Convergence rates in the central limit theorem for weighted sums of Bernoulli random fields

We prove moment inequalities for a class of functionals of i.i.d. random fields. We then derive rates in the central limit theorem for weighted sums of such randoms fields via an approximation by $m$-dependent random fields.


Introduction and main results
1.1. Goal of the paper. In its simplest form, the central limit theorem states that if (X i ) i 1 is an independent identically distributed (i.i.d.) sequence of centered random variables having variance one, then the sequence n −1/2 n i=1 X i n 1 converges in distribution to a standard normal random variable. If X 1 has a finite moment of order three, Berry [Ber41] and Esseen [Ess42] gave the following convergence rate: where C is a numerical constant and N has a standard normal distribution. The question of extending the previous result to a larger class of sequence have 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 [Jir16].
In this paper, we will focus on random fields, that is, collection of random variables indexed by Z d and more precisely in Bernoulli random fields, which are defined as follows.
Definition 1.1. Let d 1 be an integer. The random field (X n ) n∈Z d is said to be Bernoulli if there exist an i.i.d. random field (ε i ) i∈Z d and a measurable function f : R Z d → R such that X n = f (ε n−i ) i∈Z d for each n ∈ Z d .
We are interested in the asymptotic behavior of the sequence (S n ) n 1 defined by where b n := (b n,i ) i∈Z d is an element of ℓ 2 Z d . Under appropriated conditions on the dependence of the random field (X i ) i∈Z d and the sequence of weights (b n ) n 1 that will be specified later, the sequence (S n / b n 2 ) n 1 converges in law to a normal distribution [KVW16]. 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.
In order to establish this kind of results, 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 [Ros70]. 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 a random field consisting of i.i.d. random variables. The method consists in approximating the considered random field by an m-dependent one, and in controlling the approximation with the help of the established moment inequality. In the one dimensional case, probability and moment inequalities have been established in [LXW13] 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 Subsections 1.2, we give the material which is necessary to understand the moment inequality stated in Theorem 1.4. We then give the results on convergence rates in Subsection 1.3 (for weighted sums, sums on subsets of 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.
1.2. Backgroud. The following version of Rosenthal's inequality is due to Johnson, Schechtman and Zinn [JSZ85]: if (Y i ) n i=1 are independent centered random variables with a finite moment of order p 2, then It was first estalish without explicit constant in Theorem 3 of [Ros70]. Various extension of Rosenthal-type inequalities have been obtained under mixing conditions [Sha95,Rio00] or projective conditions [PUW07,Rio09,MP13]. We are interested by extensions of (1.2.1) to the setting of dependent random fields.
Throughout the paper, we shall use the following notations. 2) The coordinatewise order is denoted by , that is, (N.8) If Λ is a subset of Z d and k ∈ Z d , then Λ − k is defined as {l − k, l ∈ Λ}. (N.9) For q 1, we denote by ℓ q Z d the space of sequences a := (a i ) i∈Z d such that (N.10) For i = (i q ) d q=1 , the quantity i ∞ is defined as max 1 q d |i q |. Let (Y i ) i∈Z d be a random field. The sum i∈Z d Y i is understood as the L 1 -limit of the sequence Following [Wu05] we define the physical dependence measure.
In [EVW13,BD14], various examples of Bernoulli random fields are given, for which the physical dependence measure is either computed or estimated. Proposition 1 of [EVW13] also gives the following moment inequality: if Γ is a finite subset of Z d , (a i ) i∈Γ is a family of real numbers and p 2, then for any Bernoulli random field (X n ) n∈Z d , (1.

2.3)
This was used in [EVW13,BD14] in order to establish functional central limit theorems. Truquet [Tru10] also obtained an inequality in this spirit. If (X i ) i∈Z d is i.i.d. and centered, ( a better result in this context. In the case of linear processes, equality δ j,p Kδ j,2 holds for a constant K which does not depend on j. However, there are processes for which such an inequality does not hold. Example 1.3. We give an example of a random field such that there is no constant K such that δ j,p Kδ j,2 holds for all j ∈ Z d . Let p 2 and let (ε i ) i∈Z d be an i.i.d. random field and for each k ∈ Z d , let f k : R → R be a function such that the random variable Z k := f k (ε 0 ) is centered and has a finite moment of order p, and Consequently, having the ℓ p -norm instead of the ℓ 2 -norm of the (a i ) 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.
where for j 1, We can formulate a version of inequality (1.3.1) where the right hand side is expressed in terms of the coefficients of physical dependence measure. The obtained result is not directly comparable to (1.2.3) because of the presence of the ℓ p -norm of the coefficients. Corollary 1.5. Let ε i , i ∈ Z d be an i.i.d. set of random variables. Then for any measurable function f : R Z d → R such that X j := f (X j−i ) i∈Z d has a finite moment of order p 2 and is centered, and any Bernoulli random field and for any positive integer n, let b n := (b n,i ) i∈Z d be an element of ℓ 2 Z d . We are interested in the asymptotic behavior of the sequence (S n ) n 1 defined by In [KVW16], Corollary 2.6 gives the following result: under a Hannan type condition on the random field (X i ) i∈Z d and under the following condition on the weights: for any q ∈ [d], the series i∈Z d |Cov (X 0 , X i )| converges and with the sequence (S n / b n ℓ 2 ) n 1 converges in distribution to a centered normal distribution with variance σ 2 . The argument relies on an approximation by an m-dependent random field. The purpose of the next theorem is to give a general speed of convergence. In order to measure it, we define The following quantity will also play an important role in the estimation of convergence rates.
Bernoulli random field with a finite moment of order p and for any positive integer n, let b n := (b n,i ) i∈Z d be an element of ℓ 2 Z d such that for any n 1, the set k ∈ Z d , b n,k = 0 is finite and nonempty, lim n→+∞ b n ℓ 2 = +∞ and (1.3.5) holds for any q ∈ [d]. Assume that for some positive α and β, the following series are convergent: Let S n be defined by (1.3.4), Assume that i∈Z d |Cov (X 0 , X i )| is finite and that σ be given by (1.3.6) is positive. Let γ > 0 and let Then for each n n 0 , (1.3.11) In particular, there exists a constant κ such that for all n n 0 , Remark 1.7. If (1.3.5), lim n→+∞ b n ℓ 2 = +∞ and the family (δ i,2 ) i∈Z d is summable, then the sequence (ε n ) n 1 converges to 0 hence n 0 is well-defined. However, it is not clear to us whether the finiteness of C 2 (α) combined with (1.3.5) and lim n→+∞ b n ℓ 2 = +∞ imply that j∈Z d |E [X 0 X j ]| is finite. Nevertheless, we can show an analogous result in terms of δ i,p coefficients by changing the following in the statement of Theorem 1.6: (1) the definition of C 2 (α) should be replaced by (2) the definition of C p (β) should be replaced by (1.3.14) In this case, the convergence of i∈Z d |Cov (X 0 , X i )| holds (cf. Proposition 2 in [EVW13]).
Recall notation (N.8). Let (Λ n ) n 1 be a sequence of subsets of Z d . The choice b n,j = 1 if j ∈ Λ n and 0 otherwise yields the following corollary for set-indexed partial sums.
Corollary 1.8. Let (X i ) i∈Z d be a centered Bernoulli random field with a finite moment of order p 2, p ′ := min {p, 3} and let (Λ n ) n 1 be a sequence of subset of Z d such that |Λ n | → +∞ and for any k ∈ Z d , lim n→+∞ |Λ n ∩ (Λ n − k)| / |Λ n | = 1. Assume that the series defined in (1.3.9) are convergent for some positive α and β, that i∈Z d |Cov (X 0 , X i )| is finite and that σ defined by (1.3.6) is positive. Let γ > 0 and n 0 be defined by (1.3.10). There exists a constant κ such that for any n n 0 , We consider now the following regression model: 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: (A) The probability kernel K fulfills R d K (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 speed of convergence of (nh n ) to a normal distribution by the use of the quantity (1.3.20) Two other quantities will be involved, namely, Assume that i∈Z d |Cov (X 0 , X i )| is finite and that σ := j∈Z d Cov (X 0 , X j ) > 0. Let n 1 ∈ N be such that for each n n 1 , Let n 0 be the smallest integer for which for all n n 0 , Then there exists a constant κ such that for each n max {n 0 , n 1 }, . (1.3.27) Lemma 1 in [EMS10] shows that under (1.3.18), the sequence (A n ) n 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 i1,i2 ) i1,i2Z ∈ ℓ 1 Z 2 and (ε u1,u2 ) u1,u2∈Z 2 is i.i.d., centered and ε 0,0 has a finite variance. We will focus on the case where the weights are of the form b n,i1,i2 = 1 if 1 i 1 , i 2 n and b n,i1,i2 = 0 otherwise. Mielkaitis and Paulauskas [MP11] established the following convergence rate. Denoting and assuming that E |ε 0,0 | 2+δ is finite and k1,k2∈Z the following estimate holds for ∆ ′ n : (1.3.31) In the context of Corollary 1.8, the condition on the coefficients reads as follows: Let us discuss the convergence rates in the following example. Let a k1,k2 := 2 −|k1|−|k2| and let p = 2 + δ, where δ ∈ (0, 1]. In our context, hence the convergence of j1,j2∈Z |Cov (X 0,0 , X j1,j2 )| (j 1 + j 2 ) guarantees that ε n in Corollary 1.8 is of order 1/n. Moreover, since (1.3.32) holds for all α and β, the choice of γ allows to reach rates of the form n −δ+r0 for any fixed r 0 . In particular, when δ = 1, one can reach for any fixed r 0 rates of the form n −1+r0 . In comparison, with the same assumptions, the result of [MP11] gives n −3/8 .

Proofs
2.1. Proof of Theorem 1.4. We define for j 1 and i ∈ Z d , (2.1.1) In this way, by the martingale convergence theorem, (2.1.3) Let us fix j 1. We divide Z d into blocks. For v ∈ Z d , we define Observe that the random variable i∈Av a i X i,j is measurable for the σ-algebra generated by ε u , where u satisfies (2j + 2) v q − (j + 1) u q j + 1 + (2j + 2) (v q + 1) − 1 for all q ∈ [d]. Since the family ε u , u ∈ Z d is independent, the family i∈Av a i X i,j , v ∈ E K is independent for each fixed K ⊂ [d]. (2.1.11) In order to control the last term, we use inequality (1.2.1) and bound E [X i | ε i ] q by X 0,0 q for q ∈ {1, 2}. This ends the proof of Theorem 1.4.
Proof of Corollary 1.5. The following lemma gives a control of the L q -norm of X 0,j in terms of the physical measure dependence.
Lemma 2.1. For q ∈ {2, p} and j ∈ N, the following inequality holds (2.1.12) Proof. Let j be fixed. Let us write the set of elements of Z d whose infinite norm is equal to j as {v s , 1 s N j } where N j ∈ N. We also assume that v s − v s−1 ∈ {e k , 1 k d} for all s ∈ {2, . . . , N j }. Denote , from which it follows, by Theorem 2.1 in [Rio09], that (2.1.14) Then using similar arguments as in the proof of Theorem 1 (i) in [Wu05] give the bound This ends the proof of Lemma 2.1. Now, Corollary 1.5 follows from an application of Lemma 2.1 with q = 2 and q = p respectively.
2.2. Proof of Theorem 1.6. Denote for a random variable Z the quantity (2.2.1) We say that a random field (Y i ) i∈Z d is m-dependent if the collections of random variables The proof of Theorem 1.6 will use the following tools.
(T.1) By Theorem 2.6 in [CS04], if I is a finite subset of Z d , (Y i ) i∈I an m-dependent centered random field such that E [|Y i | p ] < +∞ for each i ∈ I and some p ∈ (2, 3] and (2.2.2) (T.2) By Lemma 1 in [EMO07], for any two random variables Z and Z ′ and p 1, Let (ε u ) u∈Z d be an i.i.d. random field and let f : Let γ > 0 and n 0 defined by (1.3.10).
Let m := ([ b n ℓ 2 ] + 1) γ and let us define (2.2.4) Since the random field (ε u ) u∈Z d is independent, the following properties hold.
(P.1) The random field X (m) i i∈Z d is (2m + 1)-dependent. (P. 2) The random field X  2.3. Proof of Theorem 1.9. Since the random variables X i are centered, we derive by definition of g n (x) that ℓ 2 σ −1 (A n − 1) and using Theorem 1.4, we derive that (2.3.5) We then use Theorem 1.6 to handle ∆ ′ n (which is allowed, by (A)). Using boundedness of K, we control the ℓ p and ℓ p ′ norms by a constant times the ℓ 2 -norm. This ends the proof of Theorem 1.9.