependence has been introduced by Wu [33] (see also [34]). Letε⁰₀ be a copy ofε0 independent of (εj)_j∈Zd and define (ε^∗_j)_j∈Zd by ε^∗₀ = ε⁰₀ and ε^∗_j = εj for j 6= 0. For every p ∈ (0,+∞], if X0 ∈ L^p, the physical dependence measureδ_p,j ofX_j is given by

δp,j=kXj−X_j^∗kp

where (X_j^∗)_j∈Zd denotes the stationary random field defined by (3) with (ε^∗_j)_j∈Zd instead of (εj)_j∈Zd. Then, a random field (Xj)_j∈Zd is calledp-stableif it is defined as above and

∆_p= X

j∈Z^d

δ_p,j<+∞.

Note that p-stability implies p⁰-stability for any p⁰ ∈(0, p]. Let us also remark that when X0 is only assumed to be in L¹ and Xj−X_j^∗ ∈ L^p for all j ∈ Z^d with ∆p < +∞, Lemma 1 of [14]

implies in fact thatX0∈L^pand thus (Xj)_j∈Zdisp-stable. The following proposition is proved in El Machkouri et al. [14] for a finite set Γ⊂Z^d. Its generalization to infinite set is straightforward, as remarked in [32], using the convention thatX

j∈Γ

= lim

M→+∞

X

j∈Γ;|j|≤M

, with|j|the Euclidean norm ofj.

Proposition 2.1. Let (X_j)_j∈Zd be a centeredp-stable random field withp≥2.

(i) For all Γ⊂Z^d and for all real numbers(aj)_j∈Γ such thatX

j∈Γ

a²_j <+∞,

X

j∈Γ

ajXj

_p

≤p 2p



 X

j∈Γ

a²_j





1 2

∆p. (ii) The random field satisfies the short range property

X

k∈Z^d

|Cov(X0, Xk)|<+∞.

Let us give some examples of fields satisfying such an assumption.

Example 1. (linear random fields) This example is the first example given in [14] or [33] in dimension 1. Let (εj)_j∈Zdbe a sequence of i.i.d. centered random variables withε0∈L^pfor some p≥ 1 and consider a real filter a = (aj)_j∈Zd. When X

j∈Z^d

|aj| < +∞, one can define in L^p the centered linear random field

Xj = X

k∈Z^d

akε_j−k, j∈Z^d.

Whenp≥2, from Rosenthal’s inequality (Theorem 2.12 of [16]), this random field may be defined in L^p under the weaker assumption that X

j∈Z^d

a²_j <+∞. Note that, sincep≥1,X = (Xj)_j∈Zd is a stationary centered random field. Whenp≥2,X is a second order random field with

Cov(Xj+l, Xl) = X

k∈Z^d

akak−j.

Moreover, one has clearlyδp,j=|aj|kε0−ε⁰₀kp so X isp-stable if and only if X

j∈Z^d

|aj|<+∞.

Example 2. (functional of a linear random field) One can consider more general fields obtained as functionals of linear random fields. More precisely, let Y = (Yj)_j∈Zd be a centered L^p linear random field with filtera= (aj)_j∈Zdandp≥1, as introduced in the previous example and consider

X_j =f(Y_j), j ∈Z^d,

for some real functionf :R→R. As remarked in [14], whenf is a Lipschitz continuous function, thep-stability ofY implies thep-stability ofX. More generally, whenf isH-H¨older continuous withH ∈(0,1],X isp-stable iff(Y₀)∈L^p and

E |ε₀−ε⁰₀|^pH1/p X

j∈Z^d

|a_j|^H<+∞.

In particular, this allows the processY to be heavy-tailed.

Example 3. (functional of a Gaussian linear random field) Another interesting example is given by considering forY a Gaussian stationary process, assumingε0∼ N(0,1). NormalizingY in such a way that Var(Yj) = X

k∈Z^d

a²_k= 1, one can consider the first Wiener chaos (or Gaussian space)H1, defined as the closed linear subspace of L² generated by the family{Y_j, Y_j^∗;j ∈ Z^d}. Then, for everyq≥2, the symbolHq stands for theqthWiener chaos, defined as the closed linear subspace ofL²generated by the family{Hq(Yj), Hq(Y_j^∗);j ∈Z^d}, whereHq is theqth Hermite polynomial given by

(4) Hq(x) = (−1)^qe^x

2 2 d^q

dx^q e^−x

2 2

.

We write by convention H₀ =R. In this framework, we have for allq, q⁰ ≥1 andj, j⁰∈Z^d, (see Lemma 1.1.1 of [24]),

Cov

Hq(fYj), Hq⁰(Yfj⁰)

=q!δq,q⁰Cov(fYj,Yfj⁰)^q,

whereδq,q⁰ is the Kronecker symbol andYe is eitherY or Y^∗. It follows straightforwardly that kHq(Yj)−Hq(Y_j^∗)k²₂= 2q! Var(Yj)^q−Cov(Yj, Y_j^∗)^q

≤q!qkYj−Y_j^∗k²₂,

since Var(Y_j) = Var(Y_j^∗) = 1. Moreover using Hypercontractivity properties (see [21] p.65) one also has

kHq(Yj)−Hq(Y_j^∗)kp ≤ (p−1)^q/2kHq(Yj)−Hq(Y_j^∗)k2

≤ (p−1)^q/2p

q!qkYj−Y_j^∗k2.

It follows that 2-stability of Y implies p-stability of Hq(Y) = (Hq(Yj))_j∈

Z^d, for q ≥ 1. More generally, considering f(Y) = (f(Yj))_j∈

Z^d such thatf(Y0)∈L² andE(f(Y0)) = 0, according to the chaos expansion (see Theorem 1.1.1 of [24]) one has for allj∈Z^d the following equality inL²

(5) f(fY_j) =

+∞

X

q=1

c_q(f)H_q(fY_j),

withcq(f) =_q!¹E(f(Y0)Hq(Y0)) satisfying

+∞

X

q=1

q!cq(f)²<+∞. Assuming thatf is such that

(6)

+∞

X

q=1

pqq!(p−1)^q/2|c_q(f)|<+∞,

we obtain that, on the one hand (5) holds also inL^p, and on the other hand that kf(Yj)−f(Y_j^∗)kp≤

+∞

X

q=1

pqq!(p−1)^q/2|cq(f)|

!

kYj−Y_j^∗k2.

Therefore, under Assumption (6), the 2-stability ofY implies also thep-stability off(Y).

Example 4. (Volterra field) Another example, quoted in [14], is given by Volterra fields defined by X_j = X

k,l∈Z^d

a_k,lε_j−kε_j−l,

where (εj)_j∈Zd is a sequence of i.i.d. centered random variables with ε0 ∈ L^p for some p ≥ 2 and a= (ak,l)_k,l∈Zd is a sequence vanishing on the diagonal (ak,k = 0 for all k ∈Z^d) and such that X

k,l∈Z^d

a²_k,l<+∞. These assumptions ensure thatX_j is well defined onL² and (X_j)_j∈Zd is a stationary second order centered random field with

Cov(Xj+j⁰, Xj⁰) = X

k,l∈Z^d

ak,l(a_k−j,l−j+a_l−j,k−j).

One can easily computeXj−X_j^∗= (ε0−ε⁰₀)



 X

l∈Z^d

(aj,l+al,j)ε_j−l



. Then, by Rosenthal inequality

(see Theorem 2.12 of [16]), one can findCp>0 such that

E(|X_j−X_j^∗|^p)≤C_pE(|ε₀−ε⁰₀|^p)





kε₀k^p_pX

l∈Z^d

|a_j,l+a_l,j|^p+kε₀k^p₂



 X

l∈Z^d

|a_j,l+a_l,j|²





p/2



.

Hence,Xj−X_j^∗∈L^p and

kXj−X_j^∗kp≤C_p^1/pkε0−ε⁰₀kp(kε0kp+kε0k2)



 X

l∈Z^d

|aj,l+al,j|²





1/2

,

using



 X

l∈Z^d

|a_j,l+a_l,j|^p





1/p

≤



 X

l∈Z^d

|a_j,l+a_l,j|²





1/2

, since p≥2. Then under the additional assumption that

(7) X

j∈Z^d



 X

l∈Z^d

|aj,l+al,j|²





1/2

<+∞,

we obtain that ∆_p < +∞ which implies thatX₀ ∈ L^p and (X_j)_j∈Zd is p-stable. Finally, let us remark that, in particular, (7) is implied by the assumption that X

k,l∈Z^d

|ak,l|<+∞.

3. A central limit theorem for weighted sums

In this section we establish a central limit theorem for normalized weighted sums of 2-stable random fields. This situation arises for example when one works with linear random fieldsY_i =

X

j∈Z^d

a_i−jX_j for which the partial sum X

i∈Γn

Y_i on Γ_n ={0, . . . , n}^d can be written as X

j∈Z^d

b_n,jX_j

wherebn,j= X

i∈Γn

a_i−j. It is the case in Wang [32] or in Peligrad and Utev [26] for linear random sequences.

It also arises when one considers the smoothed partial sum process indexed by sets Sn(A) = X

j∈Γn

λ(nA∩Rj)Xj, where λis the Lebesgue measure onR^d and Rj is the unit cube with lower cornerj, as in [1], [9] or [14]. In this paper we are interested in the second situation but replacing the Lebesgue measure by different ones. Let us start with the following general result.

Let`²(Z^d) be the space of all sequencesa= (a_j)_j∈Zdindexed byZ^dsuch thatkak²:= X

j∈Z^d

a²_j <

+∞. Fork∈Z^d, we denote by τk the shift of vectorkon elements of`²(Z^d) defined by (τka)j :=

a_j+k,j∈Z^d.

Theorem 3.1. Let (X_j)_j∈Zd be a centered 2-stable random field (i.e. ∆₂ < +∞) and b_n = (b_n,j)_j∈Zd,n∈N, be sequences of real numbers in `²(Z^d). Then S_n = X

j∈Z^d

b_n,jX_j is well defined inL². If further

(i) sup

j∈Z^d

|bn,j| kbnk −−−−→

n→∞ 0,

(ii) for alle∈Z^d with |e|= 1, kτ_eb_n−b_nk kbnk −−−−→

n→∞ 0, thenkb_nk⁻²Var(S_n)→σ² asn→+∞and

S_n kbnk

−−−−→D

n→∞ N(0, σ²), whereσ²= X

k∈Z^d

Cov(X₀, X_k)and−^D→means convergence in distribution.

Remark. Note that Wang [32] in Theorem 2 proved a similar result. The dependence assumption is the same but our assumptions(i)and(ii)on the coefficients do not involve averaged coefficients as in Definition 1 of [32]. However whenkb_nk → ∞, our conditions(i)and(ii) are equivalent to Wang’s ones. Its proof is based on an approximation bym-dependent random fields, a coefficient- averaging procedure inspired by Peligrad and Utev [26], and a big/small blocking summation in order to apply a central limit theorem for triangular array of weighted i.i.d. random variables.

Our proof is inspired by El Machkouri et al. [14] and based on a theorem of Heinrich [18] for mn-dependent random fields.

Proof. 1)By stationarity, we have Var(Sn) = X

k∈Z^d

X

j∈Z^d

bn,jbn,kCov(Xj, Xk)

= X

k∈Z^d



 X

j∈Z^d

bn,jbn,j+k



Cov(X0, Xk).

(8)

Thus, using Cauchy-Schwarz inequality, Var(S_n) ≤ kb_nk² X

k∈Z^d

|Cov(X₀, X_k)| which is finite by Proposition 2.1. By assumption(ii)and triangular inequalities, we obtain that for allk∈Z^d,

kτ_kb_n−b_nk kbnk −−−−→

n→∞ 0.

Writingkτkbn−bnk²=kτkbnk²+kbnk²−2 X

j∈Z^d

bn,jbn,j+k, we infer 1

kbnk² X

j∈Z^d

bn,jbn,j+k−−−−→

n→∞ 1.

Therefore, by (8) and the dominated convergence theorem,

(9) Var

Sn

kbnk

−−−−→

n→∞

X

k∈Z^d

Cov(X₀, X_k) =σ². 2)We will apply a theorem of Heinrich [18] form_n-dependent random field.

Theorem (Heinrich, 1988). Let(V_n)_n∈N⊂Z^d be a sequence of finite sets such that|Vn| →+∞

and (m_n)_n∈N be a sequence of positive integers such that m_n → +∞. Assume for each n∈ N, (U_n,j)_j∈Zd is anm_n-dependent random field withE(U_n,j) = 0 for allj∈Z^d which satisfies:

• Var



 X

j∈Vn

U_n,j



−−−−→

n→∞ σ²<+∞,

• there exists a constant C >0, such that X

j∈Vn

E(U_n,j² )≤C, for all n∈N,

• for allε >0,Ln(ε) =m^2d_n X

j∈Vn

E

U_n,j² 1{^|Un,j|≥εm^−2dn }

−−−−→

n→∞ 0.

Then X

j∈Vn

Un,j converges in distribution toN(0, σ²).

We will have two steps of approximations ofS_n, first by a sum on a finite setV_n and then by a sum ofm_n-dependent variables.

3)LetV_n =

j∈Z^d:|j| ≤ϕ(n) withϕ(n)→+∞such that P

j∈Vnb²_n,j kbnk² −−−−→

n→∞ 1

and setS_Vn= X

j∈Vn

b_n,jX_j. Using Proposition 2.1, we obtain

(10) kSn−SV_nk2

kbnk ≤ 2 kbnk



 X

j∈Z^d\Vn

b²_n,j





1 2

∆₂−−−−→

n→∞ 0.

4) Form ∈ Nand j ∈ Z^d, we consider theσ-field Fm,j =σ{εj+i : |i| ≤m} and we define the random variable X_m,j =E(X_j|Fm,j). We denote S_m,Vn = X

j∈Vn

b_n,jX_m,j. In the same way, we

introduceF_m,j^∗ =σ

ε^∗_j+i : |i| ≤m ,X_m,j^∗ =E(X_j^∗|F_m,j^∗ ),j∈Z^d, and we define the coefficients δ_m,p,j=kX_j−X_m,j−(X_j^∗−X_m,j^∗ )k_p and ∆_m,p= X

j∈Z^d

δ_m,p,j,

for all p∈ (0,+∞]. By Lemma 2 of El Machkouri et al. [14], ∆p < +∞ implies ∆m,p → 0 as m→+∞. Thus, by Proposition 2.1 forp= 2, we get

(11) sup

n

kS_Vn−S_m,Vnk₂

kbnk ≤2∆_m,2−−−−→

m→∞ 0

5) Fix a sequence (mn)_n∈N and define Un,j = _kb^bn,j

nkXm_n,j. We will show that assumptions of Heinrich’s theorem are satisfied.

We have X

j∈Vn

U_n,j=kbnk⁻¹S_mn,Vn and equations (9), (10), (11) show that

Var

Smn,Vn

kbnk

−−−−→

n→∞ σ². We also have, usingE(X_m²

n,0)≤E(X₀²), X

j∈Vn

E(U_n,j² )≤ 1 kbnk²

X

j∈Vn

b²_n,jE(X₀²)≤E(X₀²)<+∞.

On the other hand,

Ln(ε) = m^2d_n kbnk²

X

j∈Vn

b²_n,jE X_m²_n,j1_{bn,jm2d

n|X_mn,j|≥εkbnk}

.

Thus by Jensen inequality and for a fixed sequence (a_n)_n∈N, we get E X_m²_n,j1_{bn,jm2d

n|X_mn,j|≥εkbnk}

≤E X_j²1_{bn,jm2d

n|X_mn,j|≥εkbnk}

≤E X_j²1_{|Xj|≥an}

+a²_nP bn,jm^2d_n |Xmn,j| ≥εkbnk Therefore,

L_n(ε)≤m^2d_n P

j∈Vnb²_n,j

kbnk² E X₀²1_{|X0|≥an}

+a²_n b²_n,j kbnk²

m^4d_n ε² E(X₀²)

!

≤m^2d_n E X₀²1_{|X0|≥an} +m^6d_n

ε² sup

j∈Z^d

b²_n,j

kb_nk²a²_nE(X₀²).

By assumption(i), we can choose the sequence (an)n∈Nsuch thatan→+∞and sup

j∈Z^d

|bn,j| kbnkan→0 asn→+∞. Then we can choosem_n→+∞such thatL_n(ε)→0 asn→+∞.

Thus by Heinrich’s theorem, we have Sm_n,V_n

kb_nk

−−−−→D

n→∞ N(0, σ²), and by (10) and (11) we obtain Sn

kbnk

−−−−→D

n→∞ N(0, σ²).

Theorem 3.1 will be used in the next section to derive limit theorems for set-indexed sums of random fields.

4. Invariance principle for some set-indexed sums

Let d ≥ 1 be a fixed integer. Let h·,·i denotes the Euclidean inner product and | · | the associated Euclidean norm onR^d. For eachj∈Z^d, we denote byRj the rectangle ofR^d given by [j₁, j₁+ 1)× · · · ×[j_d, j_d+ 1) forj = (j₁, . . . , j_d). We consider aσ-finite measureµon (R^d,B(R^d)) absolutely continuous with respect to the Lebesgue measure. In this section we focus on weighted sums where the weights are indexed by a Borelian setAas

(12) b_n,j(A) =

q

µ(nA∩R_j).

We are mainly interested in two kind of measures related to Chentsov random fields (see [27]

Chapter 8). The first one is defined on R^d by µ(dx) = _|x|d−2H^dx for some H ∈ (0, d/2]. The L´evy Chentsov random field is defined by considering a random Gaussian measure with µ as control measure for H = 1/2. The second one is used as control measure for the construc- tion of Takenaka random fields on R^d×R (identified with R^d+1 in the sequel) and defined by µ(dx, dr) = r^2H−d−11_r>0dxdr for H ∈ (0,1/2). Both of these measures satisfy the following general assumptions (see Section 5).

Assumption 1. µis aσ-finite measure on(R^d,B(R^d)), absolutely continuous with respect to the Lebesgue measure, and such that:

(1) There existsβ >0 such that µ(nA) =n^βµ(A), for all n∈NandA∈ B(R^d).

(2) Let us denoteπ(x) = min

1≤i≤d|hx, eii|, for x∈R^d and(ei)_1≤i≤d the canonical basis ofR^d, (2a) lim sup

π(j)→+∞

µ(Rj)<+∞.

(2b) For all e∈Z^d with |e|= 1 one has

µ(Rj+e) =µ(Rj) + o

π(j)→+∞(µ(Rj)).

Under Assumption 1, we will establish limit theorems for the set-indexed process defined in (1) with respect to (12), that is

(13) S_n(A) = X

j∈Z^d

q

µ(nA∩R_j)X_j.

Condition(1) will guarantee the self-similarity of the limit random field. One can notice that if (1)holds thenµ(cA) =c^βµ(A), for allc≥0 andA∈ B(R^d).

In the following, we need the notion of regular Borel set. We call A a regular Borel set if λ(∂A) = 0, where∂A denotes the boundary ofAandλis the Lebesgue measure on R^d. Sinceµ is absolutely continuous with respect to the Lebesgue measure, ifAis regular, thenµ(∂A) = 0.

4.1. Central limit theorem. We have the following central limit theorem for weighted set- indexed sums of 2-stable random fields.

Theorem 4.1. Let (X_j)_j∈Zd be a centered 2-stable random field (i.e. ∆₂ < +∞) and µ be a measure onR^d satisfying Assumption 1. For all regular Borel set Ain R^d, with µ(A)<+∞, the sequence Sn(A)defined in (13)verifies

1 n^β2

S_n(A)−−−−→^D

n→∞ σN(0, µ(A)), whereσ²= X

k∈Z^d

Cov(X0, Xk).

It is a direct consequence of Theorem 3.1 and of the following proposition.

Proposition 4.2. Let µ be a measure on R^d satisfying Assumption 1. For all regular Borel set A in R^d with 0< µ(A)<+∞, set b_n,j(A) =µ(nA∩R_j)¹², j ∈Z^d, and b_n(A) = (b_n,j(A))_j∈Zd. Thenbn(A)∈`²(Z^d)for all n∈Nand

(i) kbn(A)k²=n^βµ(A), (ii) sup

j∈Z^d

b_n,j(A) kbn(A)k −−−−→

n→∞ 0,

(iii) for all e∈Z^d with |e|= 1, kτ_eb_n(A)−b_n(A)k kbn(A)k −−−−→

n→∞ 0.

Proof. Fix a regular Borel set A in R^d with 0 < µ(A) < +∞. First, observe that by (1) of Assumption 1,

kbn(A)k²= X

j∈Z^d

µ(nA∩R_j) =µ(nA) =n^βµ(A).

This shows thatb_n∈`²(Z^d) and(i).

Further, by condition(2a) of Assumption 1, one can find M >0 such that sup

π(j)>M

µ(R_j)<+∞

and thus sup

π(j)>M

b_n,j(A)

kbn(A)k →0 asn→+∞. Moreover,

sup

π(j)≤M

µ(nA∩Rj) ≤ µ



nA∩



 [

π(j)≤M

Rj









≤ n^βµ A∩

x∈R^d;π(x)≤n⁻¹(M + 1) ,

using(1)of Assumption 1. Sinceµis absolutely continuous with respect to the Lebesgue measure, by the dominated convergence theorem, we obtain sup

π(j)≤M

b_n,j(A)

kbn(A)k →0 asn→+∞, which ends to prove(ii).

Let us now check condition(iii). Lete∈Z^d with|e|= 1 be fixed and define the sets:

V1,n(A) =

j∈Z^d:Rj∪Rj+e⊂nA , V_2,n(A) =

j∈Z^d: (R_j∪R_j+e)∩nA6=∅ and (R_j∪R_j+e)∩(nA)^c6=∅ , whereA^c denotes the complement of AinR^d. We have

kτ_eb_n(A)−b_n(A)k²= X

j∈V_1,n(A)

(b_n,j+e(A)−b_n,j(A))²+ X

j∈V_2,n(A)

(b_n,j+e(A)−b_n,j(A))²

= (I_n) + (II_n) .

Let us start to deal with the term (IIn). For allε >0, we denote theε-inside-neighborhoodof

∂Aby

(14) ∂˜εA=

x∈R^d: inf

y∈∂A|x−y| ≤ε

√ d

∩A,

where√

dcomes from the euclidean norm and is here to simplify the notations. We have (IIn)≤ X

j∈V2,n(A)

µ(nA∩(Rj∪Rj+e))

=µ



nA∩ [

j∈V2,n(A)

Rj



+µ



nA∩ [

j∈V2,n(A)

Rj+e





≤2µ( ˜∂2(nA))

= 2n^βµ( ˜∂²

nA).

By regularity ofA and absolute continuity ofµ, we haveµ( ˜∂εA)→0 as ε→0, using dominated convergence theorem. Thus, (IIn) =o(n^β).

On the other hand, using (√ a−√

b)²≤ |a−b|, we have

(In) = X

j∈V1,n(A)

(µ(Rj)¹² −µ(Rj+e)¹²)²≤ X

j∈V1,n(A)

|µ(Rj)−µ(Rj+e)|

(15)

Letε >0. Remark that by(2b)of Assumption 1, there existsM >0 such that forπ(j)≥M,

|µ(Rj)−µ(Rj+e)| ≤εµ(Rj) Therefore

X

j∈V1,n(A);π(j)≥M

|µ(Rj)−µ(Rj+e)| ≤εµ(nA).

Moreover

X

j∈V1,n(A);π(j)≤M

|µ(Rj)−µ(R_j+e)| ≤2µ



nA∩



 [

π(j)≤M+1

R_j









≤2n^βµ A∩

x∈R^d;π(x)≤n⁻¹(M+ 2) ,

using(1)of Assumption 1. Sinceµis absolutely continuous with respect to the Lebesgue measure, by dominated convergence theorem, one can findnM such that for alln≥nM

µ A∩

x∈R^d;π(x)≤n⁻¹(M + 2) ≤ε.

Hence, it is now clear that (In) =o(n^β) and condition (iii)follows.

In the following we consider the set-indexed process and extend the central limit theorem to a functional central limit theorem (or invariance principle). We first discuss the finite dimensional convergence.

4.2. Finite dimensional convergence. Again, letµ be aσ-finite measure on (R^d,B(R^d)) and (Xj)_j∈Zd be a stationary R-valued random field withE(X0) = 0. For a classA of regular Borel sets ofR^d, we define theA-indexed process

Sn(A) = X

j∈Z^d

µ(nA∩Rj)¹²Xj, A∈ A.

As before, we will use the notationb_n(A) to designate theZ^d-indexed sequenceb_n,j(A) =µ(nA∩ R_j)¹², j∈Z^d.

We have the following finite dimensional convergence for the normalized set-indexed process (S_n(A))_A∈A.

Theorem 4.3. Let (Xj)_j∈Zd be a centered 2-stable random field (i.e. ∆2 < +∞), let µ be a measure on R^d satisfying Assumption 1 and let A be a class of regular Borel sets of R^d with µ(A)<+∞ for anyA∈ A. Then

1

n^β/2Sn(A)

A∈A

−−−−→f dd

n→∞ (σW(A))_A∈A,

where −−→^{f dd} means convergence for finite dimensional distributions, (W(A))_A∈A is a centered Gaussian process with covariances

Cov(W(A), W(B)) =µ(A∩B), andσ²= X

k∈Z^d

Cov(X₀, X_k).

Remark. If the classAis stable by scalar multiplications,(1)of Assumption 1 guarantees that the limit random field (W(A))_A∈A is a self-similar process of order β/2, namely (cW(A))_A∈A ^f.d.d.= c^β2(W(A))A∈Afor allc≥0.

To prove the theorem we will use the following lemma which is a complement to Proposition 4.2.

Lemma 4.4. For all regular Borel setsA andB with µ(A)<+∞andµ(B)<+∞, asn→ ∞, hbn(A), b_n(B)i=n^βµ(A∩B) +o(n^β).

Proof. We define the sets Vn,1=

j ∈Z^d:Rj ⊂n(A∩B) , V_n,2=

j ∈Z^d:R_j∩n(A∩B)6=∅ \V_n,1, Vn,3=

j ∈Z^d:Rj∩nA6=∅ andRj∩nB6=∅ \Vn,1.

We can write

hbn(A), b_n(B)i= X

j∈Vn,1

µ(n(A∩B)∩R_j) + X

j∈Vn,3

µ(nA∩R_j)¹²µ(nB∩R_j)¹²

=µ(n(A∩B))− X

j∈Vn,2

µ(n(A∩B)∩R_j) + X

j∈Vn,3

µ(nA∩R_j)¹²µ(nB∩R_j)¹². Now, ifj∈V_n,2, the setn(A∩B)∩R_j is contained in the 1-inside-neighborhood of∂(n(A∩B)) as defined in (14). Remark also that ˜∂1(n(A∩B)) =n∂˜1

n(A∩B). Therefore, X

j∈Vn,2

µ(n(A∩B)∩R_j)≤µ



n(A∩B)∩ [

j∈Vn,2

R_j





≤µ(n∂˜1

n(A∩B))

=n^βµ( ˜∂1

n(A∩B)) =o(n^β).

In the same way, ifj∈Vn,3,n(A∪B)∩Rj is contained in the 1-inside-neighborhood of∂(nA) or in the 1-inside-neighborhood of∂(nB), and

X

j∈Vn,3

µ(nA∩R_j)¹²µ(nB∩R_j)¹² ≤µ



n(A∪B)∩ [

j∈Vn,3

R_j





≤n^βµ( ˜∂¹

nA) +n^βµ( ˜∂¹

nB) =o(n^β).

Proof of Theorem 4.3. We will use the Cram´er-Wold device and for simplicity only consider the casek= 2. LetAandB be two regular Borel sets of R^dwith µ(A)<+∞andµ(B)<+∞, and fixλ₁, λ₂∈R. We have

λ1Sn(A) +λ2Sn(B) = X

j∈Z^d

cn,jXj

withc_n,j =λ₁b_n,j(A) +λ₂b_n,j(B),j ∈Z^d. We can assumeµ(A)>0 andµ(B)>0, otherwise the result follows from Theorem 4.1. Denotingc_n= (c_n,j)_j∈Zd, we can write

kcnk²=λ²₁kbn(A)k²+λ²₂kbn(B)k²+ 2λ1λ2hbn(A), bn(B)i.

Thus by Proposition 4.2 and Lemma 4.4, we get kc_nk²

n^β −−−−→

n→∞ c:=λ²₁µ(A) +λ²₂µ(B) + 2λ₁λ₂µ(A∩B)≥0.

When c = 0, since Var(λ1Sn(A) +λ2Sn(B)) ≤ kcnk²σ², the random variables n^−β2(λ1Sn(A) + λ2Sn(B)) converge to 0 inL².

Now assume c > 0. We will show that the assumptions of Theorem 3.1 are satisfied for the sequencecn = (cn,j)_j∈Zd. For all j∈Z^d, we have

|c_n,j|

kcnk ≤ |λ1|kb_n(A)k kcnk

|b_n,j(A)|

kbn(A)k +|λ2|kb_n(B)k kcnk

|b_n,j(B)|

kbn(B)k. Since the sequenceskbn(A)k

kcnk andkbn(B)k

kcnk converge respectively toµ(A)¹²c⁻¹² andµ(B)¹²c⁻¹², and so are bounded, Proposition 4.2 shows that condition(i)of Theorem 3.1 holds.

Further, since for alle∈Z^d, kτec_n−c_nk

kcnk ≤ |λ1|kτeb_n(A)−b_n(A)k

kcnk +|λ2|kτeb_n(B)−b_n(B)k kcnk , the same argument shows that condition(ii)also holds. Then Theorem 3.1 shows that

n^−βVar(λ1Sn(A) +λ2Sn(B))→σ²c

asn→+∞and

1 n^β2

(λ₁S_n(A) +λ₂S_n(B))−−−−→^D

n→∞ N(0, σ²c).

We derive the convergence 1 n^β2

(Sn(A), Sn(B))−−−−→^D

n→∞ σ(W(A), W(B)), where the Gaussian processW verifies

Cov(W(A), W(B)) = lim

n→+∞

1

2n^βσ²[Var(Sn(A) +Sn(B))−Var(Sn(A))−Var(Sn(B))]

=1

2[µ(A) +µ(B) + 2µ(A∩B)−µ(A)−µ(B)]

=µ(A∩B).

4.3. Invariance principle. We keep the notations of the preceding section. We will establish an invariance principle for the set-indexed process (Sn(A))_A∈Aunder an entropy condition on the classA.

We equip the Borel σ-field B(R^d) with the pseudo-metricρ defined by ρ(A, B) = µ(A4B)¹². For a classA of Borel sets, we define the covering numberN(A, ρ, ε) as the smallest number of ρ-balls of radiusεneeded to coverAandN(A, ρ, ε) = +∞if there is no such finite covering. The entropyH(A, ρ, ε) is the logarithm ofN(A, ρ, ε). We also denote byC(A) the space of continuous functions onAequipped with the supremum norm.

Theorem 4.5. Letµbe a measure onR^d satisfying Assumption 1 and letAbe a class of regular Borel sets of R^d such that µ(A) < +∞ for any A ∈ A. Assume that one of the two following conditions holds:

(i) there exists p≥2 such that the centered random field(Xj)_j∈Zd is p-stable (i.e. ∆p <+∞) and

(16)

Z 1 0

N(A, ρ, ε)^1pdε <+∞;

(ii) for allp≥2, the centered random field(X_j)_j∈Zd isp-stable withsup

p≥2

∆_p<+∞and

(17)

Z 1 0

H(A, ρ, ε)¹²dε <+∞;

then

n^−β2Sn(A)

A∈A converges in distribution in C(A) to(σW(A))_A∈A, where (W(A))_A∈A is the centered Gaussian process with covariances

Cov(W(A), W(B)) =µ(A∩B), andσ²= X

k∈Z^d

Cov(X₀, X_k).

Remark. In Section 5, we present some applications to obtain particular limit processes. To this aim, it is possible to only use condition (i) of the theorem. Nevertheless, the entropy condition (17) is weaker and is the standard condition that we can expect in such an invariance principle.

Proof. The finite dimensional convergence of the process (n^−β2Sn(A))A∈Ais a direct consequence of Theorem 4.3. To prove the tightness, we will use the following lemma.

Lemma 4.6. If(Xj)_j∈Zd is a p-stable random field for somep≥2, then for all Borel sets Aand B,

n^−β2kSn(A)−Sn(B)kp≤p

2p∆pρ(A, B).

Proof. We haveS_n(A)−S_n(B) = X

j∈Z^d

(b_n,j(A)−b_n,j(B))X_j and by Proposition 2.1, n^−β2kSn(A)−Sn(B)kp≤p

2p∆pn^−β2kbn(A)−bn(B)k.

Now,

n^−βkbn(A)−b_n(B)k²=µ(A) +µ(B)−2n^−βhbn(A), b_n(B)i, while

hbn(A), bn(B)i= X

j∈Z^d

µ(nA∩Rj)¹²µ(nB∩Rj)¹²

≥ X

j∈Z^d

µ(n(A∩B)∩R_j) =µ(n(A∩B)).

Thus,n^−βkb_n(A)−b_n(B)k²≤µ(A) +µ(B)−2µ(A∩B) =ρ(A, B)²and the proof of the lemma

is complete.

First, assume condition(i) holds. Note that (16) implies thatN(A, ρ, ε)<+∞for allε >0, i.e. (A, ρ) is totally bounded. By Theorem 11.6 in Ledoux and Talagrand [21], as a consequence of Lemma 4.6 and condition (16), we obtain that for allε >0 there exists δ >0 such that for all n≥1,

E sup

A,B∈A, ρ(A,B)<δ

n^−β2|S_n(A)−S_n(B)|

!

< ε.

This last property implies that the process (n^−β2S_n(A))_A∈A is tight in C(A) and completes the proof of the theorem under(i).

Now, assume condition (ii) holds. Let us introduce the Young function ψ2, defined on R+

by ψ₂(x) = exp(x²)−1. Its associated Orlicz space is composed by random variables X such that Eψ₂(|X|/a) < +∞ for some a > 0 and is equipped with the norm kXkψ2 = inf{a > 0 : Eψ₂(|X|/a)≤1}. It is well known (see Lemma 4 in [14]) that there exists a positive constant C such that for every random variableX,

kXkψ2 ≤Csup

p≥2

kXkp

√p .

Thus, using Lemma 4.6, we obtain

(18) n^−β2kSn(A)−Sn(B)kψ₂≤C√ 2 sup

p≥2

∆pρ(A, B).

Now, applying Theorem 11.6 in [21], the inequality (18) and the condition (17) show that the process (n^−β2Sn(A))_A∈Ais tight inC(A) and completes the proof of the theorem under(ii).

To apply Theorem 4.5 to particular classes of Borel sets, we will need an upper bound of the associated covering number (in order to establish condition (16)). In the next section, we introduce the so-called Vapnik-Chervonenkis dimension which provides a useful tool to derive such bounds.

4.4. Vapnik-Chervonenkis dimension and covering numbers. LetEbe a set andAa class of subsets ofE. IfC is a finite subset ofE of cardinalityk, we denote

A ∩C={A∩C: A∈ A}

and we say thatAshattersCif Card(A ∩C) = 2^k. The classAhas a finiteVapnik-Chervonenkis dimension if there exists k ∈ N such that no set of cardinality k can be shattered by A. The VC-dimensionV(A) ofAis the smallestkwith this property.

For example the VC-dimension of the collection of all intervals of the form (−∞, t] in Ris 2.

More generally, the VC-dimension of the collection of all rectangles (−∞, t1]×. . .×(−∞, td] in R^d isd+ 1.

In some situations, a bound of the covering number can be derive from the Vapnik-Chervonenkis dimension: