Mohamed El Machkouri

Dalibor Volný

Wei Biao

A central limit theorem for stationary random fields

Keywords: AMS Subject Classifications (2000): 62G05, 62G07, 60G60 Central limit theorem, spatial processes, mdependent random fields, weak mixing

This paper establishes a central limit theorem and an invariance principle for a wide class of stationary random fields under natural and easily verifiable conditions. More precisely, we deal with random fields of the form

variables and g is a measurable function. Such kind of spatial processes provides a general framework for stationary ergodic random fields. Under a short-range dependence condition, we show that the central limit theorem holds without any assumption on the underlying domain on which the process is observed. A limit theorem for the sample auto-covariance function is also established.

Introduction

Central limit theory plays a fundamental role in statistical inference of random fields. There have been a substantial literature for central limit theorems of random fields under various dependence conditions. See [START_REF] Bolthausen | On the central limit theorem for stationary mixing random fields[END_REF], [START_REF] Bradley | On the spectral density and asymptotic normality of weakly dependent random fields[END_REF], [START_REF] Bulinski | Central limit theorem for random fields and applications[END_REF], [START_REF] Bulinskiȋ | Various mixing conditions and the asymptotic normality of random fields[END_REF], [START_REF] Chen | A uniform central limit theorem for nonuniform φ-mixing random fields[END_REF], [START_REF] Dedecker | A central limit theorem for stationary random fields[END_REF], [START_REF] Guyon | Vitesse de convergence du théorème de la limite centrale pour des champs faiblement dépendants[END_REF], [START_REF] Jenish | Central limit theorems and uniform laws of large numbers for arrays of random fields[END_REF], [START_REF] Maltz | On the central limit theorem for nonuniform φ-mixing random fields[END_REF], [START_REF] Nakhapetyan | An approach to the proof of limit theorems for dependent random variables[END_REF], [START_REF] Neaderhouser | Limit theorems for multiply indexed mixing random variables, with application to Gibbs random fields[END_REF], [START_REF] Neaderhouser | Some limit theorems for random fields[END_REF], [START_REF] Paulauskas | On Beveridge-Nelson decomposition and limit theorems for linear random fields[END_REF], [START_REF] Perera | Geometry of Z d and the central limit theorem for weakly dependent random fields[END_REF], among others. However, many of them require that the underlying random fields have very special structures such as Gaussian, linear, Markovian or strong mixing of various types. In applications those structural assumptions can be violated, or not easily verifiable.

In this paper we consider stationary random fields which are viewed as nonlinear transforms of independent and identically distributed (iid) random variables. Based on that representation we introduce dependence measures and establish a central limit theorem and an invariance principle. We assume that the random field (X i ) i∈Z d has the form

X i = g ε i-s ; s ∈ Z d , i ∈ Z d , (1)
where (ε j ) j∈Z d are iid random variables and g is a measurable function. In the one-dimensional case (d = 1) the model ( 1) is well known and includes linear as well as many widely used nonlinear time series models as special cases. In Section 2 based on (1) we shall introduce dependence measures. It turns out that, with our dependence measure, central limit theorems and moment inequalities can be established in a very elegant and natural way. The rest of the paper is organized as follows. In Section 3 we present a central limit theorem and an invariance principle for

S Γ = i∈Γ X i ,
where Γ is a finite subset of Z d which grows to infinity. The proof of our Theorem 1 is based on a central limit theorem for m n -dependent random fields established by Heinrich [START_REF] Heinrich | Asymptotic behaviour of an empirical nearest-neighbour distance function for stationary Poisson cluster process[END_REF]. Unlike most existing results on central limit theorems for random fields which require certain regularity conditions on the boundary of Γ, Heinrich's central limit theorem (and consequently our Theorem 1) has the very interesting property that no condition on the boundary of Γ is needed, and the central limit theorem holds under the minimal condition that |Γ| → ∞, where |Γ| the cardinal of Γ. This is a very attractive property in spatial applications in which the underlying observation domains can be quite irregular. As an application, we establish a central limit theorem for sample auto-covariances. Section 3 also present an invariance principle. Proofs are provided in Section 4.

Examples and Dependence Measures

In (1), we can interpret (ε s ) s∈Z d as the input random field, g is a transform or map and (X i ) i∈Z d as the output random field. Based on this interpretation, we define dependence measure as follows: let (ε ′ j ) j∈Z d be an iid copy of (ε j ) j∈Z d and consider for any positive integer n the coupled version X * i of X i defined by

X * i = g ε * i-s ; s ∈ Z d , where for any j in Z d , ε * j = ε j if j = 0 ε ′ 0 if j = 0.
Recall that a Young function ψ is a real convex nondecreasing function defined on R + which satisfies lim t→∞ ψ(t) = ∞ and ψ(0) = 0. We define the Orlicz space L ψ as the space of real random variables Z defined on the probability space (Ω, F , P) such that E[ψ(|Z|/c)] < +∞ for some c > 0. The Orlicz space L ψ equipped with the so-called Luxemburg norm . ψ defined for any real random variable Z by

Z ψ = inf{ c > 0 ; E[ψ(|Z|/c)] ≤ 1 }
is a Banach space. For more about Young functions and Orlicz spaces one can refer to Krasnosel'skii and Rutickii [START_REF] Krasnosel | Convex Functions and Orlicz Spaces[END_REF].

Following Wu [START_REF] Wu | Nonlinear system theory: another look at dependence[END_REF], we introduce the following dependence measures which are directly related to the underlying processes.

Definition 1 (Physical dependence measure). Let ψ be a Young function and i in Z d be fixed. If X i belongs to L ψ , we define the physical dependence

measure δ i,ψ by δ i,ψ = X i -X * i ψ . If p ∈]0, +∞] and X i belongs to L p , we denote δ i,p = X i -X * i p .
Definition 2 (Stability). We say that the random field X defined by

(1) is p-stable if ∆ p := i∈Z d δ i,p < ∞.
As an illustration, we give some examples of p-stable spatial processes.

Example 1. (Linear random fields) Let (ε i ) i∈Z d be i.i.d random variables with ε i in L p , p ≥ 2. The linear random field X defined for any k in Z d by

X k = s∈Z d a s ε k-s
is of the form (1) with a linear functional g. For any i in

Z d , δ i,p = |a i | ε 0 - ε ′ 0 p . So, X is p-stable if i∈Z d |a i | < ∞.
Clearly, if K is a Lipschitz continuous function, under the above condition, the subordinated process

Y i = K(X i ) is also p-stable since δ i,p = O(|a i |).
Example 2. (Volterra field) Another class of nonlinear random field is the Volterra process which plays an important role in the nonlinear system theory (Casti [5], Rugh [START_REF] Rugh | Nonlinear system theory[END_REF]): consider the second order Volterra process

X k = s 1 ,s 2 ∈Z d a s 1 ,s 2 ε k-s 1 ε k-s 2 ,
where a s 1 ,s 2 are real coefficients with a s 1 ,s

2 = 0 if s 1 = s 2 and ε i in L p , p ≥ 2. Let A k = s 1 ,s 2 ∈Z d (a 2 s 1 ,k + a 2 k,s 2 ) and B k = s 1 ,s 2 ∈Z d (|a s 1 ,k | p + |a k,s 2 | p ).
By the Rosenthal inequality, there exists a constant C p > 0 such that

δ k,p = X k -X * k p ≤ C p A 1/2 k ε 0 2 ε 0 p + C p B 1/p k ε 0 2 p .

Main Results

To establish a central limit theorem for S Γ we need the following moment inequality. With the physical dependence measure, it turns out that the moment bound can have an elegant and concise form.

Proposition 1. Let Γ be a finite subset of Z d and (a i ) i∈Γ be a family of real numbers. For any p ≥ 2, we have

i∈Γ a i X i p ≤ 2p i∈Γ a 2 i 1 2 ∆ p
where ∆ p = i∈Z d δ i,p .

In the sequel, for any i in Z d , we denote δ i in place of δ i,2 .

Proposition 2. If ∆ 2 := i∈Z d δ i < ∞ then k∈Z d |E(X 0 X k )| < ∞. More- over, if (Γ n ) n≥1 is
lim n→+∞ |Γ n | -1 E(S 2 Γn ) = k∈Z d E(X 0 X k ). (2)

Central Limit Theorem

Our first main result is the following central limit theorem.

Theorem 1. Let (X i ) i∈Z d be the stationary centered random field defined by

(1) satisfying ∆ 2 := i∈Z d δ i < ∞. Assume that σ 2 n := E S 2 Γn → ∞. Let (Γ n ) n≥1 be a sequence of finite subsets of Z d such that |Γ n | → ∞, then the Levy distance L[S Γn / |Γ n |, N(0, σ 2 n /|Γ n |)] → 0 as n → ∞. (3)
We emphasize that in Theorem 1 no condition on the domains Γ n is imposed other than the natural one

|Γ n | → ∞. Applying Proposition 2, if |∂Γ n |/|Γ n | goes to zero and σ 2 := k∈Z d E(X 0 X k ) > 0 then S Γn |Γ n | L -----→ n→+∞ N (0, σ 2 ).
Theorem 1 can be applied to the mean estimation problem: suppose that a stationary random field X i with unknown mean µ = EX i is observed on the domain Γ. Then µ can be estimated by the sample mean μ = S Γ /|Γ| and a confidence interval for μ can be constructed if there is a consistent estimate for var(S Γ )/|Γ|.

Interestingly, the Theorem can also be applied to the estimation of autocovariance functions. For

k ∈ Z d let γ k = cov(X 0 , X k ) = E(X 0 X k ) -µ 2 . ( 4
)
Assume X i is observed over i ∈ Γ and let Ξ = {i ∈ Γ : i + k ∈ Γ}. Then γ k can be estimated by γk = 1 |Ξ| i∈Ξ X i X i+k -μ2 . (5)
To apply Theorem 1, we need to compute the physical dependence measure for the process Y i := X i X i+k , i ∈ Z d . It turns out that the dependence for Y i can be easily obtained from that of X i . Note that

δ i,p/2 (Y ) = X i X i+k -X * i X * i+k p/2 ≤ X i X i+k -X i X * i+k p/2 + X i X * i+k -X * i X * i+k p/2 ≤ X i p δ i+k,p + δ i,p X * i+k p = X 0 p (δ i+k,p + δ i,p ). Hence, if ∆ 4 = i∈Z d δ i,4 < ∞, we have i∈Z d δ i,2 (Y ) < ∞ and the central limit theorem for i∈Ξ X i X i+k /|Ξ| holds if |Ξ| → ∞.

Invariance Principles

Now, we are going to see that an invariance principle holds too. If A is a collection of Borel subsets of [0, 1] d , define the smoothed partial sum process

{S n (A) ; A ∈ A} by S n (A) = i∈{1,...,n} d λ(nA ∩ R i )X i (6)
where

R i =]i 1 -1, i 1 ] × ...×]i d -1, i d ]
is the unit cube with upper corner at i, λ is the Lebesgue measure on R d and X i is defined by [START_REF] Bolthausen | On the central limit theorem for stationary mixing random fields[END_REF]. We equip the collection A with the pseudo-metric ρ defined for any A, B in A by ρ(A, B) = λ(A∆B). To measure the size of A one considers the metric entropy: denote by H(A, ρ, ε) the logarithm of the smallest number N(A, ρ, ε) of open balls of radius ε with respect to ρ which form a covering of A. The function H(A, ρ, .) is the entropy of the class A. Let C(A) be the space of continuous real functions on A, equipped with the norm . A defined by

f A = sup A∈A |f (A)|.
A standard Brownian motion indexed by A is a mean zero Gaussian process W with sample paths in C(A) and Cov(W (A), W (B)) = λ(A ∩ B). From Dudley [START_REF] Dudley | Sample functions of the Gaussian process[END_REF] we know that such a process exists if

1 0 H(A, ρ, ε) dε < +∞. (7)
We say that the invariance principle or functional central limit theorem (FCLT) holds if the sequence {n -d/2 S n (A) ; A ∈ A} converges in distribution to an A-indexed Brownian motion in the space (A). The first weak convergence results for Q d -indexed partial sum processes were established for i.i.d. random fields and for the collection

Q d of lower-left quadrants in [0, 1] d , that is to say the collection {[0, t 1 ] × . . . × [0, t d ] ; (t 1 , . . . , t d ) ∈ [0, 1] d }.
They were proved by Wichura [START_REF] Wichura | Inequalities with applications to the weak convergence of random processes with multi-dimensional time parameters[END_REF] under a finite variance condition and earlier by Kuelbs [START_REF] Kuelbs | The invariance principle for a lattice of random variables[END_REF] under additional moment restrictions. When the dimension d is reduced to one, these results coincide with the original invariance principle of Donsker [START_REF] Donsker | An invariance principle for certain probability limit theorems[END_REF]. Dedecker [START_REF] Dedecker | Exponential inequalities and functional central limit theorems for random fields[END_REF] gave an L ∞ -projective criterion for the process {n -d/2 S n (A) ; A ∈ A} to converge in the space C(A) to a mixture of A-indexed Brownian motions when the collection A satisfies only the entropy condition [START_REF] Dedecker | A central limit theorem for stationary random fields[END_REF]. This projective criterion is valid for martingaledifference bounded random fields and provides a sufficient condition for φmixing bounded random fields. For unbounded random fields, the result still holds provided that the metric entropy condition on the class A is reinforced (see [START_REF] Machkouri | Kahane-Khintchine inequalities and functional central limit theorem for stationary random fields[END_REF]). It is shown in [START_REF] Machkouri | Contre-exemple dans le théorème central limite fonctionnel pour les champs aléatoires réels[END_REF] that the FCLT may be not valid for p-integrable martingale-difference random fields (0 ≤ p < +∞) but it still holds if the conditional variances of the martingale-difference random field are assumed to be bounded a.s. (see [START_REF] Machkouri | Invariance principles for standardnormalized and self-normalized random fields[END_REF]). In this paper, we are going to establish the FCLT for random fields of the form (1) (see Theorem 2).

Following [START_REF] Van Der Vaart | Weak convergence and empirical processes with applications to statistics[END_REF], we recall the definition of Vapnik-Chervonenkis classes (V Cclasses) of sets: let C be a collection of subsets of a set X . An arbitrary set of n points F n := {x 1 , ..., x n } possesses 2 n subsets. Say that C picks out a certain subset from F n if this can be formed as a set of the form C ∩ F n for a C in C. The collection C is said to shatter F n if each of its 2 n subsets can be picked out in this manner. The VC-index V (C) of the class C is the smallest n for which no set of size n is shattered by C. Clearly, the more refined C is, the larger is its index. Formally, we have

V (C) = inf n ; max x 1 ,...,xn ∆ n (C, x 1 , ..., x n ) < 2 n where ∆ n (C, x 1 , ..., x n ) = # {C ∩ {x 1 , ..., x n } ; C ∈ C}. Two classical exam- ples of V C-classes are the collection Q d = [0, t] ; t ∈ [0, 1] d and Q ′ d = [s, t] ; s, t ∈ [0, 1] d ,
s ≤ t with index d + 1 and 2d + 1 respectively (where s ≤ t means s i ≤ t i for any 1 ≤ i ≤ d). Fore more about Vapnik-Chervonenkis classes of sets, one can refer to [START_REF] Van Der Vaart | Weak convergence and empirical processes with applications to statistics[END_REF].

Let β > 0 and h β = ((1β)/β) 1 β 1 1 {0<β<1} . We denote by ψ β the Young function defined by ψ β (x) = e (x+h β ) βe h β β for any x in R + .

Theorem 2. Let (X i ) i∈Z d be the stationary centered random field defined by (1) and let A be a collection of regular Borel subsets of [0, 1] d . Assume that one of the following condition holds:

(i) The collection A is a Vapnik-Chervonenkis class with index V and there exists p > 2(V -1) such that X 0 belongs to L p and ∆ p := i∈Z d δ i,p < ∞.

(ii) There exists θ > 0 and 0 < q < 2 such that E[exp(θ|X 0 | β(q) )] < ∞ where β(q) = 2q/(2q) and ∆ ψ β(q) := i∈Z d δ i,ψ β(q) < ∞ and such that the class A satisfies the condition

1 0 (H(A, ρ, ε)) 1/q dε < +∞. (8)
(iii) X 0 belongs to L ∞ , the class A satisfies the condition (7) and ∆ ∞ :=

i∈Z d δ i,∞ < ∞.
Then the sequence of processes {n -d/2 S n (A) ; A ∈ A} converges in distribution in C(A) to σW where W is a standard Brownian motion indexed by A and σ 2 = k∈Z d E(X 0 X k ).

Proofs

Proof of Proposition 1. Let τ : Z → Z d be a bijection. For any i ∈ Z, for any

j ∈ Z d , P i X j := E(X j |F i ) -E(X j |F i-1 ) (9)
where

F i = σ ε τ (l) ; l ≤ i . Lemma 1.
For any i in Z and any j in Z d , we have P i X j p ≤ δ j-τ (i),p .

Proof of Lemma 1.

P i X j p = E(X j |F i ) -E(X j |F i-1 ) p = E(X 0 |T j F i ) -E(X 0 |T j F i-1 ) p where T j F i = σ ε τ (l)-j ; l ≤ i = E g ((ε -s ) s∈Z d ) |T j F i -E g (ε -s ) s∈Z d \{j-τ (i)} ; ε ′ τ (i)-j |T j F i p ≤ g ((ε -s ) s∈Z d ) -g (ε -s ) s∈Z d \{j-τ (i)} ; ε ′ τ (i)-j p = g (ε j-τ (i)-s ) s∈Z d -g (ε j-τ (i)-s ) s∈Z d \{j-τ (i)} ; ε ′ 0 p = X j-τ (i) -X * j-τ (i) p = δ j-τ (i),p .
The proof of Lemma 1 is complete.

For all j in Z d ,

X j = i∈Z P i X j .
Consequently,

a j P i X j 2 p   1 2 ≤   2p i∈Z j∈Γ |a j | P i X j p 2   1 2 (10)
By the Cauchy-Schwarz inequality, we have

j∈Γ |a j | P i X j p 2 ≤ j∈Γ a 2 j P i X j p × j∈Γ P i X j p
and by Lemma 1,

j∈Z d P i X j p ≤ j∈Z d δ j-τ (i),p = ∆ p . So, we obtain j∈Γ a j X j p ≤ 2p∆ p j∈Γ a 2 j i∈Z P i X j p 1 2
.

Applying again Lemma 1, for any j in Z d , we have

i∈Z P i X j p ≤ i∈Z δ j-τ (i),p = ∆ p , Finally, we derive j∈Γ a j X j p ≤ 2p j∈Γ a 2 j 1 2 ∆ p .
The proof of Proposition 1 is complete.

Proof of Proposition 2. Let k in Z d be fixed. Since X k = i∈Z P i X k where P i is defined by ( 9) and E((P i X 0 )(P j X k )) = 0 if i = j, we have

E(X 0 X k ) = i∈Z E((P i X 0 )(P i X k )).
Thus, we obtain

k∈Z d |E(X 0 X k )| ≤ i∈Z P i X 0 2 k∈Z d P i X k 2 .
Applying again Lemma 1, we derive

k∈Z d |E(X 0 X k )| ≤ ∆ 2 2 < ∞.
In the other part, since (X k ) k∈Z d is stationary, we have

|Γ n | -1 E(S 2 Γn ) = k∈Z d |Γ n | -1 |Γ n ∩ (Γ n -k)|E(X 0 X k ) where Γ n -k = {i -k ; i ∈ Γ n }. Moreover |Γ n | -1 |Γ n ∩ (Γ n -k)||E(X 0 X k )| ≤ |E(X 0 X k )| and k∈Z d |E(X 0 X k )| < ∞. Since lim n→+∞ |Γ n | -1 |Γ n ∩ (Γ n -k)| = 1
, applying the Lebesgue convergence theorem, we derive

lim n→+∞ |Γ n | -1 E(S 2 Γn ) = k∈Z d E(X 0 X k ).
The proof of Proposition 2 is complete.

Proof of Theorem 1. We first assume that lim inf n σ 2 n /|Γ n | > 0. Let (m n ) n≥1 be a sequence of positive integers going to infinity. In the sequel, we denote X j = E (X j |F mn (j)) where F mn (j) = σ(ε j-s ; |s| ≤ m n ). By factorization, there exists a measurable function h such that X j = h(ε j-s ; |s| ≤ m n ). So, we have

X * j = h(ε * j-s ; |s| ≤ m n ) = E X * j |F * mn (j) (11)
where F * mn (j) = σ(ε * j-s ; |s| ≤ m n ). We denote also for any j in Z d ,

δ (mn) j,p = (X j -X j ) -(X j -X j ) * p .
The following result is a direct consequence of Proposition 1.

Proposition 3. Let Γ be a finite subset of Z d and (a i ) i∈Γ be a family of real numbers. For any n in N * and any p ∈ [2, +∞], we have

j∈Γ a j (X j -X j ) p ≤ 2p i∈Γ a 2 i 1 2 ∆ (mn) p where ∆ (mn) p = j∈Z d δ (mn) j,p .
We need also the following lemma.

Lemma 2. Let p ∈]0, +∞] be fixed. If ∆ p < ∞ then ∆ (mn) p → 0 as n → ∞. Proof of Lemma 2. Let j in Z d be fixed. Since (X j -X j ) * = X * j -X * j , we have δ (mn) j,p = (X j -X j ) -(X j -X j ) * p ≤ X j -X * j p + X j -X * j p = δ j,p + E(X j |F mn (j) ∨ F * mn (j)) -E(X * j |F * mn (j) ∨ F mn (j)) p ≤ 2δ j,p .
Moreover, lim n→+∞ δ (mn) j,p = 0. Finally, applying the Lebesgue convergence theorem, we obtain lim n→+∞ ∆ (mn) p = 0. The proof of Lemma 2 is complete.

Let (Γ n ) n≥1 be a sequence of finite subsets of Z d such that lim n→+∞ |Γ n | = ∞ and lim inf n σ 2 n
|Γn| > 0 and recall that ∆ 2 is assumed to be finite. Combining Proposition 3 and Lemma 2, we have

lim sup n→+∞ S n -S n 2 σ n = 0. (12)
We are going to apply the following central limit theorem due to Heinrich ([15], Theorem 2).

Theorem 3 [START_REF] Heinrich | Asymptotic behaviour of an empirical nearest-neighbour distance function for stationary Poisson cluster process[END_REF]). Let (Γ n ) n≥1 be a sequence of finite subsets of Z d with |Γ n | → ∞ as n → ∞ and let (m n ) n≥1 be a sequence of positive integers. For each n ≥ 1, let {U n (j), j ∈ Z d } be an m n -dependent random field with EU n (j) = 0 for all j in Z d . Assume that E j∈Γn U n (j)

2 → σ 2
as n → ∞ with σ 2 < ∞. Then j∈Γn U n (j) converges in distribution to a Gaussian random variable with mean zero and variance σ 2 if there exists a finite constant c > 0 such that for any n ≥ 1,

j∈Γn EU 2 n (j) ≤ c
and for any ε > 0 it holds that

lim n→+∞ L n (ε) := m 2d n j∈Γn E U 2 n (j) 1 1 |Un(j)|≥εm -2d n = 0. Since lim inf n σ 2 n |Γn| > 0, there exists c 0 > 0 and n 0 ∈ N such that |Γn| σ 2 n ≤ c 0 for any n ≥ n 0 . Consider S n = i∈Γn X i , S n = i∈Γn X i and U n (j) := X j
σn . We have

E j∈Γn U n (j) 2 = E(S 2 n ) -σ 2 n σ 2 n + 1.
So, for any n ≥ n 0 we derive

σ 2 n -E(S 2 n ) σ 2 n = 1 σ 2 n E j∈Γn (X j -X j ) j∈Γn (X j + X j ) ≤ 1 σ 2 n j∈Γn (X j -X j ) 2 j∈Γn (X j + X j ) 2 ≤ 2|Γ n |∆ (mn) 2 σ 2 n 4∆ 2 + 2∆ (mn) 2 ≤ 4c 0 ∆ (mn) 2 2∆ 2 + ∆ (mn) 2 -----→ n→+∞ 0.
Consequently,

lim n→+∞ E j∈Γn U n (j) 2 = 1.
Moreover, for any n ≥ n 0 ,

j∈Γn EU 2 n (j) = |Γ n |E(X 2 0 ) σ 2 n ≤ c 0 E(X 2 0 ) < ∞.
Let ε > 0 be fixed. We have

L n (ε) ≤ c 0 m 2d n E X 2 0 1 1 |X 0 |≥ εσn m 2d n ≤ c 0 m 2d n E X 2 0 1 1 |X 0 |≥ εσn m 2d n ≤ c 0 m 2d n σ n P |X 0 | ≥ εσ n m 2d n + c 0 m 2d n E X 2 0 1 1 {|X 0 |≥ √ σn} ≤ c 0 E(X 2 0 )m 6d n ε 2 σ n + c 0 m 2d n ψ( √ σ n ) where ψ(x) = E X 2 0 1 1 {|X 0 |≥x} . Lemma 3. If the sequence (m n ) n≥1 is defined for any integer n ≥ 1 by m n = min ψ √ σ n -1 4d
, σ

1 12d n if ψ( √ σ n ) = 0 and by m n = σ 1 12d n if ψ( √ σ n ) = 0 where [ . ] is the integer part function then m n → ∞, m 6d n σ n → 0 and m 2d n ψ ( √ σ n ) → 0. Proof of Lemma 3. Since σ n → ∞ and ψ( √ σ n ) → 0, we derive m n → ∞. Moreover, m 6d n σ n ≤ 1 √ σ n → 0 and m 2d n ψ ( √ σ n ) ≤ ψ ( √ σ n ) → 0.
The proof of Lemma 3 is complete.

Consequently, we obtain lim n→∞ L n (ε) = 0. So, applying Theorem 3, we derive that

S n σ n Law -----→ n→+∞ N (0, 1). (13)
Combining ( 12) and ( 13), we deduce

S n σ n Law -----→ n→+∞ N (0, 1).
Hence (3) holds if lim inf n σ 2 n /|Γ n | > 0. In the general case, we argue as follows: If (3) does not hold then there exists a subsequence n ′ → ∞ such that

L S n ′ |Γ n ′ | , N 0, σ 2 n ′ |Γ n ′ | converges to some l in ]0, +∞]. (14)
Assume that

L S n ′′ |Γ n ′′ | , N 0, σ 2 n ′′ |Γ n ′′ | converges to 0. (15)
Since ( 15) contradicts ( 14), we have

σ 2 n ′ |Γ n ′ | converges to zero. Consequently S n ′ / |Γ n ′ | converges to zero in probability and L S n ′ √ |Γ n ′ | , N 0, σ 2 n ′ |Γ n ′ | con-
verges to 0 which contradicts again [START_REF] Guyon | Vitesse de convergence du théorème de la limite centrale pour des champs faiblement dépendants[END_REF]. Consequently, (3) holds. The proof of Theorem 1 is then complete.

Proof of Theorem 2. As usual, we have to prove the convergence of the finite-dimensional laws and the tightness of the partial sum process

{n -d/2 S n (A) ; A ∈ A} in C(A). For any Borel subset A of [0, 1] d , we de- note by Γ n (A) the finite subset of Z d defined by Γ n (A) = nA ∩ Z d . We say that A is a regular Borel set if λ(∂A) = 0. Proposition 4. Let A be a regular Borel subset of [0, 1] d with λ(A) > 0. We have lim n→+∞ |Γ n (A)| n d = λ(A) and lim n→+∞ |∂Γ n (A)| |Γ n (A)| = 0. Moreover, if ∆ 2 is finite then lim n→+∞ n -d/2 S n (A) -S Γn(A) 2 = 0 (16)
where S Γn(A) = i∈Γn(A) X i .

Proof of Proposition 4. The first part of Proposition 4 is the first part of Lemma 2 in Dedecker [START_REF] Dedecker | Exponential inequalities and functional central limit theorems for random fields[END_REF]. So, we are going to prove only the second part. Let n be a positive integer. Arguing as in Dedecker [START_REF] Dedecker | Exponential inequalities and functional central limit theorems for random fields[END_REF], we have

S n (A) -S Γn(A) = i∈Wn a i X i (17)
where

a i = λ(nA ∩ R i ) -1 1 i∈Γn(A) and W n is the set of all i in {1, .., n} d such that R i ∩ (nA) = ∅ and R i ∩ (nA) c = ∅. Noting that |a i | ≤ 1 and applying Proposition 1, we obtain S n (A) -S Γn(A) 2 ≤ 2∆ 2 i∈Wn a 2 i ≤ 2∆ 2 |W n |. (18)
Following the proof of Lemma 2 in [START_REF] Dedecker | Exponential inequalities and functional central limit theorems for random fields[END_REF], we have |W n | = o(n d ) and we derive [START_REF] Jenish | Central limit theorems and uniform laws of large numbers for arrays of random fields[END_REF]. The proof of Proposition 4 is complete.

The convergence of the finite-dimensional laws follows from Proposition 4 and Theorem 1.

So, it suffices to establish the tightness property.

Proposition 5. Assume that Assumption (i), (ii) or (iii) in Theorem 2 holds. Then for any x > 0, we have

lim δ→0 lim sup n→+∞ P    sup A,B∈A ρ(A,B)<δ n -d/2 S n (A) -n -d/2 S n (B) > x    = 0. (19)
Proof of Proposition 5. Let A and B be fixed in A and recall that ρ(A, B) = λ(A∆B). We have

S n (A) -S n (B) = i∈Λn a i X i where Λ n = {1, ..., n} d and a i = λ(nA ∩ R i ) -λ(nB ∩ R i ). Applying Propo- sition 1, we have n -d/2 S n (A) -S n (B) p ≤ ∆ p 2p n d i∈Λn λ(n(A∆B) ∩ R i ) 1 2
≤ 2p∆ p ρ(A, B).

(20) Assume that Assumption (i) in Theorem 2 holds. Then there exists a positive constant K such that for any 0 < ε < 1, we have (see Van der Vaart and Wellner [START_REF] Van Der Vaart | Weak convergence and empirical processes with applications to statistics[END_REF], Theorem 2.6.4)

N(A, ρ, ε) ≤ KV (4e) V 1 ε 2(V -1)
where N(A, ρ, ε) is the smallest number of open balls of radius ǫ with respect to ρ which form a covering of A. So, since p > 2(V -1), we have

The condition ( 19) is then satisfied under Assumption (i) in Theorem 2 and the sequence of processes {n -d/2 S n (A) ; A ∈ A} is tight in C(A).

Now, we assume that Assumption (ii) in Theorem 2 holds. The following technical lemma can be obtained using the expansion of the exponential function. Combining Lemma 4 with (20), for any 0 < q < 2, there exists C q > 0 such that n -d/2 S n (A) -S n (B) ψq ≤ C q ∆ ψ β(q) ρ(A, B) [START_REF] Neaderhouser | Some limit theorems for random fields[END_REF] where β(q) = 2q/(2q). Applying Theorem 11.6 in Ledoux and Talagrand [START_REF] Ledoux | Probability in Banach spaces[END_REF], for each positive ǫ there exists a positive real δ, depending on ǫ and on the value of the entropy integral (8) but not on n, such that (22) holds. The condition ( 19) is then satisfied and the process {n -d/2 S n (A) ; A ∈ A} is tight in C(A).

′ 2 n

 2 | does not converge to zero. Then there exists a subsequence n ′′ such that lim inf n σ ′′ |Γ n ′′ | > 0. By the first part of the proof of Theorem 1, we obtain

1 p

 1 dε < +∞.

( 21 )

 21 Combining[START_REF] Maltz | On the central limit theorem for nonuniform φ-mixing random fields[END_REF] and[START_REF] Nakhapetyan | An approach to the proof of limit theorems for dependent random variables[END_REF] and applying Theorem 11.6 in Ledoux and Talagrand[START_REF] Ledoux | Probability in Banach spaces[END_REF], we infer that the sequence {n -d/2 S n (A) ; A ∈ A} satisfies the following property: for each positive ǫ there exists a positive real δ, depending on ǫ and on the value of the entropy integral[START_REF] Nakhapetyan | An approach to the proof of limit theorems for dependent random variables[END_REF] but not on n,|n -d/2 S n (A)n -d/2 S n (B)|   < ǫ.

Lemma 4 .

 4 Let β be a positive real number and Z be a real random variable. There exist positive universal constants A β and B β depending only on β such that A β sup p>2 Z p p 1/β ≤ Z ψ β ≤ B β sup p>2 Z p p 1/β .

Finally, if

  Assumption (iii) in Theorem 2 holds then combining Lemma 4 with (20), there exists C > 0 such thatn -d/2 S n (A)n -d/2 S n (B) ψ 2 ≤ C∆ ∞ ρ(A, B).

  a sequence of finite subsets of Z d such that |Γ n | goes to infinity and |∂Γ n |/|Γ n | goes to zero then

Applying again Theorem 11.6 in Ledoux and Talagrand [19], we obtain the tightness of the process {n -d/2 S n (A) ; A ∈ A} in C(A). The proofs of Proposition 5 and Theorem 2 are complete.

Acknowledgments. The authors thank an anonymous referee for his\her constructive comments and Olivier Durieu and Hermine Biermé for pointing us a mistake in the first version of the proof of Theorem 1.


Claire Tinker

Andrew David

Keywords: Muslim schools, Collective Identity, Education, Muslims, Claimsmaking, Multiculturalism Muslim Schools, Collective Identity, Muslim, Claims-making, Multiculturalism, Education

Constructions of collective Muslim identity by advocates

of Muslim schools in Britain.

Background

Muslim identity

Expositions of Muslim identity grapple with a tension between commonalities and diversities. On the one hand, it appears self-evident that all Muslims share some thing(s); the thing(s) that enable them to identify as Muslims (or allow other people to label them as such). On the other hand, there is unarguable diversity within the group 'Muslims'; not only national, ethnic, cultural, linguistic, political and sectarian variation, but also cross-cutting differences relating to class, gender, age/generation, migration, sexuality, disability, etc. The dualism of homogeneity and heterogeneity is not, of course, solely a problem when conceptualizing collective Muslim identity, but nevertheless it is prominent 3 in scholarship on 'Muslims in the West'. Indeed, many authors adopt a position in which they reveal or defend the existence of heterogeneity (e.g. [START_REF] Modood | Multiculturalism: a civic idea[END_REF], and communicate a clear message: all Muslims are not the same. The prominence and tenor of this message can, at least in part, be explained by the intellectual impact of the 'anti-essentialist critique' [START_REF] Modood | Anti-Essentialism, Multiculturalism and the 'Recognition' of Religious Groups[END_REF]) and an antiracist undertone that accompanies concerns about prejudice and discrimination against Muslims.

Muslim identity can be analysed as a form of religious identity [START_REF] Peek | Becoming Muslim: The Development of a Religious Identity[END_REF]. In his discussion of British Muslims, however, Meer (2008, p. 66) claims that 'rather than being scripturally informed, we can view Muslim identity as a quasi-ethnic sociological formation'. He justifies this conceptualization due to the blurred boundary between religion and ethnicity in relation to minority group experiences and political mobilizations. Moreover, it also emphasizes the intersection between Muslim identity as self-assigned and externally ascribed. Modood (2007, p. 134) identifies that the salient dimensions of identity for Muslims in Britain include: 'a sense of family', 'community membership and heritage', 'simple precepts about self, compassion, justice and the afterlife' and 'collective political advancement'. He notes, nevertheless, that in the current British context Muslims may often be positioned (or position themselves) as having an identity marked by 'difference ' (p. 109).

A sociological lens focuses attention onto the political, social and economic processes and contexts involved in the creation and construction of identities (e.g. [START_REF] Schwedler | Islamic Identity: Myth, Menace, or Mobilizer?[END_REF]). In contemporary Britain, the politics of Muslim identity appear to be in flux. Modood displaying and promoting 'a particular Muslim identity ' (2007, p.140). For example, Kahani-Hopkins and Hopkins (2002, p. 294) describe the different ways that two Muslim political organizations invoke the concept of da'wah -'the injunction to invite people to Islam ' (p. 288).

If more individualized identity constructions are considered, however, this fluidity appears to be the norm. [START_REF] Dwyer | Veiled Meanings: young British Muslim women and the negotiation of differences[END_REF]Dwyer ( , 2000)), following Hall (1992), conceptualizes identity as 'contextual and relational positionings ' (1999: 6), and views identity 'as a process that is always in progress ' (2000: 475). Her analysis of young British South-Asian Muslim women attends to their multiple and intersecting identities, the contingent nature of 'difference', and how individuals subvert, resist and rework dominant meanings in their identity constructions. Hopkins (2007, p. 61) reveals considerable variation in the 'strength, nature and meaning' of individuals' affiliations to Muslim identity. This may be contingent on circumstance; for example, [START_REF] Ahmad | The making and representation of Muslim identity in Britain: conversations with British Muslim 'elites[END_REF] report that the sense of 'otherness' felt by some Muslims was heightened by political tensions.

Muslim claims-making

Muslims (individually and as a group) have become increasingly socially visible.

The protests associated with The Satanic Verses in the 1980s marked the rise in a politicized Muslim identity [START_REF] Modood | British Asian Muslims and the Rushdie affair[END_REF]. By the end of the 1990s the Runnymede [START_REF] Trust | Beyond Westphalia: Nations and Diasporas -the Case of the Muslim Umma[END_REF] was reporting discrimination against Muslims, and it was being argued that: 'the Muslim community has become increasingly a symbol of identification and collective action, particularly around issues of separate schools, religious education and religious discrimination.' (Alexander During this same time period, analysis of survey data suggested that many Muslims were prioritizing religious identity over other identifications [START_REF] Modood | Ethnic minorities in Britain: diversity and disadvantage. The fourth national survey of ethnic minorities[END_REF]). In addition, qualitative research provided evidence of situations in which Muslim identities were given primacy over other aspects of identity (see Modood 2007, p108-9). Of course, evidence of prioritization does not preclude the salience of other identities. As noted above, other social differentiations and contextual factors influence the expression of Muslim identities, including 'the dominant narratives of national identity' (Dwyer 2000, p.476). Indeed, in the context of debates about primacy and allegiance, it is the construction and potential salience of the compound identity 'British Muslim' that has come to the fore [START_REF] Modood | Multiculturalism: a civic idea[END_REF].

Muslim identities have been mobilized for collective action [START_REF] Statham | Political Mobilisation by Minorities in Britain: a negative feedback of 'race relations'?[END_REF]. As Statham et al. (2005, p. 439) note, however, various overlapping (and potentially competing) categories of collective identity are evident in group demands: 'The same migrant group may mobilize as 'Pakistanis', 'Asians'…, 'Muslims', 'British Muslims' or as 'ethnic minorities''. Statham et al.'s (2005) analysis of British newspaper coverage nevertheless showed that two thirds of demands were based on religious forms of identification, and of these the majority (61.4%) were from groups identifying themselves as Muslim or Islamic. This evidence of a pattern of group claims-making is somewhat tempered by the aforementioned studies of press coverage. It does, however, demonstrate that collective Muslim identity is not only a focus for mobilization, but also it is part of a media narrative in which Muslims are sometimes presented as a group with common needs. Meer (2007, p. 55) argues that mobilizations for Muslim schools can be understood as 'an addition to -or continuation of -a historical settlement between earlier religious minorities, the established church and the state'. The contemporary existence of faith schools reflects the significant historical role of religious organizations in the development of the British education system. This predominantly involved the Protestant and Catholic churches [START_REF] Chadwick | The Anglican Perspective on Church Schools[END_REF], although Jewish schools also have a heritage spanning four centuries [START_REF] Miller | Meeting the Challenge: the Jewish schooling phenomenon in the UK[END_REF]. Faith schools exist either 'independently' or within the state sector (where public funding has attendant policy and practice requirements). In Britain there are over 7000 faith schools in the state sector, although very few of these are non-Christian (in England this includes eleven Muslim schools) [START_REF] Bolton | Faith schools: admissions and performance[END_REF]. In addition, there are 140 independent Muslim schools in England and two in Wales. 1 The Muslim Council of Britain (2007) estimates that there are over 400,000 school-aged Muslim children in Britain, and that only approximately 4% of these attend an independent school.

Muslim schools

There is a broad spectrum of Muslim schools. Pupil numbers range from 5 to over 1800, and while some larger schools are purpose-built, many smaller The eleven state-funded Muslim schools are required to employ fully qualified teachers and follow the National Curriculum for England. Those in the independent sector (like all independent schools) need not follow these requirements, but nevertheless some choose to (Tinker 2009) while others claim they have insufficient resources to do so (Walford 2001).

Muslims schools are a relatively recent addition to the educational landscape; the first independent Muslim school opened in 1979 [START_REF] Dooley | Muslim Private Schools[END_REF], and the first to enter the state sector was in 1998. This extension of state support to Muslim schools was mired by dispute and controversy. The first Muslim school admitted into the state sector had endured multiple rejections during a fifteen year campaign (Parker-Jenkins 2002), and it was not alone in experiencing opposition from central government [START_REF] Mcloughlin | A Part of the Community'? The politics of representation and a Muslim school's application for state funding[END_REF].

Consecutive Labour governments from 1997 promoted policies to increase the number of Christian and minority faith schools [START_REF] Jackson | Should the State Fund Faith Based Schools? A review of the arguments[END_REF]. Nevertheless, the idea of state-funded Muslim schools has continued to generate unease and debate.

Objections to initial applications for state support centred on resources and local necessity, and arguments about segregation in the context of multicultural education policies. It has been argued, however, that resistance ran much deeper. McLoughlin (1998, p. 451) claims that underlying perceptions of 'Islam and Muslim schools as necessarily 'separatist' and 'fundamentalist'' meant that Christian schools did not face similar objections. Dwyer (1993, p. 149) similarly argued that the Muslim schools were subject to public and political distrust in ways that their Jewish and Catholic counterparts were not, because they were 'seen to offer a particular threat to the hegemonic culture'. Substantial differences in socio-historical context complicate direct comparisons between faith schools. Nevertheless, even since the shift to an ostensibly supportive policy environment, particular concerns about Muslim schools have persisted in relation to social cohesion, extremism and gender roles [START_REF] Meer | Muslim Schools in Britain: Challenging mobilisations or logical developments[END_REF].

There is a continuing perception that Muslim schools are culturally 'out of place' or hazardous.

A key justification for Muslim schools (including in the state-sector) is the need for equality in an educational system which includes other faith schools.

In addition, arguments have been made about the rights, choices or needs of parents, children and/or 'the Muslim community' [START_REF] Hewer | Schools for Muslims[END_REF][START_REF]Equal Access to State Funding: the case of Muslim schools in Britain[END_REF][START_REF] Jackson | Should the State Fund Faith Based Schools? A review of the arguments[END_REF]Tinker 2006Tinker , 2009;;[START_REF] Meer | Muslim Schools in Britain: Challenging mobilisations or logical developments[END_REF]McCreery et al. 2007).

These include: having educational institutions that are organized and run in ways that reflect religious (and/or cultural) values and practices (including gendered practices); communicating or inculcating particular religious and cultural knowledge, practices or values (which are viewed as marginalized if not wholly absent in the curricula of mainstream schools); enabling children to avoid the prejudice and discrimination that is perceived to occur in mainstream schooling and which is argued to undermine their self-confidence and self-identity; and, improving children's attainment. The rationale of protecting or supporting Muslim identity is highly prominent in these accounts. The promotion of Muslim identity (as an end in itself, rather than as an outcome of a principle of justice or an instrumental desire to enhance educational outcomes) is, however, controversial. Secular interest groups, and some politicians, journalists and

Methodology

The findings reported in this paper originate from research (undertaken by CT) examining the debate about the state-funding of Muslim schools (Tinker 2006(Tinker , 2009)). A qualitative research design was favoured to enable an in-depth consideration of the outlooks of research participants on this potentially controversial topic [START_REF] Mason | Qualitative researching[END_REF].

A purposive sampling technique ensured that participants with relevant views and experiences were recruited [START_REF] Ritchie | Designing and Selecting Samples[END_REF]. Thirty-six interviews were organized with representatives from four stakeholder groups: political parties; interest groups; head teachers; and parents of Muslim school-children.

Individuals within the four groups were selected in different ways. Documentary and on-line searches were used to identify representatives from four political parties and eleven interest groups (eight of which were 'Muslim interest groups', ranging from those focussed on education, through those with a general remit to serve the needs of Muslims, to those with overtly political aims). Head teachers of four independent Muslim schools were identified through contact with the Association of Muslim Schools, while eleven head teachers of state-maintained schools with a high proportion of Muslim pupils were identified through Local Education Authorities. Twelve parents were identified using a 'snowball' sample, following recommendations from head teachers or other parents.

Semi-structured interviews focused the data collection on key issues, but allowed respondents' some freedom to direct the discussion [START_REF] Kvale | InterViews[END_REF].

Interview 'guides' varied between stakeholder groups, but common topics included the state-funding of faith schools in general and Muslim schools in particular. Interviews were conducted between July 2003 andMarch 2004 (by CT), each lasting approximately one hour. Interviews usually took place in the respondents' home or workplace, and were conducted in English.

Interviews were transcribed in full and imported into NVivo data analysis software package. CT constructed a list of codes from preliminary interviews, and applied these to subsequent data, expanding and compressing the coding frame as necessary. A 'thematic analysis' (Gomm 2004, p. 189) was developed, and the findings reported here comprise four themes relating to the justifications for Muslim schools that involved ideas about Muslim identity.

The research was conducted in accordance with the British Sociological Association (2002) statement on ethical practice. With so few state-funded Muslim schools in Britain, and a limited number of stakeholder organizations, there was a risk that some respondents may be identifiable. This limit on confidentiality was discussed with relevant interviewees, and it was agreed that Certain restrictions in the study can be noted. The issue of 'interviewer effect' in this research is discussed in detail elsewhere (Tinker and Armstrong 2008). Many respondents (25/36) were Muslim and the interviewer is not, and in this respect, it is open to question whether the commonly recounted narratives of threat and/ or attack against Muslim identity would have been similarly generated in an interview with 'matched' religious affiliations. Conducting the interviews in English may have affected the potential pool of interviewees, and the possibilities of exploring deeper thoughts and feelings with respondents who spoke English as an additional language. Qualitative research rarely aims to have statistical generalizablility [START_REF] Mason | Qualitative researching[END_REF]. Thus, while the findings reported here chart the range of concerns expressed by this sample of respondents, questions about the representativeness of these opinions (among others who comprise the stakeholder group but were not in this sample) would require further research of a different design.

Findings

The overarching theme that emerged from the data analysis was a perception that Muslim identity was under threat and/or attack, and that separate Muslim schools were an appropriate means of protection or defence. Four sub-themes are presented below; these represent the range of ways in which ideas about identity were used by respondents to support the case for Muslim schools. This view reveals a fear that children's religious identity or affiliation may be confused if they are 'taught all the wrong things'; and it implies such contradictions would be avoided if children attended Muslim schools.

A lack of religious knowledge was also a concern for the above-quoted respondent. Here, he invoked the views of parents to support his opinion that Muslim schools best serve the interests of Muslims: […] for the parents, it is important that their children is getting religious education for longer. Which is necessary, because if you look at those other boys who go to those [non-Muslim] schools, ask them about their religion, 'I don't know, I don't know'. So, what they will be in their practical lives? Will they be Muslim, will they be Christian, will they be Jews, or nothing? Most of them will be nothing. This viewpoint, that religious instruction is a necessary part of education, appears to be rooted in a fear that a lack of knowledge of Islam may result in a child losing their Muslim identity and becoming 'nothing'.

Secular or Western values

A second theme that emerged from the data analysis, most commonly expressed by head teachers of independent Muslim schools, related to the values, morals and behaviours inculcated by schools. Typically, the concern was that collective Muslim identity may be lost or undermined because aspects of secular or Western values are different and threaten 'good' 'Islamic' morals and behaviour.

One head teacher of an independent Muslim school invoked the views of parents (and then Muslims generally) when he argued that aspects of identity differ between Muslims and 'secularized people'.

When they see most secularized people, for example they see the dress code of the average British child, it sends a warning signal, it's not dressed according to the Sharia. When they see the drinking, it's not according to Sharia. All these factors build a particular picture, a particular identity in the minds of Muslims. And they see the Muslim identity as distinctly different from the one that they see upon the street or on the television.

Other interviewees, such as this head teacher of a different independent Muslim school, viewed these allegedly secular values as threatening: Muslim would not put his parents in an old people's home because he would understand that his obligation is to look after his parents […] It is notable that the examples of 'Islamic manner and behaviours' or 'the Islamic personality' claimed by these respondents are actions or attitudes that many non-Muslims would also consider 'good'. These viewpoints suggest that Muslim schools were viewed as a way of instilling collective identity rooted in 'good' morals and behaviour, but they also imply a perception on the part of these respondents that Islam has a monopoly on these or that non-Muslims are immoral and/or badly-behaved.

A particular expression of the concern about values and the loss of identity related to gender. The head teacher quoted at the start of this sub-section reported that he struggles to convince some parents to allow their daughters to go to co-educational institutions after leaving his school:

[…] in a sense they are afraid that if they allow their girls to go on into mixed education schools or colleges that they'll lose their own identity, become embroiled in things which are not Islamic. […] They feel that the girls should not really be educated in [non-Muslim] schools because they'll be lost to Western society and so forth.

While this viewpoint is evidence of a perceived gender-specific identity threat from being 'embroiled in things which are not Islamic', the notion of loss is also striking; not only a loss of 'their own identity', but also that they will be 'lost to Western society'. In this respondent's opinion, not only was her child subjected to discrimination because he expressed a Muslim identity in his clothing, but also the school offered no protection and subsequently blamed the problem on her.

It is in the context of such experiences that one head teacher argued that state-funded Muslim schools, such as his, were necessary to create: 'a sense of self-esteem and self-confidence in the children'. Indeed, he argued that restoring self-confidence to young Muslims was essential not only to ensure 'they will be able to participate fully and positively in the world outside', but also because those who lacked it were at risk of radicalization:

Without that self-confidence, our experience and observations and analysis is that they are very prey to a sort of identity crisis, […] and then they possibly seek out some sort of strong, perhaps fundamentalist or extremist position which gives them a strong sense of identity.

For this respondent, having Muslim schools offered denigrated Muslim children a route out of an 'identity crisis' that alternatively risked losing them to the 'strong sense of identity' offered by fundamentalism or extremism. Indeed, he argued in particular that active support of Muslim school in the state sector was the best way to encourage the development of a secure British Muslim identity.

Discordant identities

A fourth theme that emerged from the data analysis, most commonly expressed by parents and head teachers, related to the potential for discord between sources of identity. In summary, some respondents viewed plural identities as a cause of unhappiness, confusion or conflict, and Muslim schools as source of a stable and coherent identity. This respondent believed that he could help to protect his children from isolation by sending them to a school that offered 'a strong sense of identity', which appeared to mean an identity that emphasized the collective: belonging, affiliation, pride in being part of a group and knowledge of the existence of 'other Muslims'. A potential conflict over a dual identity, nevertheless, remains evident in his juxtaposition of a Muslim identity to pride and belonging to 'their country'.

Another source of identity conflict was potential differences in expectations and behaviours at home and school. One head teacher of a statefunded Muslim school argued:

There is the possibility that children will grow up being very mixed up.

They have one way of being at home and another way of being at school and that can cause problems.

The differences in the 'way of being' remained vague in this respondent's account (it is surmised that the source of differences were aspects of belief and behaviour reported above, which are further evident in the extract which follows below). Nevertheless there is concern that these variations impact on identity, with a risk of children being 'mixed up' if the differences are great.

A representative of an interest group that politically promotes the establishment of an Islamic state argued that 'the role of an Islamic school is to complement the role of a parent in educating children'. This view was echoed by a parent whose child attended an independent Muslim school, who suggested that conflict was avoided as the school reinforced parental messages:

Things like when we teach them eating manners and things like that. […] I would say eat with your right hand, say 'bismillah' which means in the name of Allah. I, as a parent, would do that and that would be reinforced if they went to an Islamic school from a young age [because] the teacher would say 'can we all say our prayers together', 'can you make sure that you go and wash your hands and that you're all eating with the right hand'. This solution to a perceived home-school conflict portrays an idealized harmony between parental and institutional viewpoints, but nevertheless it demonstrates an underlying concern with difference in religious (and cultural) beliefs and practice and posits 'an Islamic school' as a means to a coherent Muslim identity. The respondents' arguments for Muslim schools draw on and reproduce ideas about collective Muslim identity. This evidence can be explained using the concept of 'strategic essentialism' [START_REF] Spivak | Other Worlds: Essays in Cultural Politics[END_REF], whereby those identifying with a particular group promote an idea of intra-group unity in order to further what they consider to be their, or the groups', interests. Employing this concept as an analytical tool, however, can leave the analyst open to an accusation that they are unfairly representing organizations or individuals by implying that an idea of group unity is being used instrumentally. Such an accusation is a particular concern for this study, as slyness and inscrutability are common Orientalist stereotypes (Said 1978) and the Runnymede [START_REF] Trust | Beyond Westphalia: Nations and Diasporas -the Case of the Muslim Umma[END_REF] argues that portrayals of Islam as an instrumentalist political ideology (and subsequent representations of Muslims as manipulative and insincere) can be manifestations of Islamophobia. To clarify, it is not suggested that that respondents' in this study were disingenuous when they invoked ideas about collective Muslim identity, and it is worth re-iterating that such ideas were largely mobilized defensively, in social circumstances that respondents perceived as threatening.

Discussion

The data reported above, however, also revealed an example of essentializing perpetrated by those who perceive themselves as resisting discrimination, which raises difficult questions (Thompson 2008). Negatively judgmental group stereotypes of immoral Western or secular 'others' were expressed by a few respondents as they marked out their identity in relative or oppositional terms. Bauman (1989, p. 64) discusses the 'antagonism generated by the human practices of identity-seeking and boundary-drawing', and for Jenkins (2008, p. 48) [START_REF] Modood | Multiculturalism and Groups[END_REF] argues that essentializing that excludes groups, or casts them as inferior, is different from that which is used to resist such activities. Such a division, however, is difficult to sustain. A claim for Muslims schools underpinned by negative stereotypes of non-Muslims is a claim based on stereotypes, regardless of whether the aim is to protect Muslim pupils against prejudice and discrimination. To the extent that strategic essentialism is an ideological activity, the distorted or misleading portrayals of 'other' groups or individuals that occur under its rubric could be considered ethnocentric, or even racist. 2 Whether such outlooks are commonly generated, encouraged or legitimated by strategic essentialism is a question for further research.

Avoiding essentialism

Reporting these expressions of collective identity raises critical questions about the extent to which such findings essentialize, homogenize and/or reify the group under discussion. Recent sociological debates about multiculturalism have addressed such questions, and while the protagonists commonly reject conceptualizations of groups as static or homogenous, they offer different ideas about the relationship between individuals and groups. Meer and Modood (2009, p. 4) distinguish between formulations of multiculturalism that 'invalidate 'group' identities' and those that are 'inclusive of 'groupings''. Three theoretical positions -that range from weak to strong in their conceptions of groups -are outlined below. They are used to explain different aspects of the expression and mobilization of collective Muslim identity in the data, while avoiding the risk of portraying these as essential group characteristics. advocates 'multiculturalism without groups'. [START_REF] Gilroy | After Empire: Multiculture or Postcolonial Melancholia[END_REF] champions the potential of convivial, cosmopolitan 'multiculture' against continuing racism and 'the endless narrative of immigration as invasion ' (p. 166). With respect to group identity, Gilroy argues that '[t]he radical openness that brings conviviality alive makes a nonsense of closed, fixed, and reified identity and turns attention toward the always-unpredictable mechanisms of identification ' (2004, p. xi). This stance draws the analytical focus toward the processes and contexts of identifications (and away from the 'content' of collective identities). [START_REF] Hempel | What's it worth to you? The questionable value of instrumentalist approaches to ethnic identification[END_REF] contends that explanations of 'ethnic identifications' need to examine the conditions under which identities and interests converge. In this study, the convergence of identities and interests was evident in respondents support for Muslim schools based on defensive narratives about threats to, and protection of, Muslim identity. The conditions for such expressions include 3 : the minority status of the religion and its adherents (in terms of religious affiliation and often in terms of ethnicity); Britain's dominant secular (post-Christian) culture; prominent global and national political contexts; and, perceptions and experiences of prejudice and discrimination. They also, however, include a dominant socio-political context that promotes discourses of rights, equality and choice (including in relation to education), and where demands for equitable and 'culturally-sensitive' service provisions are commonplace. Other conditions relate to the responsibilities and concerns of parents and head teachers, and the organizational agendas of interest groups (and of schools). Gilroy's injunction to attend to the mechanisms of identification is thus analytically useful for the present study as it emphasizes the ' (2007, p. 8), but opposes 'forms of multiculturalism that cede power and authority to cultural groups' as intra-group disagreement is masked when representatives become the 'definitive voice of 'their' group ' (2008, p. 558). While promoting an agency-centred version of multiculturalism, she recognizes nonetheless that 'reified notions of culture are widely deployed by groups' (Squires 2008, p. 536). Phillips' approach to the intersection of individuals and groups is summarized thus by Squires (2008, p. 536): 'her primary normative concern […] is with individuals, and the rights of individuals vis-à-vis groups; but individuals who nonetheless experience structural inequalities on the basis of group membership'. This approach draws attention to the construction of collective identities through experiences of structural inequality. In the findings reported above, some respondents attributed experiences of marginalization (i.e. threat and denigration) to their affiliation to the group 'Muslim', and used these to explain the need for the protection believed to be on offer in Muslim schools. In this respect, collective Muslim identity was thus in part defined by marginalized social status (which, as argued above, should be understood within broader contexts of minority/majority relations of power). Phillips's approach thus highlights that constructions of collective identity can be rooted in subjective experiences of inequality (which those affiliated with the group may deploy in their interests). The third and 'strongest' conception of groups appears in communitarian versions of multiculturalism [START_REF] Parekh | Rethinking multiculturalism: cultural diversity and political theory[END_REF][START_REF] Modood | Multiculturalism: a civic idea[END_REF]). It emphasizes 'strong ethnic, religious or cultural identities' in order to address the circumstances facing particular groups, 'not least subjectively conceived ethno-religious minority groupings' (Meer and Modood 2009, p. 14 and 4). One danger of strong conceptualizations is that groups start to appear inflexible and homogenous. [START_REF] Modood | Multiculturalism: a civic idea[END_REF]2008) answers this problem by applying Wittgenstein's idea about 'family resemblance' to groupings, and thereby suggests that groups can be coherent without the presence of definitive (essential) characteristics. The groupings are phenomenological -people are perceived (by themselves or others) as belonging to or being excluded from groups categorized by labels such as Muslim, Jewish, Sikh, etc. Underlying these labels are ideas about knowledge, practice and values, which when enacted, mark attachment and affiliation to (or exclusion from) the group. This analytical position accurately reflects certain ideas about collective identity expressed by respondents in the present study. Specifically, when respondents explained about belonging to the group Muslim, they often invoked ideas about shared knowledge, values or practice. There was, nevertheless, sometimes a blurred boundary between religion and culture, which coheres with [START_REF] Meer | The politics of voluntary and involuntary identities: are Muslims in Britain an ethnic, racial or religious minority?[END_REF] claim that Muslim identity is a 'quasi-ethnic sociological formation'. These 'strong' conceptualizations of collective identities are thus helpful as they prioritize respondents' conceptualization of their identities, even if -analytically speaking -their outlooks may be considered essentialist. Theories with varying approaches to 'groups' (but all with anti-essentialist undertones) were used to understand different elements of the data: that identity constructions are context-bound processes; that subjective experience of structural inequality can inform affiliation to groups; and, that group members may experience and express their affiliations in essentialist terms. The forms of collective Muslim identity reported in this study may be thus regarded as context-dependent constructions that reflected an intersection between perceived social conditions and interests, and which drew upon, shaped and promoted ideas about commonality and collective needs. 2 Miles (1989, p. 42) defines racism as an ideology which portrays 'human beings, and the social relations between human beings, in a distorted or misleading manner '. 3 This is not to suggest that all Muslims, or even all the respondents, share a common stance toward these conditions.

Conclusion

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  . 441) The terrorist atrocities which followed the millennium cast a very different light on these nascent socio-political engagements; Muslims became perceived less as 'victims of social inequality' and more as 'potential aggressors'. Indeed, studies of British press coverage show that Muslims have been increasingly visible and often portrayed negatively(Poole 2002;[START_REF] Meer | Get off your knees". Print media public intellectuals and Muslims in Britain[END_REF].

  homes, offices or above mosques[START_REF] Hartas | In good faith: schools, religion, and public funding[END_REF].

  argue that religious identities should not be promoted by state-sanctioned education. Heterogeneity in the construction and expression of Muslim identities must be recognized. Arguably, however, this marginalizes questions about the coherence and stability of collective Muslim identity. This paper explores how collective Muslim identity is constructed and expressed in the contentious arena of claims for Muslims schools. It seeks to examine what it is about collective Muslim identity that is most stable or most commonly attested to; and what might be the dominant formations and meanings in collective Muslim identity in this context.

  anonymized and related descriptors would aim to limit identifiability.

  commonly aired by representatives of 'Muslim interest groups' was that Muslim identity may be confused, lost or denigrated because of the content of the curriculum in mainstream schools, both what was included and what was omitted. There were three perceived shortfalls in the curriculum: that it might contravene religious teaching, lack information about Islam or lack information about the achievements of Muslims. A representative of an interest group that promotes Islam and serves the needs of Muslims argued that aspects of the curriculum, such as sex education and evolutionary theory, might contravene Islam: Children spend about 7 or 8 hours at school, and they're taught all the wrong things. By wrong things I mean they are against the religion. They are teaching about Darwinist theory, about this family planning, sex and other things. […] So when they come back after school it's a very hard job to clear those cobwebs in their minds.

  It was the systematic under-representation of the experiences, contributions and perspectives of non-European individuals and groups that most concerned a respondent at an interest group that campaigns and lobbies on behalf of 'the Muslim community': […] the national curriculum […] is very Eurocentric. I think this has to change, to take into account the contribution other cultures and civilizations made to our civilization. […] In these faith schools this young person will be told of the contribution their faith, their culture, their civilization has made to the wider civilization. They are not simply useless people.This view, which connects self-confidence and collective identity, reveals a fear of a denigrated identity if Muslim pupils view themselves as part of a group of 'useless people'. The respondent contrasts this with opportunities afforded by would purportedly reflect more positively 'their faith, their culture, their civilization'.

  scares us to be quite frank with you. And these are the ways that we try to safeguard ourselves, having Islamic schools, etcetera.The views of these respondents suggest a perceived contrast between Islamic and secular values, and position Muslim schools as a means safeguarding collective identity.It was also repeatedly claimed that pupils in mainstream state schools exhibit anti-social behaviour (with specific references to underage drinking, drug-taking, swearing and vandalism), and this was contrasted with the behaviour instilled by Muslim schools. For example, the head teacher of a third independent Muslim school argued that the school aimed to: […] bring the children up in an Islamic environment, and equip the children with Islamic manner and behaviour. Islamic manner and behaviour tells the children to be kind to the neighbours, to encourage them not to mess around, not to throw your banana skin in the street, or smoke cigarettes. Similarly, a representative of an interest group that politically promotes the establishment of an Islamic state argued the role of Muslim schools was to: […] nurture the Islamic personality first and foremost. What that means essentially is a person's thoughts, his inclinations, that these are in line with Islam. […] To give it a practical slant […], this would mean that a

  youngest child when he was little he went to the state school he used to wear little hats. […] Every day the boys in the playground would take his hat and put it on the roof, and the annoying thing was that I would get a complaint from the school that the caretaker had to keep on getting up. I said 'look you've got to respect the fact that he is wearing that, […] you have got to educate other children not to keep throwing it on the top, not to keep telling me not to make him wear it, that's not fair'.

  One fear was that Muslim pupils attending schools where there are few other Muslim children would result in them not 'fitting in'. For example, a parent of children at an independent Muslim school recalled his childhood 'identity crisis', the attendant feelings of isolation and how this influenced his decision about his children's education: I didn't really know who I was […]. At one time I thought well I'm English, I'm not Muslim I'm English, and I wanted to be as much English as I could, more English than the English, and I wanted to kind of pretend I had nothing else. And then I [...] swung the other way, I am Muslim and my dad is from Egypt although I've only been to Egypt a couple of times but I tried to, you know… But then I realized eventually that I am actually a bit of both. […] It took me a long time to sort that out, and Ifeel that part of the reason was because I didn't have[…] anyone that I , I didn't have any guidance on the matter, I was kind of alone[…]. I just didn't want that for my children, I wanted my children to feel a strong sense of identity, that they belonged, that they were Muslim and they could be proud to be Muslim and there were other Muslims in this country and they felt happy about that. But at the same time, you know, I wanted them to have a good education and feel that this is their country and feel proud of their country as well.

  paper questions were raised about collective Muslim identity (its common formations and attendant meanings) and the construction of claims for Muslim schools. Advocates of Muslim schools interviewed in this study commonly expressed a perception that Muslims were under threat and/or attack. This was sometimes infused by meanings that revealed fears about loss of identity, or even the loss of individuals from the group. Muslim identity was often seen as being challenged, denigrated and undermined by Western or secular ideas, values and behaviours. Threats came from perceived (overt and institutionalized) anti-Muslim prejudice, and from everyday 'non-Islamic' values and behaviours. As such, fears particularly focused on 'external' threats, relating to assimilation into Western, secular society, or from the identity-crises that might arise from majority/minority relations of power. Less frequently, fears extended to 'internal' threats -the loss of 'moderate' Muslims to 'radical' or 'extreme' Islam. In this threatening social context, Muslim schools offered various means of defence (that were differently emphasized by stakeholder groups). In sum, schools could instil or reinforce knowledge about Islam and related practices or modes of behaviour, and/or they could protect Muslim pupils from anti-Muslim prejudice (or extremism) and offer them a sense of worth. It is thus suggested that two constructions of collective Muslim identity were invoked in the context of advocacy for Muslim schools. The first related to knowledge, values and practices expressed in relation to religion but which sometimes blurred into ethnicity. The second centred on life experiences relating to denigrated or discordant social identities.

  processes and contexts framing respondents' expressions of collective Muslim identity. An analytical position in which groups are more prominent is offered by Anne Phillips in Multiculturalism without Culture. Phillips aims to 'address inequalities between cultural groups

  , advocates of Muslim schools in Britain invoked and mobilized ideas about collective Muslim identity that were based on their perception that they shared with other Muslims either knowledge values and practice and/or life experiences relating to denigrated or discordant social identities. It is not claimed, however, that these are essential characteristics of collective Muslim identity. Throughout the paper, the importance of heterogeneity and context has been reiterated. The collective Muslim identity expressed by the interviewees converged with their shared interest for a separate school provision in social conditions in which they perceived themselves to be under threat and/or attack.

  . 2008 'Multiculturalism without Multiple Cultures?', Social and Legal Studies, vol. 17, pp. 543-547 TINKER, C. 2006 'Islamophobia, Social Cohesion and Autonomy: Challenging the arguments against state funded Muslim schools in Britain', Muslim Education Quarterly, vol. 23, pp. 4-19 TINKER, C. 2009 'Rights, social cohesion and identity: arguments for and against state-funded Muslim schools in Britain', Race, Ethnicity and Education, vol. 12, 539-553 TINKER, C. and ARMSTRONG, N. 2008 'From the Outside Looking in: How an Awareness of Difference Can Benefit the Qualitative Research Process', The Qualitative Report, vol. 13, pp. 53-60 WALFORD, G. 2001 'Funding for Religious Schools in England and the Netherlands: can the piper call the tune?', Research Papers in Education, vol. 16, pp. 359-80 CLAIRE TINKER is a Research Fellow in the Department of Social Sciences, Bath Spa University. ADDRESS: Newton Park Campus, Bath, BA2 9BN, UK. E-mail: clairetinker@hotmail.co.uk ANDREW SMART is Senior Lecturer in the Department of Social Sciences, Bath Spa University. Park Campus, Bath, BA2 9BN, UK. E-mail: a.smart@bathspa.ac.uk 1 Personal communication: Association of Muslim Schools (10/05/2010).

  argues that there is ongoing debate about 'what it is to be a Muslim' and a 'proliferation of Muslim organizations' each

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URL: http://mc.manuscriptcentral.com/rers ethnic@surrey.ac.ukEthnic and Racial Studies

Acknowledgements

CT was supported by the Economic and Social Research Council (PTA-026-27-1361) and the Centre for Ethnicity and Citizenship, University of Bristol.

Prejudice and discrimination

Concern over prejudice and discrimination were widespread among the sample, but most commonly expressed by parents and head teachers. Characteristically, the fear was that Muslims were under attack and afforded little protection, and that this created risks to children's self-esteem or self-confidence.

The fear that Muslim children experience prejudice was commonly rooted in experience. For example, the parent of child who attended a mainstream state school claimed that her son's religious views had been unfairly criticized by both fellow pupils and teachers:

[…] the teacher would be putting him down, you know, 'you're not supposed to believe that and you're not supposed to say this'. […] Not that he believed in any of the terrorists or anything like that, it's just the religion itself, that's what people were picking on. […] They said it's wrong, and he will say 'you can't tell me that my religion is wrong'. He says 'anyway we don't believe in killing', because they said 'your religion is all about killing'.

Such perceptions, of attacks on belief, underpinned some parents' desire for defensive or protective measures. This parent therefore sent her child to an independent Muslim school after having negative experiences in a mainstream state school: