In its simplest form, the central limit theorem states that if ${({X_{i}})}_{i\geqslant 1}$ is an independent identically distributed (i.i.d.) sequence of centered random variables having variance one, then the sequence ${({n^{-1/2}}{\sum _{i=1}^{n}}{X_{i}})}_{n\geqslant 1}$ converges in distribution to a standard normal random variable. If ${X_{1}}$ has a finite moment of order three, Berry [2] and Esseen [12] gave the following convergence rate: 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 ${X_{i}}$ 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 ${\mathbb{Z}^{d}}$ and more precisely in Bernoulli random fields, which are defined as follows.

We are interested in the asymptotic behavior of the sequence ${({S_{n}})}_{n\geqslant 1}$ defined by where ${b_{n}}:={({b_{n,\boldsymbol{i}}})}_{\boldsymbol{i}\in {\mathbb{Z}^{d}}}$ is an element of ${\ell ^{2}}({\mathbb{Z}^{d}})$. Under appropriated conditions on the dependence of the random field ${({X_{\boldsymbol{i}}})}_{\boldsymbol{i}\in {\mathbb{Z}^{d}}}$ and the sequence of weights ${({b_{n}})}_{n\geqslant 1}$ that will be specified later, the sequence ${({S_{n}}/\| {b_{n}}{\| _{2}})}_{n\geqslant 1}$ 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 $\operatorname{BL}(\theta )$-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 ${\mathbb{Z}^{d}}$ 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.

The following version of Rosenthal’s inequality is due to Johnson, Schechtman and Zinn [14]: if ${({Y_{i}})_{i=1}^{n}}$ are independent centered random variables with a finite moment of order $p\geqslant 2$, then where $\| Y{\| _{q}}:={(\mathbb{E}[|Y{|^{q}}])^{1/q}}$ for $q\geqslant 1$.

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.

Let ${({Y_{\boldsymbol{i}}})}_{\boldsymbol{i}\in {\mathbb{Z}^{d}}}$ be a random field. The sum ${\sum _{\boldsymbol{i}\in {\mathbb{Z}^{d}}}}{Y_{\boldsymbol{i}}}$ is understood as the ${\mathbb{L}^{1}}$-limit of the sequence ${({S_{k}})}_{k\geqslant 1}$ where ${S_{k}}={\sum _{\boldsymbol{i}\in {\mathbb{Z}^{d}},\| \boldsymbol{i}{\| _{\infty }}\leqslant k}}{Y_{\boldsymbol{i}}}$.

Following [27] we define the physical dependence measure.

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 ${\mathbb{Z}^{d}}$, ${({a_{\boldsymbol{i}}})}_{\boldsymbol{i}\in \varGamma }$ is a family of real numbers and $p\geqslant 2$, then for any Bernoulli random field ${({X_{\boldsymbol{n}}})}_{\boldsymbol{n}\in {\mathbb{Z}^{d}}}$, This was used in [11, 3] in order to establish functional central limit theorems. Truquet [26] also obtained an inequality in this spirit. If ${({X_{\boldsymbol{i}}})}_{\boldsymbol{i}\in {\mathbb{Z}^{d}}}$ is an i.i.d. and centered random field, (3) would give while Rosenthal’s inequality (3) would give what is a better result in this context.

In the case of linear processes, equality ${\delta _{\boldsymbol{j},p}}\leqslant K{\delta _{\boldsymbol{j},2}}$ holds for a constant K which does not depend on $\boldsymbol{j}$. However, there are processes for which such an inequality does not hold.

Consequently, having the ${\ell ^{p}}$-norm instead of the ${\ell ^{2}}$-norm of the ${({a_{\boldsymbol{i}}})}_{\boldsymbol{i}\in \varGamma }$ is more suitable.

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

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 ${\ell ^{p}}$-norm of the coefficients.

Let ${({X_{\boldsymbol{j}}})}_{\boldsymbol{j}\in {\mathbb{Z}^{d}}}=f({({\varepsilon _{\boldsymbol{j}-\boldsymbol{i}}})}_{\boldsymbol{i}\in {\mathbb{Z}^{d}}})$ be a centered square integrable Bernoulli random field and for any positive integer n, let ${b_{n}}:={({b_{n,\boldsymbol{i}}})}_{\boldsymbol{i}\in {\mathbb{Z}^{d}}}$ be an element of ${\ell ^{2}}({\mathbb{Z}^{d}})$. We are interested in the asymptotic behavior of the sequence ${({S_{n}})}_{n\geqslant 1}$ defined by Let us denote for $\boldsymbol{k}\in {\mathbb{Z}^{d}}$ the map ${\tau _{\boldsymbol{k}}}:{\ell ^{2}}({\mathbb{Z}^{d}})\to {\ell ^{2}}({\mathbb{Z}^{d}})$ defined by

In [15], Corollary 2.6 gives the following result: under a Hannan type condition on the random field ${({X_{\boldsymbol{i}}})}_{\boldsymbol{i}\in {\mathbb{Z}^{d}}}$ and under the condition on the weights that for any $q\in [d]$, the series ${\sum _{\boldsymbol{i}\in {\mathbb{Z}^{d}}}}|\operatorname{Cov}({X_{\mathbf{0}}},{X_{\boldsymbol{i}}})|$ converges, and the sequence ${({S_{n}}/\| {b_{n}}{\| _{{\ell ^{2}}}})}_{n\geqslant 1}$ converges in distribution to a centered normal distribution with variance ${\sigma ^{2}}$, where 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 The following quantity will also play an important role in the estimation of convergence rates:

Recall the notation (N.8). Let ${({\varLambda _{n}})}_{n\geqslant 1}$ be a sequence of subsets of ${\mathbb{Z}^{d}}$. The choice ${b_{n,\boldsymbol{j}}}=1$ if $\boldsymbol{j}\in {\varLambda _{n}}$ and 0 otherwise yields the following corollary for set-indexed partial sums.

We consider now the following regression model: where $g:{[0,1]^{d}}\to \mathbb{R}$ is an unknown smooth function and ${({X_{\boldsymbol{i}}})}_{\boldsymbol{i}\in {\mathbb{Z}^{d}}}$ is a zero mean stationary Bernoulli random field. Let K be a probability kernel defined on ${\mathbb{R}^{d}}$ and let ${({h_{n}})}_{n\geqslant 1}$ be a sequence of positive numbers which converges to zero and which satisfies

We estimate the function g by the kernel estimator ${g_{n}}$ defined by We make the following assumptions on the regression function g and the probability kernel K:

We measure the rate of convergence of ${({(n{h_{n}})^{d/2}}({g_{n}}(\boldsymbol{x})-\mathbb{E}[{g_{n}}(\boldsymbol{x})]))}_{n\geqslant 1}$ to a normal distribution by the use of the quantity Two other quantities will be involved, namely, and

Lemma 1 in [10] shows that under (25), the sequence ${({A_{n}})}_{n\geqslant 1}$ goes to 1 as n goes to infinity and that the integer ${n_{1}}$ is well defined.

We now consider the case of linear random fields in dimension 2, that is, where ${({a_{{i_{1}},{i_{2}}}})}_{{i_{1}},{i_{2}}\mathbb{Z}}\in {\ell ^{1}}({\mathbb{Z}^{2}})$ and ${({\varepsilon _{{u_{1}},{u_{2}}}})}_{{u_{1}},{u_{2}}\in {\mathbb{Z}^{2}}}$ is an i.i.d. centered random field and ${\varepsilon _{0,0}}$ has a finite variance. We will focus on the case where the weights are of the form ${b_{n,{i_{1}},{i_{2}}}}=1$ if $1\leqslant {i_{1}},{i_{2}}\leqslant n$ and ${b_{n,{i_{1}},{i_{2}}}}=0$ otherwise.

Mielkaitis and Paulauskas [18] established the following convergence rate. Denoting and assuming that $\mathbb{E}[|{\varepsilon _{0,0}}{|^{2+\delta }}]$ is finite and the following estimate holds for ${\varDelta ^{\prime }_{n}}$: In the context of Corollary 2, the condition on the coefficients reads as follows: where $p=2+\delta $. Let us compare (37) with (39). Let $s:=\max \{1+\alpha ,2-2/p+\beta \}$. When $s\geqslant 2$, (39) implies (37). However, this implication does not hold if $s\mathrm{<}3/2$. Indeed, let $r\in (s+1,5/2)$ and define ${a_{{k_{1}},{k_{2}}}}:={k_{1}^{-r}}$ if ${k_{1}}={k_{2}}\geqslant 1$ and ${a_{{k_{1}},{k_{2}}}}:=0$ otherwise. Then (39) holds whereas (37) does not.

Let us discuss the convergence rates in the following example. Let ${a_{{k_{1}},{k_{2}}}}:={2^{-|{k_{1}}|-|{k_{2}}|}}$ and let $p=2+\delta $, where $\delta \in (0,1]$. In our context, hence the convergence of ${\sum _{{j_{1}},{j_{2}}\in \mathbb{Z}}}|\operatorname{Cov}({X_{0,0}},{X_{{j_{1}},{j_{2}}}})|({j_{1}}+{j_{2}})$ guarantees that ${\varepsilon _{n}}$ 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 ${n^{-\delta +{r_{0}}}}$ for any fixed ${r_{0}}$. In particular, when $\delta =1$, one can reach for any fixed ${r_{0}}$ rates of the form ${n^{-1+{r_{0}}}}$. In comparison, with the same assumptions, the result of [18] gives ${n^{-3/8}}$.

We define for $j\geqslant 1$ and $\boldsymbol{i}\in {\mathbb{Z}^{d}}$, In this way, by the martingale convergence theorem, hence Let us fix $j\geqslant 1$. We divide ${\mathbb{Z}^{d}}$ into blocks. For $\boldsymbol{v}\in {\mathbb{Z}^{d}}$, we define and if K is a subset of $[d]$, we define Therefore, the following inequality takes place

Observe that the random variable ${\sum _{\boldsymbol{i}\in {A_{\boldsymbol{v}}}}}{a_{\boldsymbol{i}}}{X_{\boldsymbol{i},j}}$ is measurable with respect of the σ-algebra generated by ${\varepsilon _{\boldsymbol{u}}}$, where $\boldsymbol{u}$ satisfies $(2j+2){v_{q}}-(j+1)\leqslant {u_{q}}\leqslant j+1+(2j+2)({v_{q}}+1)-1$ for all $q\in [d]$. Since the family $\{{\varepsilon _{\boldsymbol{u}}},\boldsymbol{u}\in {\mathbb{Z}^{d}}\}$ is independent, the family $\{{\sum _{\boldsymbol{i}\in {A_{\boldsymbol{v}}}}}{a_{\boldsymbol{i}}}{X_{\boldsymbol{i},j}},\boldsymbol{v}\in {E_{K}}\}$ is independent for each fixed $K\subset [d]$. Using inequality (3), it thus follows that By stationarity, one can see that $\| {X_{\boldsymbol{i},j}}{\| _{q}}=\| {X_{\mathbf{0},j}}{\| _{q}}$ for $q\in \{2,p\}$, hence the triangle inequality yields By Jensen’s inequality, for $q\in \{2,p\}$, and using ${\sum _{i=1}^{N}}{x_{i}^{1/q}}\leqslant {N^{\frac{q-1}{q}}}{({\sum _{i=1}^{N}}{x_{i}})^{1/q}}$, it follows that

Combining (43), (46) and (50), we derive that In order to control the last term, we use inequality (3) and bound $\| \mathbb{E}[{X_{\boldsymbol{i}}}\mid {\varepsilon _{\boldsymbol{i}}}]{\| _{q}}$ by $\| {X_{\mathbf{0},0}}{\| _{q}}$ for $q\in \{1,2\}$. This ends the proof of Theorem 1.

Denote for a random variable Z the quantity

We say that a random field ${({Y_{\boldsymbol{i}}})}_{\boldsymbol{i}\in {\mathbb{Z}^{d}}}$ is m-dependent if the collections of random variables $({Y_{\boldsymbol{i}}},\boldsymbol{i}\in A)$ and $({Y_{\boldsymbol{i}}},\boldsymbol{i}\in B)$ are independent whenever

The proof of Theorem 2 will use the following tools.

Let ${({\varepsilon _{\boldsymbol{u}}})}_{\boldsymbol{u}\in {\mathbb{Z}^{d}}}$ be an i.i.d. random field and let $f:{\mathbb{R}^{{\mathbb{Z}^{d}}}}\to \mathbb{R}$ be a measurable function such that for each $\boldsymbol{i}\in {\mathbb{Z}^{d}}$, ${X_{\boldsymbol{i}}}=f({({\varepsilon _{\boldsymbol{i}-\boldsymbol{u}}})}_{\boldsymbol{u}\in {\mathbb{Z}^{d}}})$. Let $\gamma \mathrm{>}0$ and ${n_{0}}$ defined by (17).

Let $m:={([\| {b_{n}}{\| _{{\ell ^{2}}}}]+1)^{\gamma }}$ and let us define Since the random field ${({\varepsilon _{\boldsymbol{u}}})}_{\boldsymbol{u}\in {\mathbb{Z}^{d}}}$ is independent, the following properties hold. Define ${S_{n}^{(m)}}:={\sum _{\boldsymbol{i}\in {\mathbb{Z}^{d}}}}{b_{n,\boldsymbol{i}}}{X_{\boldsymbol{i}}^{(m)}}$. An application of (T.2) to $Z:={S_{n}^{(m)}}\| {b_{n}}{\| _{{\ell ^{2}}}^{-1}}{\sigma ^{-1}}$ and ${Z^{\prime }}:=({S_{n}}-{S_{n}^{(m)}})\| {b_{n}}{\| _{{\ell ^{2}}}^{-1}}{\sigma ^{-1}}$ yields Moreover, hence, by (P.1) and (T.1) applied with ${Y_{\boldsymbol{i}}}:={X_{\boldsymbol{i}}^{(m)}}/\| {S_{n}^{(m)}}{\| _{2}}$, ${p^{\prime }}$ instead of p and $2m+1$ instead of m, we derive that where By (P.2) and the reversed triangular inequality, the term $(I)$ can be bounded as and by (P.3) with $q=2$, we obtain that By (15), we have and we eventually get Since $n\geqslant {n_{0}}$, we derive, in view of (17),

In order to bound $(II)$, we argue as in [28] (p. 456). Doing similar computations as in [9] (p. 272), we obtain that where ${a_{n}}:=\| {S_{n}^{(m)}}{\| _{2}}{\sigma ^{-1}}\| {b_{n}}{\| _{{\ell ^{2}}}^{-1}}$. Observe that for any n, by (P.3), and using again (P.3) combined with Theorem 1 for $p=q=2$, This leads to the estimate and since $n\geqslant {n_{0}}$, we derive, in view of (17),

The estimate of $(III)$ relies on (P.3): hence The combination of (64), (71), (80) and (82) gives (18).

Since the random variables ${X_{\boldsymbol{i}}}$ are centered, we derive by definition of ${g_{n}}(\boldsymbol{x})$ that We define and ${b_{n,\boldsymbol{i}}}=0$ otherwise. Denote ${b_{n}}={({b_{n,\boldsymbol{i}}})}_{\boldsymbol{i}\in {\mathbb{Z}^{d}}}$ and $\| {b_{n}}{\| _{{\ell ^{2}}}}:={({\sum _{\boldsymbol{i}\in {\mathbb{Z}^{d}}}}{b_{n,\boldsymbol{i}}^{2}})^{1/2}}$. In this way, by (83) and (28), Applying (T.2) to and using Theorem 1, we derive that where We then use Theorem 2 to handle ${\varDelta ^{\prime }_{n}}$ (which is allowed, by (A)). Using boundedness of K, we control the ${\ell ^{p}}$ and ${\ell ^{{p^{\prime }}}}$ norms by a constant times the ${\ell ^{2}}$-norm. This ends the proof of Theorem 3.