VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA121 10.15559/18-VMSTA121 Research Article Convergence rates in the central limit theorem for weighted sums of Bernoulli random fields GiraudoDavidedavide.giraudo@rub.de Ruhr-Universität Bochum, Fakultät für Mathematik, IB 2/167, Universitätsstr. 150, 44780 Bochum, Germany 2019 2112201862251267 20122017 19102018 20102018 © 2019 The Author(s). Published by VTeX2019 Open access article under the CC BY license.

Moment inequalities for a class of functionals of i.i.d. random fields are proved. Then rates are derived in the central limit theorem for weighted sums of such randoms fields via an approximation by m-dependent random fields.

 Random fields moment inequalities central limit theorem 60F05 60G60 This research was supported by the grand SFB 823. Introduction and main results Goal of the paper

In its simplest form, the central limit theorem states that if (Xi)i⩾1 is an independent identically distributed (i.i.d.) sequence of centered random variables having variance one, then the sequence (n−1/2∑i=1nXi)n⩾1 converges in distribution to a standard normal random variable. If X1 has a finite moment of order three, Berry [2] and Esseen [12] gave the following convergence rate: supt∈R|P{n−1/2 ∑i=1nXi⩽t}−P{N⩽t}|⩽CE[|X1|3]n−1/2, where C is a numerical constant and N has the standard normal distribution. The question of extending the previous result to a larger class of sequences has received a lot of attention. When Xi can be represented as a function of an i.i.d. sequence, optimal convergence rates are given in [13].

In this paper, we will focus on random fields, that is, collections of random variables indexed by Zd and more precisely in Bernoulli random fields, which are defined as follows.

Let d⩾1 be an integer. The random field (Xn)n∈Zd is said to be Bernoulli if there exist an i.i.d. random field (εi)i∈Zd and a measurable function f:RZd→R such that Xn=f((εn−i)i∈Zd) for each n∈Zd .

We are interested in the asymptotic behavior of the sequence (Sn)n⩾1 defined by Sn:=∑i∈Zdbn,iXi, where bn:=(bn,i)i∈Zd is an element of ℓ2(Zd) . Under appropriated conditions on the dependence of the random field (Xi)i∈Zd and the sequence of weights (bn)n⩾1 that will be specified later, the sequence (Sn/‖bn‖2)n⩾1 converges in law to a normal distribution [15]. The goal of this paper is to provide bounds of the type Berry–Esseen in order to give convergence rates in the central limit theorem.

This type of question has been addressed for the so-called BL(θ) -dependent random fields [5], martingale differences random fields [19], positively and negatively dependent random fields [4, 20] and mixing random fields [1, 6].

In order to establish results of this kind, we need several ingredients. First, we need convergence rates for m-dependent random fields. Second, a Bernoulli random field can be decomposed as the sum of an m-dependent random field and a remainder. The control of the contribution of the remainder is done by a moment inequality in the spirit of Rosenthal’s inequality [24]. One of the main applications of such an inequality is the estimate of the convergence rates in the central limit theorem for random fields that can be expressed as a functional of an i.i.d. random field. The method consists of approximating the considered random field by an m-dependent one and controlling the approximation with the help of the established moment inequality. In the one-dimensional case, probability and moment inequalities have been established in [16] for maxima of partial sums of Bernoulli sequences. The techniques used therein permit to derive results for weighted sums of such sequences.

The paper is organized as follows. In Subsection 1.2, we give the material which is necessary to understand the moment inequality stated in Theorem 1. We then give the results on convergence rates in Subsection 1.3 (for weighted sums, sums on subsets of Zd and in a regression model) and compare the obtained results in the case of linear random fields with some existing ones. Section 2 is devoted to the proofs.

 Background

The following version of Rosenthal’s inequality is due to Johnson, Schechtman and Zinn [14]: if (Yi)i=1n are independent centered random variables with a finite moment of order p⩾2 , then ‖ ∑i=1nYi‖p⩽14.5plogp(( ∑i=1n‖Yi‖22)1/2+( ∑i=1n‖Yi‖pp)1/p), where ‖Y‖q:=(E[|Y|q])1/q for q⩾1 .

It was first establish without explicit constant in Theorem 3 of [24].

Various extensions of Rosenthal-type inequalities have been obtained under mixing conditions [25, 22] or projective conditions [21, 23, 17]. We are interested in extensions of (3) to the setting of dependent random fields.

Throughout the paper, we shall use the following notations.

For a positive integer d, the set {1,…,d} is denoted by [d] .

The coordinatewise order is denoted by ≼, that is, for i=(iq)q=1d∈Zd and j=(jq)q=1d∈Zd , i≼j means that ik⩽jk for any k∈[d] .

For q∈[d] , eq denotes the element of Zd whose qth coordinate is 1 and all the others are zero. Moreover, we write 0=(0,…,0) and 1=(1,…,1) .

For n=(nk)k=1d∈Nd , we write the product ∏k=1dnq as |n| .

The cardinality of a set I is denoted by |I| .

For a real number x, we denote by [x] the unique integer such that [x]⩽x<[x]+1 .

We write Φ for the cumulative distribution function of the standard normal law.

If Λ is a subset of Zd and k∈Zd , then Λ−k is defined as {l−k,l∈Λ} .

For q⩾1 , we denote by ℓq(Zd) the space of sequences a:=(ai)i∈Zd such that ‖a‖ℓq:=(∑i∈Zd|ai|q)1/q<+∞ .

For i=(iq)q=1d , the quantity ‖i‖∞ is defined as max1⩽q⩽d|iq| .

Let (Yi)i∈Zd be a random field. The sum ∑i∈ZdYi is understood as the L1 -limit of the sequence (Sk)k⩾1 where Sk=∑i∈Zd,‖i‖∞⩽kYi .

Following [27] we define the physical dependence measure.

Let (Xi)i∈Zd:=(f((εi−j))j∈Zd)i∈Zd be a Bernoulli random field, p⩾1 and (εu′)u∈Zd be an i.i.d. random field which is independent of the i.i.d. random field (εu)u∈Zd and has the same distribution as (εu)u∈Zd . For i∈Zd , we introduce the physical dependence measure δi,p:=‖Xi−Xi∗‖p, where Xi∗=f((εi−j∗)j∈Zd) and εu∗=εu if u≠0 , ε0∗=ε0′ .

In [11, 3], various examples of Bernoulli random fields are given, for which the physical dependence measure is either computed or estimated. Proposition 1 of [11] also gives the following moment inequality: if Γ is a finite subset of Zd , (ai)i∈Γ is a family of real numbers and p⩾2 , then for any Bernoulli random field (Xn)n∈Zd , ‖∑i∈ΓaiXi‖p⩽(2p∑i∈Γai2)1/2·∑j∈Zdδj,p. This was used in [11, 3] in order to establish functional central limit theorems. Truquet [26] also obtained an inequality in this spirit. If (Xi)i∈Zd is an i.i.d. and centered random field, (3) would give ‖∑i∈ΓaiXi‖p⩽C(∑i∈Γai2)1/2‖X1‖p, while Rosenthal’s inequality (3) would give ‖∑i∈ΓaiXi‖p⩽C(∑i∈Γai2)1/2‖X1‖2+C(∑i∈Γ|ai|p)1/p‖X1‖p, what is a better result in this context.

In the case of linear processes, equality δj,p⩽Kδj,2 holds for a constant K which does not depend on j . However, there are processes for which such an inequality does not hold.

We give an example of a random field for which there is no constant K such that δj,p⩽Kδj,2 holds for all j∈Zd . Let p⩾2 and let (εi)i∈Zd be an i.i.d. random field and for each k∈Zd , let fk:R→R be a function such that the random variable Zk:=fk(ε0) is centered and has a finite moment of order p, and ∑k∈Zd‖Zk‖22<+∞ . Define Xn:=limN→+∞∑−N1≼j≼N1fk(εn−k) , where the limit is taken in L2 . Then Xi−Xi∗=fi(ε0)−fi(ε0′) , hence δi,2 is of order ‖Zi‖2 while δi,p is of order ‖Zi‖p .

Consequently, having the ℓp -norm instead of the ℓ2 -norm of the (ai)i∈Γ is more suitable.

 Mains results

We now give a Rosenthal-like inequality for weighted sums of Bernoulli random fields in terms of the physical dependence measure.

Let (εi)i∈Zd be an i.i.d. random field. For any measurable function f:RZd→R such that Xj:=f((Xj−i)i∈Zd) has a finite moment of order p⩾2 and is centered, and any (ai)i∈Zd∈ℓ2(Zd) , ‖∑i∈ZdaiXi‖p⩽14.5plogp(∑i∈Zdai2)1/2 ∑j=0+∞(4j+4)d/2‖X0,j‖2+14.5plogp(∑i∈Zd|ai|p)1/p ∑j=0+∞(4j+4)d(1−1/p)‖X0,j‖p, where for j⩾1 , X0,j=E[X0∣σ{εu,‖u‖∞⩽j}]−E[X0∣σ{εu,‖u‖∞⩽j−1}] and X0,0=E[X0∣σ{ε0}] .

We can formulate a version of inequality (8) where the right-hand side is expressed in terms of the coefficients of the physical dependence measure. The obtained result is not directly comparable to (5) because of the presence of the ℓp -norm of the coefficients.

Let {εi,i∈Zd} be an i.i.d. set of random variables. Then for any measurable function f:RZd→R such that Xj:=f((Xj−i)i∈Zd) has a finite moment of order p⩾2 and is centered, and any (ai)i∈Zd∈ℓ2(Zd) , ‖∑i∈ZdaiXi‖p⩽214.5plogp(∑i∈Zdai2)1/2 ∑j=0+∞(4j+4)d/2(∑‖i‖∞=jδi,22)1/2+214.5plogpp−1(∑i∈Zd|ai|p)1/p ∑j=0+∞(4j+4)d(1−1/p)(∑‖i‖∞=jδi,p2)1/2.

Let (Xj)j∈Zd=f((εj−i)i∈Zd) be a centered square integrable Bernoulli random field and for any positive integer n, let bn:=(bn,i)i∈Zd be an element of ℓ2(Zd) . We are interested in the asymptotic behavior of the sequence (Sn)n⩾1 defined by Sn:=∑i∈Zdbn,iXi. Let us denote for k∈Zd the map τk:ℓ2(Zd)→ℓ2(Zd) defined by τk((xi)i∈Zd):=(xi+k)i∈Zd.

In [15], Corollary 2.6 gives the following result: under a Hannan type condition on the random field (Xi)i∈Zd and under the condition on the weights that for any q∈[d] , 1‖bn‖ℓ2‖τeq(bn)−bn‖ℓ2=0, the series ∑i∈Zd|Cov(X0,Xi)| converges, and the sequence (Sn/‖bn‖ℓ2)n⩾1 converges in distribution to a centered normal distribution with variance σ2 , where σ:=(∑i∈ZdCov(X0,Xi))1/2. The argument relies on an approximation by an m-dependent random field.

The purpose of the next theorem is to give a general rates of convergence. In order to measure it, we define Δn:=supt∈R|P{Sn‖bn‖ℓ2⩽t}−Φ(t/σ)|. The following quantity will also play an important role in the estimation of convergence rates: εn:=∑j∈Zd|E[X0Xj]|∑i∈Zd|bn,ibn,i+j‖bn‖ℓ2−1|.

Let p>2 }2$]]>, p′:=min{p,3} and let (Xj)j∈Zd=(f((εj−i)i∈Zd))j∈Zd be a centered Bernoulli random field with a finite moment of order p and for any positive integer n, let bn:=(bn,i)i∈Zd be an element of ℓ2(Zd) such that for any n⩾1 , the set {k∈Zd,bn,k≠0} is finite and nonempty, limn→+∞‖bn‖ℓ2=+∞ and (12) holds for any q∈[d] . Assume that for some positive α and β, the following series are convergent: C2(α):= ∑i=0+∞(i+1)d/2+α‖X0,i‖2andCp(β):= ∑i=0+∞(i+1)d(1−1/p)+β‖X0,i‖p.

Let Sn be defined by (11).

Assume that ∑i∈Zd|Cov(X0,Xi)| is finite and that σ be given by (13) is positive. Let γ>0 }0$]]> and let n0:=inf{N⩾1∣∀n⩾N,σ2+εn−29(log2)−1C2(α)([‖bn‖ℓ2]γ)−α⩾σ/2}. Then for each n⩾n0 , Δn˜⩽150(29([‖bn‖ℓ2]+21)γ+21)(p′−1)d‖X0‖p′p′(‖bn‖ℓp′‖bn‖ℓ2)p′(σ/2)−p′+(2|εn|σ2+80(log2)−1‖bn‖ℓ2−γασ2C2(α)2)(2πe)−1/2+(14.5pσlogp4d/2‖bn‖ℓ2−γαC2(α))pp+1+(‖bn‖ℓpσ‖bn‖ℓ214.5plogp4d(1−1/p)‖bn‖ℓ2−γβCp(β))pp+1. In particular, there exists a constant κ such that for all n⩾n0 , Δn⩽κ(‖bn‖ℓ2γ(p′−1)d−p′‖bn‖ℓp′p′+|εn|)+κ(‖bn‖ℓ2−γαpp+1+‖bn‖ℓppp+1‖bn‖ℓ2−pp+1(γβ+1)).

If (12), limn→+∞‖bn‖ℓ2=+∞ and the family (δi,2)i∈Zd is summable, then the sequence (εn)n⩾1 converges to 0 hence n0 is well defined. However, it is not clear to us whether the finiteness of C2(α) combined with (12) and limn→+∞‖bn‖ℓ2=+∞ imply that ∑j∈Zd|E[X0Xj]| is finite. Nevertheless, we can show an analogous result in terms of coefficients δi,p with the following changes in the statement of Theorem 2:

the definition of C2(α) should be replaced with C2(α):=2 ∑j=0+∞(j+1)d/2+α(∑‖i‖∞=jδi,22)1/2;

the definition of Cp(β) should be replaced with Cp(β):=2(p−1) ∑j=0+∞(j+1)d(1−1/p)+β(∑‖i‖∞=jδi,22)1/2.

In this case, the convergence of ∑i∈Zd|Cov(X0,Xi)| holds (cf. Proposition 2 in [11]).

Recall the notation (N.8). Let (Λn)n⩾1 be a sequence of subsets of Zd . The choice bn,j=1 if j∈Λn and 0 otherwise yields the following corollary for set-indexed partial sums.

Let (Xi)i∈Zd be a centered Bernoulli random field with a finite moment of order p⩾2 , p′:=min{p,3} and let (Λn)n⩾1 be a sequence of subset of Zd such that |Λn|→+∞ and for any k∈Zd , limn→+∞|Λn∩(Λn−k)|/|Λn|=1 . Assume that the series defined in (16) are convergent for some positive α and β, that ∑i∈Zd|Cov(X0,Xi)| is finite and that σ defined by (13) is positive. Let γ>0 }0$]]> and n0 be defined by (17). There exists a constant κ such that for any n⩾n0 , supt∈R|P{∑i∈ΛnXi|Λn|1/2⩽t}−Φ(t/σ)|⩽κ(|Λn|q+∑j∈Zd|E[X0Xj]|||Λn∩(Λn−j)||Λn|−1|), where q:=max{γ(p′−1)d−p′2+1;−γαp2(p+1);2−p−pγβ2(p+1)}.

We consider now the following regression model: Yi=g(in)+Xi,i∈Λn:={1,…,n}d, where g:[0,1]d→R is an unknown smooth function and (Xi)i∈Zd is a zero mean stationary Bernoulli random field. Let K be a probability kernel defined on Rd and let (hn)n⩾1 be a sequence of positive numbers which converges to zero and which satisfies limn→+∞nhn=+∞andlimn→+∞nhnd+1=0.

We estimate the function g by the kernel estimator gn defined by gn(x)=∑i∈ΛnYiK(x−i/nhn)∑i∈ΛnK(x−i/nhn),x∈[0,1]d. We make the following assumptions on the regression function g and the probability kernel K:

The probability kernel K fulfills ∫RdK(u)du=1 , is symmetric, non-negative, supported by [−1,1]d . Moreover, there exist positive constants r, c and C such that for any x,y∈[−1,1]d , |K(x)−K(y)|⩽r‖x−y‖∞ and c⩽K(x)⩽C .

We measure the rate of convergence of ((nhn)d/2(gn(x)−E[gn(x)]))n⩾1 to a normal distribution by the use of the quantity Δn˜:=supt∈R|P{(nhn)d/2(gn(x)−E[gn(x)])⩽t}−Φ(tσ‖K‖2)|. Two other quantities will be involved, namely, An:=(nhn)d/2(∑i∈ΛnK2(x−i/nhn))1/2‖K‖L2(Rd)−1(∑i∈ΛnK(x−i/nhn))−1/2 and εn:=∑j∈Zd|E[X0Xj]|(∑i∈Λn∩(Λn−j)K(x−i/nhn)K(x−(i−j)/nhn)∑k∈ΛnK2(x−k/nhn)−1).

Let p>2 }2$]]>, p′:=min{p,3} and let (Xj)j∈Zd=(f((εj−i)i∈Zd))j∈Zd be a centered Bernoulli random field with a finite moment of order p. Assume that for some positive α and β, the following series are convergent: C2(α):= ∑i=0+∞(i+1)d/2+α‖X0,i‖2andCp(β):= ∑i=0+∞(i+1)d(1−1/p)+β‖X0,i‖p.

Let gn(x) be defined by (26), (hn)n⩾1 be a sequence which converges to 0 and satisfies (25).

Assume that ∑i∈Zd|Cov(X0,Xi)| is finite and that σ:=∑j∈ZdCov(X0,Xj)>0 }0$]]>. Let n1∈N be such that for each n⩾n1 , 12⩽(nhn)−dK(x−i/nhn)⩽32 and 12‖K‖L2(Rd)⩽(nhn)−dK2(x−i/nhn)⩽32‖K‖L2(Rd). Let n0 be the smallest integer for which for all n⩾n0 , σ2+εn−29(log2)−1C2(α)([(∑i∈ΛnK(1hn(x−in))2)1/2]γ)−α⩾σ/2.

Then there exists a constant κ such that for each n⩾max{n0,n1} , Δn⩽κ|An−1|pp+1+|εn|+κ(nhn)d2(γ(p′−1)d−p′+2)+(nhn)−d2γαpp+1+(nhn)2d−p(γβ+1)2(p+1).

Lemma 1 in [10] shows that under (25), the sequence (An)n⩾1 goes to 1 as n goes to infinity and that the integer n1 is well defined.

We now consider the case of linear random fields in dimension 2, that is, Xj1,j2=∑i1,i2∈Zai1,i2εj1−i1,j2−i2, where (ai1,i2)i1,i2Z∈ℓ1(Z2) and (εu1,u2)u1,u2∈Z2 is an i.i.d. centered random field and ε0,0 has a finite variance. We will focus on the case where the weights are of the form bn,i1,i2=1 if 1⩽i1,i2⩽n and bn,i1,i2=0 otherwise.

Mielkaitis and Paulauskas [18] established the following convergence rate. Denoting Δn′:=supr⩾0|P{|1n ∑i1,i2=1nXi1,i2|⩽r}−P{|N|⩽r}| and assuming that E[|ε0,0|2+δ] is finite and ∑k1,k2∈Z(|k1|+1)2(|k2|+1)2ak1,k22<+∞, the following estimate holds for Δn′ : Δn′=O(n−r),r:=12min{δ,1−13+δ}. In the context of Corollary 2, the condition on the coefficients reads as follows: ∑i=0+∞(i1+α+i2−2/p+β)(∑(j1,j2):‖(j1,j2)‖∞=iaj1,j22)1/2<+∞, where p=2+δ . Let us compare (37) with (39). Let s:=max{1+α,2−2/p+β} . When s⩾2 , (39) implies (37). However, this implication does not hold if s<3/2 . Indeed, let r∈(s+1,5/2) and define ak1,k2:=k1−r if k1=k2⩾1 and ak1,k2:=0 otherwise. Then (39) holds whereas (37) does not.

Let us discuss the convergence rates in the following example. Let ak1,k2:=2−|k1|−|k2| and let p=2+δ , where δ∈(0,1] . In our context, ||Λn∩(Λn−j)||Λn|−1|⩽n2−(n−j1)(n−j2)n2⩽j1+j2n, hence the convergence of ∑j1,j2∈Z|Cov(X0,0,Xj1,j2)|(j1+j2) guarantees that εn in Corollary 2 is of order 1/n . Moreover, since (39) holds for all α and β, the choice of γ allows to reach rates of the form n−δ+r0 for any fixed r0 . In particular, when δ=1 , one can reach for any fixed r0 rates of the form n−1+r0 . In comparison, with the same assumptions, the result of [18] gives n−3/8 .

 Proofs Proof of Theorem <xref rid="j_vmsta121_stat_004">1</xref>

We define for j⩾1 and i∈Zd , Xi,j=E[Xi∣σ(εu,‖u−i‖∞⩽j)]−E[Xi∣σ(εu,‖u−i‖∞⩽j−1)]. In this way, by the martingale convergence theorem, Xi−E[Xi∣εi]=limN→+∞ ∑j=1NXi,j, hence ‖∑i∈ZdaiXi‖p⩽ ∑j=1+∞‖∑i∈ZdaiXi,j‖p+‖∑i∈ZdaiE[Xi∣εi]‖p. Let us fix j⩾1 . We divide Zd into blocks. For v∈Zd , we define Av:= ∏q=1d([(2j+2)vq,(2j+2)(vq+1)−1]∩Z), and if K is a subset of [d] , we define EK:={v∈Zd,vqis even if and only ifq∈K}. Therefore, the following inequality takes place ‖∑i∈ZdaiXi,j‖p⩽∑K⊂[d]‖∑v∈EK∑i∈AvaiXi,j‖p.

Observe that the random variable ∑i∈AvaiXi,j is measurable with respect of the σ-algebra generated by εu , where u satisfies (2j+2)vq−(j+1)⩽uq⩽j+1+(2j+2)(vq+1)−1 for all q∈[d] . Since the family {εu,u∈Zd} is independent, the family {∑i∈AvaiXi,j,v∈EK} is independent for each fixed K⊂[d] . Using inequality (3), it thus follows that ‖∑v∈EK∑i∈AvaiXi,j‖p⩽14.5plogp(∑v∈EK‖∑i∈AvaiXi,j‖22)1/2+14.5plogp(∑v∈EK‖∑i∈AvaiXi,j‖pp)1/p. By stationarity, one can see that ‖Xi,j‖q=‖X0,j‖q for q∈{2,p} , hence the triangle inequality yields ‖∑v∈EK∑i∈AvaiXi,j‖p⩽14.5plogp‖X0,j‖2(∑v∈EK(∑i∈Av|ai|)2)1/2+14.5plogp‖X0,j‖p(∑v∈EK(∑i∈Av|ai|)p)1/p. By Jensen’s inequality, for q∈{2,p} , (∑i∈Av|ai|)q⩽|Av|q−1∑i∈Av|ai|q⩽(2j+2)d(q−1)∑i∈Av|ai|q and using ∑i=1Nxi1/q⩽Nq−1q(∑i=1Nxi)1/q , it follows that ∑K⊂[d]‖∑v∈EK∑i∈AvaiXi,j‖p⩽14.5plogp‖X0,j‖2(∑i∈Zdai2)1/2(4j+4)d/2+14.5plogp‖X0,j‖p(∑i∈Zd|ai|p)1/p(4j+4)d(1−1/p).

Combining (43), (46) and (50), we derive that ‖∑i∈ZdaiXi‖p⩽14.5plogp ∑j=1+∞‖X0,j‖2(∑i∈Zdai2)1/2(4j+4)d/2+14.5plogp ∑j=1+∞‖X0,j‖p(∑i∈Zd|ai|p)1/p(4j+4)d(1−1/p)+‖∑i∈ZdaiE[Xi∣εi]‖p. In order to control the last term, we use inequality (3) and bound ‖E[Xi∣εi]‖q by ‖X0,0‖q for q∈{1,2} . This ends the proof of Theorem 1.

The following lemma gives a control of the Lq -norm of X0,j in terms of the physical measure dependence.

For q∈{2,p} and j∈N , the following inequality holds ‖X0,j‖q⩽(2(q−1)∑i∈Zd,‖i‖∞=jδi,q2)1/2.

Let j be fixed. Let us write the set of elements of Zd whose infinite norm is equal to j as {vs,1⩽s⩽Nj} where Nj∈N . We also assume that vs−vs−1∈{ek,1⩽k⩽d} for all s∈{2,…,Nj} .

Denote Fs:=σ(εu,‖u‖∞⩽j,εvt,1⩽t⩽s), and F0:=σ(εu,‖u‖∞⩽j) . Then X0,j=∑s=1NjE[X0∣Fs]−E[X0∣Fs−1] , from which it follows, by Theorem 2.1 in [23], that ‖X0,j‖q2⩽(q−1) ∑s=1Nj‖E[X0∣Fs]−E[X0∣Fs−1]‖q2. Then arguments similar as in the proof of Theorem 1 (i) in [27] give the bound ‖E[X0∣Fs]−E[X0∣Fs−1]‖q⩽δvs,q+δvs−1,q . This ends the proof of Lemma 1.  □

Now, Corollary 1 follows from an application of Lemma 1 with q=2 and q=p respectively.  □

 Proof of Theorem <xref rid="j_vmsta121_stat_006">2</xref>

Denote for a random variable Z the quantity δ(Z):=supt∈R|P{Z⩽t}−Φ(t)|.

We say that a random field (Yi)i∈Zd is m-dependent if the collections of random variables (Yi,i∈A) and (Yi,i∈B) are independent whenever inf{‖a−b‖∞,a∈A,b∈B}>m. }m.\]]]>

The proof of Theorem 2 will use the following tools.

By Theorem 2.6 in [7], if I is a finite subset of Zd , (Yi)i∈I an m-dependent centered random field such that E[|Yi|p]<+∞ for each i∈I and some p∈(2,3] and Var(∑i∈IYi)=1 , then δ(∑i∈IYi)⩽75(10m+1)(p−1)d∑i∈IE[|Yi|p].

By Lemma 1 in [8], for any two random variables Z and Z′ and p⩾1 , δ(Z+Z′)⩽2δ(Z)+‖Z′‖ppp+1.

Let (εu)u∈Zd be an i.i.d. random field and let f:RZd→R be a measurable function such that for each i∈Zd , Xi=f((εi−u)u∈Zd) . Let γ>0 }0$]]> and n0 defined by (17).

Let m:=([‖bn‖ℓ2]+1)γ and let us define Xi(m):=E[Xi∣σ(εu,i−m1≼u≼i+m1)]. Since the random field (εu)u∈Zd is independent, the following properties hold.

The random field (Xi(m))i∈Zd is (2m+1) -dependent.

The random field (Xi(m))i∈Zd is identically distributed and ‖Xi(m)‖p′⩽‖X0‖p′ .

For any (ai)i∈Zd∈ℓ2(Zd) and q⩾2 , the following inequality holds: ‖∑i∈Zdai(Xi−Xi(m))‖q⩽14.5qlogq(∑i∈Zdai2)1/2∑j⩾m(4j+4)d/2‖X0,j‖2+14.5qlogq(∑i∈Zd|ai|q)1/q∑j⩾m(4j+4)d(1−1/q)‖X0,j‖q. In order to prove (59), we follow the proof of Theorem 1 and start from the decomposition Xi−Xi(m)=limN→+∞∑j=mNXi,j instead of (42).

 Define Sn(m):=∑i∈Zdbn,iXi(m) . An application of (T.2) to Z:=Sn(m)‖bn‖ℓ2−1σ−1 and Z′:=(Sn−Sn(m))‖bn‖ℓ2−1σ−1 yields Δn⩽2δ(Sn(m)σ‖bn‖ℓ2)+σ−pp+11‖bn‖ℓ2pp+1‖Sn−Sn(m)‖ppp+1. Moreover, δ(Sn(m)σ‖bn‖ℓ2)=supt∈R|P{Sn(m)σ‖bn‖ℓ2⩽t}−Φ(t)| =supu∈R|P{Sn(m)‖Sn(m)‖2⩽u}−Φ(u‖Sn(m)‖2σ‖bn‖ℓ2)| ⩽δ(Sn(m)‖Sn(m)‖2)+supu∈R|Φ(u‖Sn(m)‖2σ‖bn‖ℓ2)−Φ(u)|, hence, by (P.1) and (T.1) applied with Yi:=Xi(m)/‖Sn(m)‖2 , p′ instead of p and 2m+1 instead of m, we derive that Δn⩽(I)+(II)+(III) where (I):=150(20m+21)(p′−1)d∑i∈Zd|bn,i|p′‖Xi(m)‖p′p′‖Sn(m)‖2−p′, (II):=2supu∈R|Φ(u‖Sn(m)‖2σ‖bn‖ℓ2)−Φ(u)|and (III):=σ−pp+11‖bn‖ℓ2pp+1‖Sn−Sn(m)‖ppp+1. By (P.2) and the reversed triangular inequality, the term (I) can be bounded as (I)⩽150(20m+21)(p′−1)d‖X0‖p′p′‖bn‖ℓp′p′(‖Sn‖2−‖Sn−Sn(m)‖2)−p′ and by (P.3) with q=2 , we obtain that (‖Sn‖2−‖Sn−Sn(m)‖2)−p′⩽(‖Sn‖2−29(log2)−1m−α‖bn‖ℓ2C2(α))−p′. By (15), we have ‖Sn‖22‖bn‖ℓ22=σ2+εn, and we eventually get (I)⩽150(20m+21)(p′−1)d‖X0‖p′p′(‖bn‖ℓp′‖bn‖ℓ2)p′·(σ2+εn−29(log2)−1m−αC2(α))−p′. Since n⩾n0 , we derive, in view of (17), (I)⩽150(20m+21)(p′−1)d‖X0‖p′p′(‖bn‖ℓp′‖bn‖ℓ2)p′(σ/2)−p′.

In order to bound (II) , we argue as in [28] (p. 456). Doing similar computations as in [9] (p. 272), we obtain that (II)⩽(2πe)−1/2(infk⩾1ak)−1|an2−1|, where an:=‖Sn(m)‖2σ−1‖bn‖ℓ2−1 . Observe that for any n, by (P.3), an⩾‖Sn‖2−‖Sn−Sn(m)‖2σ‖bn‖ℓ2⩾σ2+εn−29(log2)−1C2(α)m−ασ and using again (P.3) combined with Theorem 1 for p=q=2 , |an2−1|=|‖Sn(m)‖22σ2‖bn‖ℓ22−1| ⩽|‖Sn‖22σ2‖bn‖ℓ22−1|+|‖Sn(m)‖22−‖Sn‖22|σ2‖bn‖ℓ22 ⩽|εn|σ2+|‖Sn(m)‖2−‖Sn‖2|(‖Sn(m)‖2+‖Sn‖2)σ2‖bn‖ℓ22 ⩽|εn|σ2+‖Sn(m)−Sn‖2(‖Sn(m)‖2+‖Sn‖2)σ2‖bn‖ℓ22 ⩽|εn|σ2+40(log2)−1m−ασ2C2(α)2. This leads to the estimate (II)⩽(2πe)−1/2σ2+εn−29(log2)−1C2(α)m−α(|εn|σ+40(log2)−1m−ασC2(α)2), and since n⩾n0 , we derive, in view of (17), (II)⩽(2|εn|σ2+80(log2)−1‖bn‖ℓ2−γασ2C2(α)2)(2πe)−1/2.

The estimate of (III) relies on (P.3): (III)⩽σ−pp+1(14.5plogp∑j⩾m(4j+4)d/2‖X0,j‖2)pp+1+σ−pp+1‖bn‖ℓ2−pp+1‖bn‖ℓppp+1(14.5plogp∑j⩾m(4j+4)d(1−1/p)‖X0,j‖p)pp+1 hence (III)⩽(14.5pσlogp4d/2‖bn‖ℓ2−γαC2(α))pp+1+(‖bn‖ℓpσ‖bn‖ℓ214.5plogp4d(1−1/p)‖bn‖ℓ2−γβCp(β))pp+1. The combination of (64), (71), (80) and (82) gives (18).

 Proof of Theorem <xref rid="j_vmsta121_stat_009">3</xref>

Since the random variables Xi are centered, we derive by definition of gn(x) that (nhn)d/2(gn(x)−E[gn(x)])=(nhn)d/2∑i∈ΛnXiK(x−i/nhn)∑i∈ΛnK(x−i/nhn). We define bn,i=K(1hn(x−in)),i∈Λn, and bn,i=0 otherwise. Denote bn=(bn,i)i∈Zd and ‖bn‖ℓ2:=(∑i∈Zdbn,i2)1/2 . In this way, by (83) and (28), 1‖K‖L2(Rd)σ(nhn)d/2(gn(x)−E[gn(x)])=1σ∑i∈Zdbn,iXi‖bn‖ℓ2−1An. Applying (T.2) to Z=∑i∈Zdbn,iXi‖bn‖ℓ2−1andZ′=∑i∈Zdbn,iXi‖bn‖ℓ2−1σ−1(An−1) and using Theorem 1, we derive that Δn˜⩽cpΔn′+cp(σ−1C2(α)+Cp(β))pp+1|An−1|pp+1, where Δn′=supt∈R|P{Z⩽t}−Φ(tσ)|. We then use Theorem 2 to handle Δn′ (which is allowed, by (A)). Using boundedness of K, we control the ℓp and ℓp′ norms by a constant times the ℓ2 -norm. This ends the proof of Theorem 3.

 Acknowledgments

The author would like to thank the referees for many suggestions which improved the presentation of the paper.

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