Jérôme Dedecker

Florence Merlevède

Magda Peligrad

A quenched weak invariance principle

Keywords: quenched central limit theorem, weak invariance principle, strong mixing, Markov chains. Mathematical Subject Classification (2010): 60F05, 60F17, 60J05

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A quenched weak invariance principle

Introduction

Let (ξ i ) i≥0 be a Markov chain admitting an invariant probability π. Let f be a real-valued function such that π(f 2 ) < ∞ and π(f ) = 0, and let S n = f (ξ 1 ) + • • • + f (ξ n ). If the central limit theorem (CLT) holds for n -1/2 S n starting form the initial distribution π, an interesting question is to know whether it remains true for another initial distribution ν. [START_REF] Maxwell | Central limit theorem for additive fonctionals of Markov chains[END_REF] have given a projective criterion under which S n satisfies the so-called conditional CLT, which implies that the CLT holds for any initial distribution having a bounded density with respect to π. Necessary and sufficient conditions for the conditional CLT are given in [START_REF] Dedecker | Necessary and sufficient conditions for the conditional central limit theorem[END_REF], and [START_REF] Wu | Martingale approximations for sums of stationary processes[END_REF].

The question is more delicate if ν is a Dirac mass at point x. One says that the CLT is quenched if it holds for almost every starting point with respect to π. The quenched CLT implies the central limit theorem for the chain starting from an invariant probability measure π, referred as annealed CLT. The same terminologies are used for the functional central limit theorem (FCLT). For aperiodic Harris recurrent Markov chains, the quenched CLT question is solved by using Proposition 18.1.2 in [START_REF] Meyn | Markov chains and stochastic stability[END_REF]. More precisely, for an aperiodic Harris recurrent Markov chain, if the CLT holds for the initial distribution π, then it holds for any initial distribution, and hence for any starting point x (see Proposition 3.1 in Chen (1999) and its proof). In the non irreducible setting, the situation is not so clear. For instance, an example of a Markov chain with normal transition operator satisfying the annealed CLT but not the quenched is given at the end of Section 3 in [START_REF] Derriennic | The central limit theorem for Markov chains with normal transition operators, started at a point[END_REF].

This question of the quenched CLT can be formulated in the more general context of stationary sequences: it means that, on a set of measure one, the central limit theorem holds when replacing the usual expectation by the conditional expectation with respect to the past σ-algebra. Some examples of stationary processes satisfying the CLT but not the quenched CLT can be found in Volný and Woodroofe (2010a).

The first general results on the quenched CLT and FCLT are given in [START_REF] Borodin | Limit theorems for functionals of random walks[END_REF]: in the Markov chain setting, it says that the FCLT holds if there is a solution in L 2 (π) to the Poisson equation (see Gordin and Lifschitz (1978)); in a general setting it means that the FCLT is true under Gordin's condition (1969). This result has been improved by Derriennic andLin (2001, 2003), [START_REF] Zhao | Law of the iterated logarithm for stationary processes[END_REF], [START_REF] Cuny | Pointwise ergodic theorems with rate and application to limit theorems for stationary processes[END_REF], [START_REF] Cuny | Central limit theorem started at a point for additive functional of reversible Markov Chains[END_REF], [START_REF] Cuny | A quenched invariance principle for stationary processes[END_REF], Volný and Woodroofe (2010b) and Merlevède et al. (2012). In a recent paper, [START_REF] Cuny | On martingale approximations and the quenched weak invariance principle[END_REF] have proved that the FCLT is quenched under the condition of [START_REF] Maxwell | Central limit theorem for additive fonctionals of Markov chains[END_REF].

All the papers cited above use a martingale approximation in L 2 . Consequently, the projective condition obtained up to now are always expressed in terms of L 2 norms of conditional expectations. In this paper, we prove the quenched FCLT under a projective condition involving L 1 -norms, in the spirit of [START_REF] Gordin | Abstracts of Communication, T.1:A-K, International Conference on Probability Theory[END_REF]. As a consequence, we obtain that the FCLT of [START_REF] Doukhan | The functional central limit theorem for strongly mixing processes[END_REF] for strongly mixing sequences is quenched. Note that [START_REF] Doukhan | The functional central limit theorem for strongly mixing processes[END_REF] have shown that their condition is optimal in some sense for the usual FCLT, so it is also sharp for the quenched FCLT. In Section 3.1, we study the example of the non irreducible Markov chain associated to an intermittent map. Once again, we shall see through this example that our condition is essentially optimal.

Our main result, Theorem 2.1 below, is a consequence of the more general Proposition 4.1, where the conditions are expressed in terms of conditional expectations of partial sums. The proof of this proposition is done via a blocking argument followed by a two step martingale decomposition. We start with a finite number of consecutive blocks of random variables. The sum in blocks are approximated by martingales. This decomposition introduces the need of studying the normal approximation for a family of double indexed martingales. This approximation has interest in itself and is presented in Section 6.

Results

Let (Ω, A, P) be a probability space, and T : Ω → Ω be a bijective bimeasurable transformation preserving the probability P. An element A is said to be invariant if T (A) = A. We denote by I the σ-algebra of all invariant sets. The probability P is ergodic if each element of I has measure 0 or 1.

Let F 0 be a σ-algebra of A satisfying F 0 ⊆ T -1 (F 0 ) and define the nondecreasing filtration (F i ) i∈Z by F i = T -i (F 0 ). We assume that there exists a regular version P T |F 0 of T given F 0 , and for any integrable random variable f from Ω to R we write K(f ) = P T |F 0 (f ). Since P is invariant by T , for any integer k, a regular version

P T |F k of T given F k is then obtained via P T |F k (f ) = K(f • T -k ) • T k .
In the sequel, all the conditional expectations with respect to F k are obtained through these conditional probabilities. More precisely, we shall use the following notations:

E 0 (X) := E(X|F 0 ) = K(X • T -1 ) and E k (X) := E(X|F k ) = K(X • T -k-1 ) • T k . With these notations, E(f • T 2 |F 0 ) = E(K(f ) • T |F 0 ) = K 2 (
f ), and more generally, for any positive integer ℓ, E(f

• T ℓ |F 0 ) = K ℓ (f ).
Let X 0 be an F 0 -measurable, square integrable and centered random variable. Define the sequence X = (X i ) i∈Z by

X i = X 0 • T i . Let S n = X 1 + • • • + X n , and define the Donsker process W n by W n (t) = n -1/2 (S [nt] + (nt -[nt])X [nt]+1 ). Let H * the space of continuous functions ϕ from (C([0, 1]), . ∞ ) to R such that x → |(1 + x 2 ∞ ) -1 ϕ(x)| is bounded.
Our main result is the following:

Theorem 2.1. Assume that k≥0 X 0 E 0 (X k ) 1 < ∞ . (2.1)
then the series η = E(X 2 0 |I) + 2 k>0 E(X 0 X k |I) (2.2)
converges almost surely and in L 1 . Moreover, on a set of probability one, for any ϕ in H * ,

lim n→∞ E 0 (ϕ(W n )) = ϕ(z √ η)W (dz) , (2.3)
where W is the distribution of a standard Wiener process. The convergence in (2.3) also holds in L 1 .

Note that the L 1 -convergence in (2.3) has been proved in [START_REF] Dedecker | Necessary and sufficient conditions for the conditional central limit theorem[END_REF]. In this paper, we shall prove the almost sure convergence. Various classes of examples satisfying (2.1) can be found in [START_REF] Dedecker | On the functional central limit theorem for stationary processes[END_REF].

This result has an interesting interpretation in the terminology of additive functionals of Markov chains. Let (ξ n ) n≥0 be a Markov chain with values in a Polish space S, so that there exists a regular transition probability P ξ 1 |ξ 0 =x . Let P be the transition kernel defined by P (f )(x) = P ξ 1 |ξ 0 =x (f ) for any bounded measurable function f from S to R, and assume that there exists an invariant probability π for this transition kernel, that is a probability measure on S such that π(f ) = π(P (f )) for any bounded measurable function f from S to R. Let then L 2 0 (π) be the set of functions from S to R such that π(f 2 ) < ∞ and π(f

) = 0. For f ∈ L 2 0 (π) define X i = f (ξ i ).
Notice that any stationary sequence (Y k ) k∈Z can be viewed as a function of a Markov process

ξ k = (Y i ; i ≤ k), for the function g(ξ k ) = Y k .
In this setting the condition (2.1) is k≥0 π(|f P k (f )|) < ∞. Also, the random variable η defined in Theorem 2.1 is the limit almost surely and in L 1 of n -1 E(S 2 n |ξ 0 ), in such a way that η = η(ξ 0 ). By stationarity, it is also the limit in

L 1 of the sequence n -1 E((X 2 +• • •+X n+1 ) 2 |ξ 1 )
, so that η(ξ 0 ) = η(ξ 1 ) almost surely. Consequently η is an harmonic function for P in the sense that π-almost surely P (η) = η.

In the context of Markov chain the conclusion of Theorem 2.1 is also known under the terminology of FCLT started at a point. To rephrase it, let P x be the probability associated to the Markov chain started from x and let E x be the corresponding expectation. Then, for π-almost every x ∈ S, for any ϕ in H * ,

lim n→∞ E x (ϕ(W n )) = ϕ(z η(x))W (dz) . Moreover, lim n→∞ E x (ϕ(W n )) -ϕ(z η(x))W (dz) π(dx) = 0 .
We mention that in Theorem 2.1 no assumption of irreducibility nor of aperiodicity is imposed. Under the additional assumptions that the Markov chain is irreducible, aperiodic and positively recurrent, [START_REF] Chen | Limit theorems for functionals of ergodic Markov chains with general state space[END_REF] showed that the CLT holds for the stationary Markov chain under the condition k≥0 π(f P k (f )) is convergent, and the quenched CLT holds under the same condition by applying his Proposition 3.1.

Remark 2.2. Let us present an alternative condition to the criterion (2.1) in case where T is ergodic. We do not require here X 0 to be in L 2 but only in L 1 . The so-called Gordin criterion in L 1 is:

sup n∈N E 0 (S n ) 1 < ∞ and lim inf n→∞ E|S n | √ n < ∞ . (2.4)
By [START_REF] Esseen | On moment conditions for normed sums of independent variables and martingale differences[END_REF], it is known that (2.4) is equivalent to the following L 1coboundary decomposition:

X 0 = m 0 + z 0 -z 0 • T , (2.5)
where z 0 ∈ L 1 and m 0 is a F 0 -measurable random variable in L 2 such that E -1 (m 0 ) = 0 almost surely. Therefore, the criterion (2.2) leads to the annealed CLT. Note that one can easily prove that the condition (3.2) of the next section also implies (2.4). However, the condition (2.4) is not sufficient to get the annealed FCLT (see [START_REF] Volný | On the invariance principle and the law of iterated logarithm for stationary processes[END_REF]).

In addition, from Corollary 2 in Volný and Woodroofe (2010,b), it follows that (2.4) is not sufficient to get the quenched CLT either. In Proposition 5.4 of Section 5.2, we shall provide an example of stationary process for which (2.1) holds but (2.4) fails.

Applications

As a consequence of Theorem 2.1, we obtain the following corollary for a class of weakly dependent sequences. We first need some definitions. Definition 3.1. For a sequence Y = (Y i ) i∈Z , where Y i = Y 0 • T i and Y 0 is an F 0 -measurable and real-valued random variable, let for any k ∈ N,

α Y (k) = sup t∈R E(1 Y k ≤t |F 0 ) -E(1 Y k ≤t ) 1 .
Definition 3.2. Recall that the strong mixing coefficient of [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF] between two σ-algebras F and G is defined by α(F , G) = sup A∈F ,B∈G |P(A∩B)-P(A)P(B)|. For a strictly stationary sequence (Y i ) i∈Z of real valued random variables, and the σ-algebra

F 0 = σ(Y i , i ≤ 0), define then α(0) = 1 and α(k) = 2α(F 0 , σ(Y k )) for k > 0 . (3.1)
Between the two above coefficients, the following relation holds: for any positive k, α Y (k) ≤ α(k). In addition, the α-dependent coefficient as defined in Definition 3.1 may be computed for instance for many Markov chains associated to dynamical systems that fail to be strongly mixing in the sense of Rosenblatt (see Section 3.1). Definition 3.3. A quantile function Q is a function from ]0, 1] to R + , which is left-continuous and non increasing. For any nonnegative random variable Z, we define the quantile function

Q Z of Z by Q Z (u) = inf{t ≥ 0 : P(|Z| > t) ≤ u}.
Definition 3.4. Let µ be the probability distribution of a random variable X. If Q is an integrable quantile function, let Mon(Q, µ) be the set of functions g which are monotonic on some open interval of R and null elsewhere and such that Q |g(X)| ≤ Q. Let F (Q, µ) be the closure in L 1 (µ) of the set of functions which can be written as L ℓ=1 a ℓ f ℓ , where L ℓ=1 |a ℓ | ≤ 1 and f ℓ belongs to Mon(Q, µ). Corollary 3.5. Let Y 0 be a real-valued random variable with law P Y 0 , and

Y i = Y 0 • T i . Let Q be a quantile function such that k≥0 α Y (k) 0 Q 2 (u)du < ∞ .
(3.2)

Let X i = f (Y i ) -E(f (Y i ))
, where f belongs to F (Q, P Y 0 ). Then (2.1) is satisfied and consequently, the conclusion of Theorem 2.1 holds.

To prove that (3.2) implies (2.1), it suffices to apply Proposition 5.3 with m = q = 1 of [START_REF] Merlevède | Strong approximation of partial sums under dependence conditions with application to dynamical systems[END_REF].

Notice that if (α(k)) k≥0 is the usual sequence of strong mixing coefficients of the stationary sequence (X i ) i∈Z as defined in (3.1), then it follows from Corollary 3.5 that if

k≥0 α(k) 0 Q 2 |X 0 | (u)du < ∞ , (3.3

Application to functions of Markov chains associated to intermittent maps

For γ in ]0, 1[, we consider the intermittent map T γ from [0, 1] to [0, 1], which is a modification of the Pomeau-Manneville map (1980):

T γ (x) =    x(1 + 2 γ x γ ) if x ∈ [0, 1/2[ 2x -1 if x ∈ [1/2, 1] .
Recall that T γ is ergodic (and even mixing in the ergodic theoretic sense) and that there exists a unique T γ -invariant probability measure ν γ on [0, 1], which is absolutely continuous with respect to the Lebesgue measure. We denote by L γ the Perron-Frobenius operator of T γ with respect to ν γ . Recall that for any bounded measurable functions f and g, ν γ (f x(H(x))

•g •T γ ) = ν γ (L γ (f )g). Let (Y i ) i≥0 be
1-2γ 1-γ dx < ∞ . (3.4) η = ν γ ((f -ν γ (f )) 2 ) + 2 k>0 ν γ ((f -ν γ (f ))f • T k γ ) . (3.5)
Proof. To prove this corollary, it suffices to see that (3.4) implies (3.2). For this purpose, we use Proposition 1.17 in Dedecker et al. (2010) stating that there exist two positive constant B, C such that, for any n > 0, Bn (γ-1)/γ ≤ α Y (n) ≤ Cn (γ-1)/γ , together with their computations page 817.

In particular, if f is BV and γ < 1/2, we infer from Corollary 3.7 that the conclusion of Theorem 2.1 holds with η defined by (3.5) . Note also that (3.4) is satisfied if H is such that H(x) ≤ Cx -2(1-γ)/(1-2γ) (ln(x)) -b for x large enough and b > (1 -γ)/(1 -2γ). Therefore, since the density h νγ of ν γ is such that h νγ (x) ≤ Cx -γ on (0, 1], one can easily prove that if f is positive and non increasing on (0, 1), with

f (x) ≤ C x (1-2γ)/2 | ln(x)| d near 0 for some d > 1/2,
then (3.4) and the quenched FCLT hold. Notice that when f is exactly of the form f (x) = x -(1-2γ)/2 , Gouëzel (2004) proved that the central limit theorem holds for n i=1 (f (Y i )-ν γ (f )) but with the normalization n ln(n). This shows that the condition (3.4) is essentially optimal for the quenched CLT with the normalization √ n.

Some general results

In this section we develop sufficient conditions imposed to conditional expectations of partial sums for the validity of the quenched CLT and FCLT.

For any positive integers i and p, define S (i) p = S pi -S p(i-1) .

A quenched CLT

Let us introduce the following three conditions under which the quenched central limit theorem holds:

C 1 lim m→∞ lim sup p→∞ 1 √ mp m+1 i=2 E 0 |E (i-2)p (S (i) p )| = 0 a.s. C 2 there exists a T -invariant r.v. η that is F 0 -measurable and such that lim m→∞ lim sup p→∞ E 0 m i=1 1 mp E (i-1)p (S (i+1) p ) 2 -η = 0 a.s. lim m→∞ lim sup p→∞ E 0 m i=1 1 mp E (i-1)p (S (i) p + S (i+1) p ) 2 -2η = 0 a.s. C 3 for each ε > 0 lim m→∞ lim sup p→∞ 1 m m i=1 1 p E 0 (S (i) p ) 2 1 |S (i) p |/ √ p>ε √ m = 0 a.s.
lim n→∞ E 0 (f (n -1/2 S n )) = f (x √ η)g(x)dx ,
where g is the density of a standard normal.

This proposition is designed especially for the proof of Theorem 2.1. Notice that in the expression

E (i-2)p (S (i) p ) of condition C 1 there is a gap of p variables between S (i)
p and the variables used for conditioning. This gap is important for weakening the dependence and is essentially used in the proof of Theorem 2.1.

Proof of Proposition 4.1. The result will follow from Proposition 4.2 below, for double indexed arrays of random variables: Proposition 4.2. Assume that (Y n,m,i ) i≥1 is an array of random variables in L 2 adapted to an array (G n,m,i ) i≥1 of nested sigma fields. Let E n,m,i denote the conditional expectation with respect to G n,m,i . Suppose that

lim m→∞ lim sup n→∞ m+1 i=2 E|E n,m,i-2 (Y n,m,i )| = 0 , (4.1)
and that there exists σ 2 ≥ 0 such that

lim m→∞ lim sup n→∞ E m i=1 E n,m,i-1 Y 2 n,m,i+1 -σ 2 = 0 (4.2)
and

lim m→∞ lim sup n→∞ E m i=1 E n,m,i-1 (Y n,m,i + Y n,m,i+1 ) 2 -2σ 2 = 0 . (4.3)
Assume in addition that for each ε > 0

lim m→∞ lim sup n→∞ m+1 i=1 E(Y 2 n,m,i 1 |Y n,m,i |>ε ) = 0. (4.4)
Then for any continuous and bounded function f ,

lim m→∞ lim sup n→∞ E f m i=1 Y n,m,i -E(f (σN)) = 0 ,
where N is a standard Gaussian random variable.

E 0 f n -1/2 m[n/m] i=1 X i -f (x √ η)g(x)dx = 0 ,
where g is the density of a standard normal. Proposition 4.1 will then follow if we can prove that for any ε > 0,

lim m→∞ lim sup n→∞ P 0 n i=1 X i - m[n/m] i=1 X i ≥ ε √ n = 0 a.s. (4.5)
With this aim, we notice that

P 0 n i=1 X i - m[n/m] i=1 X i ≥ ε √ n ≤ P 0 m 2 max 1≤i≤n X 2 i ≥ ε 2 n .
and therefore (4.5) holds by relation (7.2) in Lemma 7.1 applied to Z i = X 2 i . It remains to prove Proposition 4.2.

Proof of Proposition 4.2. For any positive integer i, let

U n,m,i = Y n,m,i + E n,m,i (Y n,m,i+1 ) -E n,m,i-1 (Y n,m,i ) . (4.6)
To ease the notation, we shall drop the first two indexes (the pair n, m) when no confusion is possible. With this notation,

Y i = U i -E i (Y i+1 ) + E i-1 (Y i ) ,
and since we have telescoping sum,

m i=1 Y i = m i=1 U i + E 0 (Y 1 ) -E m (Y m+1 ) .
Notice that for any i ∈ {1, m + 1} and any ε > 0,

E(|E i-1 (Y i )| 2 ) ≤ ε 2 + E(Y 2 i 1 |Y i |>ε ) . (4.7)
Therefore by condition (4.4),

lim m→∞ lim sup n→∞ E E n,m,m (Y n,m,m+1 )) 2 + (E n,m.0 (Y n,m,1 )) 2 = 0 . (4.8)
The theorem will be proven if we can show that the sequence (U n,m,i ) i≥1 defined by (4.6) satisfies the conditions of Theorem 6.1. We first notice that E i-1

(U i ) = E i-1 (Y i+1
). Hence condition (6.1) is clearly satisfied under (4.1). On an other hand,

Var(U i |G i-1 ) = E i-1 Y 2 i + 2Y i E i (Y i+1 ) + E i-1 (E i (Y i+1 )) 2 -(E i-1 (Y i )) 2 -2(E i-1 (Y i ))(E i-1 (Y i+1 )) -(E i-1 (Y i+1 )) 2 . (4.9)
Notice that for any ε > 0

m i=1 E (E i-1 (Y i+1 )) 2 ≤ ε m i=1 E E i-1 (Y i+1 ) + ε m i=1 E |Y i+1 |1 |Y i+1 |>ε + m i=1 E Y 2 i+1 1 |Y i+1 |>ε ≤ ε m i=1 E E i-1 (Y i+1 ) + 2 m+1 i=2 E Y 2 i 1 |Y i |>ε . (4.10)
Similarly, for any ε > 0,

m i=1 E (E i-1 (Y i ))(E i-1 (Y i+1 )) ≤ ε m i=1 E E i-1 (Y i+1 ) + 2 m+1 i=1 E Y 2 i 1 |Y i |>ε .
In addition since

E i-1 Y 2 i + 2Y i E i (Y i+1 ) = E i-1 (Y i + Y i+1 ) 2 -E i-1 Y 2 i+1 ,
E m+1 i=1 E n,m,i-1 Y 2 n,m,i + 2Y n,m,i E n,m,i (Y n,m,i+1 ) -σ 2 = 0 . ( 4
E m i=1 E n,m,i-1 (E n,m,i (Y n,m,i+1 )) 2 -(E n,m,i-1 (Y n,m,i )) 2 = 0 . (4.12)
To prove (4.12), we first write that

m i=1 E i-1 (E i (Y i+1 )) 2 -(E i-1 (Y i )) 2 = E m (Y m+1 )) 2 -(E 0 (Y 1 )) 2 - m i=1 (E i (Y i+1 )) 2 -E i-1 (E i (Y i+1 )) 2 .
By (4.8), it follows that (4.12) will hold if we can show that

lim m→∞ lim sup n→∞ E m i=1 (E n,m,i (Y n,m,i+1 )) 2 -E n,m,i-1 (E n,m,i (Y n,m,i+1 )) 2 = 0 . (4.13)
This follows from an application of Lemma 6.2 with

d n,m,i = (E n,m,i (Y n,m,i+1 )) 2 -E n,m,i-1 (E n,m,i (Y n,m,i+1 )) 2 . Indeed m i=1 E(|d n,m,i |) ≤ 2 m+1 i=1 E(Y 2 n,m,i ) ,
and by Lemma 6.3, for any ε > 0,

m i=1 E(|d n,m,i |1 |d n,m,i |>8ε 2 ) ≤ 2 m i=1 E (E n,m,i (Y n,m,i+1 )) 2 1 (E n,m,i (Y n,m,i+1 )) 2 >4ε 2 ) ≤ 2 m i=1 E Y 2 n,m,i+1 1 |E n,m,i (Y n,m,i+1 )|>2ε ) ≤ 4 m+1 i=1 E Y 2 n,m,i 1 |Y n,m,i |>ε ) .
So condition (6.15) holds by using (4.2) and (4.4). It remains to prove that (6.3) holds. Clearly this can be achieved by using (4.4) combined with Lemma 6.3.

Finite dimensional convergence

For 0 < t 1 < • • • < t d ≤ 1, define the function π t 1 ,...,t d from C([0, 1]) to R d by π t 1 ,...,t d (x) = (x(t 1 ), x(t 2 ) -x(t 1 ), . . . , x(t d ) -x(t d-1 )). For any a in R d define the function f a from R d to R by f a (x) =< a, x
0 < t 1 < • • • < t d ≤ 1, lim n→∞ E 0 h • f a • π t 1 ,...,t d (W n ) = h • f a • π t 1 ,...,t d (z √ η)W (dz) , (4.14)
where W is the distribution of a standard Wiener process.

Proof of Proposition 4.3. Since ∞ d=1 Q d is countable, it suffices to prove that for any a ∈ R d and any t 1 , t 2 , . . . , t d rational numbers such that 0 < t 1 < • • • < t d ≤ 1, on a set of probability one, for any continuous and bounded function h, the convergence (4.14) holds. With this aim, for any ℓ ∈ {1, . . . , d}, we set t ℓ = r ℓ /s ℓ where r ℓ and s ℓ are positive integers. Let

c d = d ℓ=1 s ℓ . Rewrite t ℓ = b ℓ /c d . The b ℓ 's are then positive integers such that 0 < b 1 < • • • < b d ≤ c d .
Let m be a fixed positive integer and let p = [n/(mc d )]. Notice that for any ℓ ∈ {1, . . . , d},

[nt ℓ ] -mb ℓ < mpb ℓ ≤ [nt ℓ ] + 1 .
Therefore for any reals a 1 , • • • , a d , with the convention that t 0 = 0 and b 0 = 0,

d ℓ=1 a ℓ [nt ℓ ] i=[nt ℓ-1 ]+1 X i - d ℓ=1 a ℓ pmb ℓ i=pmb ℓ-1 +1 X i ≤ d ℓ=1 |a ℓ | (p+1)mb ℓ i=pmb ℓ +1 |X i | .
Using (7.2) of Lemma 7.1, we infer that for any ℓ ∈ {1, . . . , d} and every ε > 0,

lim n→∞ P 0 |a ℓ | √ n (p+1)mb ℓ i=pmb ℓ +1 |X i | > ε = 0 a.s.
In addition,

d ℓ=1 a ℓ W n (t ℓ ) -W n (t ℓ-1 ) - d ℓ=1 a ℓ S [nt ℓ ] -S [nt ℓ-1 ] ≤ 2 d ℓ=1 |a ℓ | max 1≤i≤n |X i | ,
implying once again by (7.2) in Lemma 7.1 that

lim n→∞ n -1/2 E 0 d ℓ=1 a ℓ W n (t ℓ ) -W n (t ℓ-1 ) - d ℓ=1 a ℓ S [nt ℓ ] -S [nt ℓ-1 ] = 0 a.s. (4.15)
From the preceding considerations, it remains to prove that, on a set of probability one, for any continuous and bounded function f ,

lim m→∞ lim sup n→∞ E 0 f n -1/2 d ℓ=1 a ℓ pmb ℓ i=pmb ℓ-1 +1 X i -E 0 (f (σ d N)) = 0 , (4.16)
where

σ 2 d = η d ℓ=1 a 2 ℓ (t ℓ -t ℓ-1 )
and N is a standard Gaussian random variable independent of F 0 . With this aim, we write that

2 = σ 2 d = η d ℓ=1 a 2 ℓ (t ℓ -t ℓ-1
), we proceed as follows. For (4.2), we write that

E 0 mb d i=1 E n,m,i-1 (Y 2 n,m,i+1 ) -σ 2 d = E 0 1 mpc d d ℓ=1 a 2 ℓ mb ℓ i=mb ℓ-1 +1 E (i-1)p ((S (i+1) p ) 2 ) -σ 2 d ≤ d ℓ=1 a 2 ℓ E 0 1 mpc d mb ℓ i=mb ℓ-1 +1 E (i-1)p ((S (i+1) p ) 2 ) -η(t ℓ -t ℓ-1 ) . Since t ℓ = b ℓ /c d , we obtain that E 0 mb d i=1 E n,m,i-1 (Y 2 n,m,i+1 ) -σ 2 d ≤ d ℓ=1 a 2 ℓ b ℓ c d E 0 1 mpb ℓ mb ℓ i=1 E (i-1)p ((S (i+1) p ) 2 ) -η + d ℓ=1 a 2 ℓ b ℓ-1 c d E 0 1 mpb ℓ-1 mb ℓ-1 i=1 E (i-1)p ((S (i+1) p ) 2 ) -η .
Condition (4.2) is then proved by using the first part of C 2 . Using similar arguments, we prove (4.3) by using the second part of C 2 .

A quenched invariance principle

Let us define the maximal version of

C 3 . For k ≤ l, let Sk,l = max k≤i≤l |S i -S k |. C 4 for any ε > 0 lim m→∞ lim sup p→∞ 1 m m i=1 1 p E 0 S2 (i-1)p,ip 1 | S(i-1)p,ip |/ √ p>ε √ m = 0 a.s.
lim n→∞ E 0 (f (W n )) = f (x √ η)W (dx) ,
where W is the distribution of a standard Wiener process.

P 0 sup (i-1)m -1 <t≤im -1 |W n (t) -W n ((i -1)m -1 )| > ε = 0 a.s. (4.18) Since sup t∈[0,1] |W n (t) -n -1/2 S [nt] | = n -1/
P 0 sup (i-1)m -1 <t≤im -1 |S [nt] -S [n(i-1)m -1 ] | > ε √ n = 0 a.s. (4.19)
Let p = [n/m], and note that, for any non negative integer i,

[nim -1 ] -i < pi ≤ [nim -1 ]. It follows that, for any integer i in [1, m],
sup

(i-1)m -1 <t≤im -1 |S [nt] -S [n(i-1)m -1 ] | ≤ S(i-1)p,ip + 1 √ n [n(i-1)m -1 ] k=[n(i-1)m -1 ]-m |X k |+ 1 √ n [nim -1 ] k=[nim -1 ]-m |X k | .
Using (7.2) of Lemma 7.1, we infer that We first prove that the series η = E(X 2 0 |I) + 2 k>0 E(X 0 X k |I) converges almost surely and in L 1 . With this aim, it suffices to prove that k≥1 E(X 0 X k |I) 1 < ∞ .

lim n→∞ 1 √ n [n(i-1)m -1 ] k=[n(i-1)m -1 ]-m E 0 (|X k |) = 0 a.s.
(5.1)

From Claim 1(b) in Dedecker and Rio (2000

), E(X 0 X k |I) = E(E(X 0 X k |F -∞ )|I) almost surely, where F -∞ = k∈Z F k . Hence E(X 0 X k |I) 1 ≤ E(X 0 X k |F -∞ ) 1 ≤ X 0 E 0 (X k ) 1 ,
which proves (5.1) by using (2.1).

We turn now to the rest of the proof. there exists a T -invariant r.v. η that is F 0 -measurable and such that for any integer i ≥ 1 lim

n→∞ E 0 1 n E (i-1)n (S (i) n ) 2 -η = 0 a.s. and for any integer i ≥ 1 lim n→∞ E 0 1 n E (i-1)n (S (i) n + S (i+1) n ) 2 -2η = 0 a.s.
More precisely, we shall prove that C * 2 holds with η defined in (2.2). We shall only prove the first part of C * 2 , the proof of the second part being similar. For any positive integer N,

S (i) n 2 n = 1 n in j=(i-1)n+1 X 2 j + 2 n in-1 j=(i-1)n+1 (in-j)∧N l=1 X j X j+l + R i,N . (5.2)
Firstly, E 0 (Z j,N ) = E(Z 0,N |I) a.s.

E 0 (|E (i-1)n (R i,N )|) ≤ 1 n in j=(i-1)n+1 E 0 l>j+N |X j E j (X l )| . Let Z j,N = l>j+N |X j E j (X l )|
Hence, lim sup

n→∞ E 0 (|E (i-1)n (R i,N )|) ≤ E(Z 0,N |I) a.s.

and consequently lim

N →∞ lim sup n→∞ E 0 (|E (i-1)n (R i,N )|) = 0 a.s. (5.3) Next, let η N = E(X 2 0 |I) + 2 N k=1 E(X 0 X k |I) and η N,K = E(X 2 0 1 |X 0 | 2 ≤K |I) + 2 N k=1 E(X 0 X k 1 |X 0 X k |≤K |I) .
By the ergodic theorem for stationary sequences,

lim n→∞ η N,K - 1 n in j=(i-1)n+1 X 2 j 1 |X j | 2 ≤K - 2 n in-1 j=(i-1)n+1 (in-j)∧N l=1 X j X j+l 1 |X j X j+l |≤K = 0 a.s.
(5.4) and by the ergodic theorem in relation (7.1) of Lemma 7.1 applied with Z j = X 2 j 1 |X j | 2 >K and with

Z j = N l=1 |X j X j+l |1 |X j X j+l |>K , lim K→∞ lim sup n→∞ E 0 1 n in j=(i-1)n+1 X 2 j 1 |X j | 2 >K + 2 n in-1 j=(i-1)n+1 (in-j)∧N l=1 |X j X j+l |1 |X j X j+l |>K = 0 a.s.
(5.5) Using (5.4), (5.5) and the dominated convergence theorem, it follows that

lim n→∞ E 0 η N - 1 n in j=(i-1)n+1 X 2 j - 2 n in-1 j=(i-1)n+1
(in-j)∧N l=1 X j X j+l = 0 a.s.

(5.6)

The first part of condition C * 2 follows from (5.2), (5.3) and (5.6), and the fact that lim N →∞ η N = η almost surely.

Next, we prove that C 1 holds. With this aim, we first notice that it suffices to prove that for any integer i ≥ 2,

lim n→∞ E(|X 0 ||I)E 0 E (i-2)n S (i) n √ n = 0 a.s.
Indeed, on the invariant set where E(|X 0 ||I) = 0 almost surely, the random variables X i 's are equal to zero almost surely. Now, using the same arguments as in the proof of Lemma 7.1, we can prove that E(|X 0 ||I) = E(E(|X 0 ||I)|F 0 ) almost surely. Hence, for any integer i ≥ 2,

E(|X 0 ||I)E 0 E (i-2)n S (i) n = E 0 E (i-2)n E(|X 0 ||I)S (i) n a.s. Now 1 √ n E 0 E (i-2)n E(|X 0 ||I)S (i) n ≤ E 0 E (i-2)n 1 n (i-1)n k=(i-2)n+1 |X k | -E(|X 0 ||I) S (i) n √ n + 1 n 3/2 (i-1)n k=(i-2)n+1 E 0 E (i-2)n |X k | S (i) n . (5.7)
Using the fact that F 0 ⊆ F (i-2)n for any i ≥ 2, and applying Cauchy-Schwarz's inequality conditionally to F 0 , the first term on right hand in (5.7) is smaller than

E 1/2 0 1 n (i-1)n k=(i-2)n+1 |X k | -E(|X 0 ||I) 2 E 1/2 0 S (i) n √ n 2 .
(5.8)

By C * 2 , lim n→∞ E 0 S (i) n √ n 2 = η a.s.
(5.9)

Since X 0 belongs to L 2 , proceeding as in the proof of (5.6), we obtain that

lim n→∞ E 0 1 n (i-1)n k=(i-2)n+1 |X k | -E(|X 0 ||I) 2 = 0 a.s.
(5.10)

From (5.8), (5.9) and (5.10), we infer that the first term on right hand in (5.7) converges to 0 almost surely as n tends to infinity. Now, for any integer k belonging to ](i -2)n, (i -1)n],

1 √ n E 0 E |X k | S (i) n F (i-2)n ≤ 1 √ n E 0 E(|X k |S (i) n |F k ) ≤ 1 √ n E 0 ∞ i=k+1 |X k E k (X i )| . Let Z k = ∞ i=k+1 |X k E k (X i )
| and note that, by assumption, Z k = Z 0 •T k belongs to L 1 . It follows that the second term on the right-hand side of (5.7) is smaller than n -3/2 (i-1)n k=(i-2)n+1 E 0 (Z k ) , which converges almost surely to 0 as n tends to infinity, by the ergodic theorem in relation (7.1) of Lemma 7.1. Hence C 1 is proved.

We turn now to the proof of C 4 . With this aim, we shall prove the following reinforcement of it:

C * 4 lim k→∞ lim sup n→∞ max 1≤i≤k E 0 S2 (i-1)n,in n 1 ∧ S(i-1)n,in √ nk = 0 a.s.
To prove C * 4 , we shall use the following maximal inequality, which is a conditional version of the inequality given in Proposition 1(a) of [START_REF] Dedecker | On the functional central limit theorem for stationary processes[END_REF]. Proposition 5.2. For any k < l and λ ≥ 0 let Γ k,l (λ) = { Sk,l > λ}. The following inequality holds

E 0 (( Sk,l -λ) 2 + ) ≤ 8 l i=k+1 E 0 (X 2 i 1 Γ k,i (λ) ) + 16 l i=k+1 E 0 (|X i 1 Γ k,i (λ) E i (S l -S i )|) .
Let us continue the proof of C * 4 . Note first that

E 0 S2 (i-1)n,in n 1 ∧ S(i-1)n,in √ nk ≤ 2εE 0 S2 (i-1)n,in n + 4 n E 0 ( S(i-1)n,in -ε √ nk) 2 + .
From Proposition 5.2 with λ = 0, we obtain that E 0 (Z j,N ) = 0 a.s.

E 0 S2 (i-1)n,in n ≤ 8 n in k=(i-1)n+1 E 0 (X 2 k ) + 16 n in k=(i-1)n+1 E 0 (Z k ) ,
E 0 ( S(i-1)n,in -ε √ nk) 2 + ≤ 4 n in j=(i-1)n+1 E 0 (X 2 j 1 Γ (i-1)n,in (ε √ nk) ) (5.13)
(5.14)

Now, for any positive M and any 0 ≤ l ≤ N,

1 n in j=(i-1)n+1 E 0 (|X j X j+l |1 Γ (i-1)n,in (ε √ nk) ) ≤ M ε 2 k E 0 S2 (i-1)n,in n + 1 n in j=(i-1)n+1 E 0 (|X j X j+l |1 |X j X j+l |>M ) . (5.15)
According to (5.11), we have that

lim k→∞ lim sup n→∞ max 1≤i≤k M ε 2 k E 0 S2 (i-1)n,in n = 0 a.s. (5.16)
Next, by the ergodic theorem in relation (7.1) of Lemma 7.1, for any positive integer i,

lim n→∞ 1 n in j=(i-1)n+1 E 0 (|X j X j+l |1 |X j X j+l |>M ) = E(|X 0 X l |1 |X 0 X l |>M |I) a.s.
and consequently lim

M →∞ lim sup k→∞ lim sup n→∞ max 1≤i≤k 1 n in j=(i-1)n+1 E 0 (|X j X j+l |1 |X j X j+l |>M ) = 0 a.s.
(5.17)

Gathering (5.13), (5.14), (5.15), (5.16) and (5.17), we infer that (5.12) holds. This ends the proof of C * 4 . Then, on a set of probability one, for any continuous and bounded function ϕ from C([0, 1]) to R, (2.3) follows by applying Proposition 4.4. To prove that (2.3) also holds for ϕ in H * , it suffices to notice that since (2.1) implies C * 4 , it entails in particular that almost surely, the sequence (n -1 max 1≤k≤n S 2 k ) n≥1 is uniformly integrable for the conditional expectation with respect to F 0 . . Proof of Proposition 5.2. It is exactly the same as to get (3.12) in the paper by [START_REF] Dedecker | On the functional central limit theorem for stationary processes[END_REF], with the only difference that the expectation is replaced by the conditional expectation with respect to F 0 . .

Some remarks on martingale approximations

The aim of this subsection is to point out that the conditions C 1 , C 2 and C 3 are satisfied if there is an almost sure conditional martingale approximation in L 2 . This is another way to see that our conditions C 1 , C 2 and C 3 lead to sharp sufficient conditions for the quenched CLT.

From the proof of Theorem 5.1, we see that, if X 1 is a martingale difference, that is E(X 1 |F 0 ) = 0 a.s., then the conditions C 1 , C * 2 and C * 4 are satisfied. The following claim is then easily deduced.

Claim 5.3. Let X 0 and d 0 be two F 0 -measurable, centered and square integrable random variables with E(d 0 • T |F 0 ) = 0 a.s., and let

X i = X 0 • T i and d i = d 0 • T i . Let S n = X 1 + • • • + X n and M n = d 1 + • • • + d n . 1. If lim n→∞ 1 n E 0 ((S n -M n ) 2 ) = 0 almost surely,
then the conditions C 1 , C 2 and C 3 are satisfied with η = E(d 2 0 |I). In particular, if the condition of [START_REF] Maxwell | Central limit theorem for additive fonctionals of Markov chains[END_REF] is satisfied

If

lim n→∞ 1 n E 0 max 1≤k≤n (S k -M k ) 2 = 0 almost surely, ( 5
n>0 E 0 (S n ) 2 n 3/2 < ∞ , (5.19)
then it follows from Cuny and Merlevède (2012) that (5.18) holds, so that the conditions C 1 , C 2 and C 4 are satisfied. We already know from Peligrad and Utev (2006) that the Maxwell and Woodroofe condition is sharp in some sense for the FCLT, and therefore for the quenched FCLT also. This shows that the conditions C 1 , C 2 and C 4 are essentially sharp for the quenched FCLT. We mention that the Maxwell and Woodroofe condition and our condition (2.1) are of independent interests. For instance, when applied to strongly mixing sequences the condition (5.19) leads to sub-optimal results as pointed out in [START_REF] Merlevède | Recent advances in invariance principles for stationary sequences[END_REF]. Obviously, the same remark is true when we apply it to α-dependent sequences as defined in Section 3.

More precisely, this gives the condition:

k≥0 (k + 1) -1/2 α Y (k) 0 Q 2 (u)du 1/2
< ∞ instead of (3.2). Hence, when applied to non necessarily bounded functions of the Markov chain associated to the intermittent map given in Section 3.1, the criterion (5. [START_REF] Gordin | Abstracts of Communication, T.1:A-K, International Conference on Probability Theory[END_REF]) is satisfied as soon as f belongs to F * (H, ν γ ) and H is such that H(x) ≤ Cx -2(1-γ)/(1-2γ) (ln(x)) -b for x large enough and b > 2(1 -γ)/(1 -2γ). Recall that by condition (3.2), we only need b > (1 -γ)/(1 -2γ). In addition, Point (v) of the main theorem in [START_REF] Durieu | Comparison between criteria leading to the weak invariance principle[END_REF] shows that one can find a stationary sequence (X i ) i∈Z adapted to an increasing and stationary filtration (F i ) i∈Z in such a way that the condition (5.19) holds but X 0 E 0 (S n ) does not converge in L 1 and so the condition (2.1) fails. Analyzing the examples given in their paper, one can also prove that there are stationary sequences for which (2.1) holds but (5.19) does not. We can even say more: there are stationary sequences for which (2.1) holds but not (5.19), neither the Gordin criterion (2.4), nor the Hannan-Heyde condition are satisfied. Recall that the Hannan-Heyde condition is the following:

E(X 0 |F -∞ ) = 0 a.s. and n≥0 E 0 (X n ) -E -1 (X n ) 2 < ∞ , (5.20)
where

F -∞ = k∈Z F k .
In what follows (Ω, A, µ) is a probability space and T : Ω → Ω a bijective bimeasurable transformation preserving the measure µ. Then (Ω, A, µ, T ) is called a dynamical system. We refer to [START_REF] Sinaȋ | A weak isomorphism of transformations with invariant measure[END_REF] for a precise definition of the entropy of a dynamical system, and for the properties of dynamical systems with positive entropy. The proof of the next proposition is given in the Appendix.

Proposition 5.4. Let (Ω, A, µ, T ) be an ergodic dynamical system with positive entropy. Let F ⊂ A be a T -invariant σ-algebra, i.e. F ⊂ T -1 (F ) and let F i = T -i (F ). There exists a F 0 -measurable and centered function f in L 2 such that, setting X i = f • T i , the condition (2.1) is satisfied but the conditions (2.4), (5.19) and (5.20) fail.

To be complete, note that a stationary sequence can be constructed in such a way that (2.4) holds but the condition (2.1) fails (see Section 5.2 in Durieu and Volný ( 2008)). Moreover, a stationary sequence can be constructed in such a way that the condition (5.20) holds but the condition (2.1) fails (see Theorem 1 in Durieu ( 2009)).

Normal approximation for double indexed arrays and auxiliary results

There are many situations when we are dealing with double indexed sequences of random variables. For instance at each point in the two dimensional space we start a random walk. Our motivation for this section comes from the fact that in our blocking procedure we introduce a new parameter, the number of blocks, m, that is kept fixed at the beginning.

The next theorem treats the martingale approximation for double arrays of random variables. Theorem 6.1. Assume that (U n,m,i ) i≥1 is an array of random variables in L 2 adapted to an array (G n,m,i ) i≥1 of nested sigma fields. Let E n,m,i denote the conditional expectation with respect to G n,m,i . Suppose that

lim m→∞ lim sup n→∞ E m i=1 E n,m,i-1 (U n,m,i ) = 0 , (6.1)
there exists σ 2 ≥ 0 such that

lim m→∞ lim sup n→∞ E m i=1 Var(U n,m,i |G n,m,i-1 ) -σ 2 = 0 , (6.2)
and for each ε > 0

lim m→∞ lim sup n→∞ m i=1 E(U 2 n,m,i 1 |U n,m,i |>ε ) = 0 . (6.3)
Then for any continuous and bounded function f ,

lim m→∞ lim sup n→∞ E f m i=1 U n,m,i -E(f (σN)) = 0 , (6.4)
where N is a standard Gaussian variable.

Proof of Theorem 6.1.

For any i ≥ 1, let d n,m,i = U n,m,i -E n,m,i-1 (U n,m,i )
. By condition (6.1), the theorem will follow if we can prove that (6.4) holds with m i=1 d n,m,i replacing m i=1 U n,m,i . If σ 2 = 0 the theorem is trivial. So we can assume without loss of generality that σ 2 = 1. In the rest of the proof, in order to ease the notations, we shall drop the first two indexes (n, m), keeping them only when it is necessary to avoid confusion. Let ε and M be positive reals fixed for the moment. For any i ≥ 1, let

V i = i ℓ=1 E ℓ-1 (d 2 ℓ ) and Y i = d i 1 |d i |≤ε 1 V i ≤M .
Notice first that

P m i=1 d i = m i=1 Y i ≤ P max 1≤i≤m d i > ε + P(V m > M) ≤ 1 ε 2 m i=1 E(d 2 i 1 |d i |>ε ) + 1 M 1 + E m i=1 Var(U i |G i-1 ) -1 .
Hence using Lemma 6.3, we get that

P m i=1 d i = m i=1 Y i ≤ 12 ε 2 m i=1 E(U 2 i 1 |U i |>ε/4 ) + 1 M 1 + E m i=1 Var(U i |G i-1 ) -1 .
Therefore using (6.3) and (6.2), it follows that for all ε > 0, lim sup

m→∞ lim sup n→∞ P m i=1 d n,m,i = m i=1 Y n,m,i ≤ 1 M . (6.5)
We notice now that since

E i-1 (d i ) = 0 a.s. and V i is G i-1 -measurable, m i=1 E i-1 (Y i ) = m i=1 1 V i ≤M E i-1 (d i 1 |d i |>ε ) .
Therefore by Lemma 6.3,

E m i=1 E i-1 (Y i ) ≤ 12 ε m i=1 E(U 2 i 1 |U i |>ε/4 ) ,
implying, by using (6.3), that for all positive reals ε and M,

lim m→∞ lim sup n→∞ E m i=1 E i-1 (Y i ) = 0 . (6.6)
Considering (6.5) and (6.6), the theorem will follow if we can show that for any continuous bounded function f , lim

M →∞ lim sup ε→0 lim sup m→∞ lim sup n→∞ E f m i=1 d * n,m,i -E(f (N)) = 0 , (6.7)
where

d * n,m,i = Y n,m,i -E n,m,i-1 (Y n,m,i ) . Let s 2 n,m = m i=1 E(d * 2 n,m,i ) ,
and notice that for any δ > 0, E |d * n,m,i | 2+2δ ) < ∞, i = 1, 2, . . . Hence, by the first theorem stated in [START_REF] Heyde | On the departure from normality of a certain class of martingales[END_REF], it follows that for any δ ∈]0, 1],

sup x∈R P m i=1 d * n,m,i ≤ x -P(s n N ≤ x) ≤ K δ s -2-2δ n,m E |d * n,m,i | 2+2δ ) + E m i=1 E n,m,i-1 (d * n,m,i ) 2 -s 2 n,m 1+δ 1/(3+2δ) , (6.8)
where K δ is a positive constant depending only on δ. Assume now that we can prove that there exists a δ in ]0, 1] such that for any positive reals ε and M,

lim sup m→∞ lim sup n→∞ m i=1 E |d * n,m,i | 2+2δ ≤ u(ε) , (6.9)
and lim sup

m→∞ lim sup n→∞ E m i=1 E n,m,i-1 (d * n,m,i ) 2 -1 1+δ ≤ v(M) , (6.10)
where u(•) and v(•) are positive functions defined on R + such that v(•) does not depend on ε, lim x→0 u(x) = 0 and lim x→∞ v(x) = 0. Then starting from (6.8) and noticing that (6.10) also implies that for any positive reals ε and M, and

lim sup m→∞ lim sup n→∞ |s 2 n,m -1| 1+δ ≤ v(M) , (6.11)
we infer that (6.7) will hold. Indeed, by standard arguments, we will get (6.7) for every continuous function f with compact support and then (6.7) for every continuous and bounded function f by using the fact that every probability measure is tight. Hence, to end the proof of the theorem, it remains to prove that (6.9) and (6.10) hold. With this aim, we first notice that

m i=1 E |d * i | 2+2δ ≤ 4(2ε) 2δ m i=1 E(d 2 i ) ≤ 4(2ε) 2δ 1 + E m i=1 Var(U i |G i-1 ) -1 .
Hence, using condition (6.2), (6.9) follows with u(ε) = 4(2ε) 2δ . It remains to prove (6.10). With this aim, using the convexity inequality: (a + b) p ≤ 2 p-1 (a b + b p ) (p ≥ 1, a > 0 and b > 0), we first write that

E m i=1 E i-1 ((d * i ) 2 ) -1 1+δ ≤ 2 δ E m i=1 E i-1 (Y 2 i ) -1 1+δ + 2 δ E m i=1 E i-1 (Y i ) 2 1+δ
. (6.12)

Now, since V n,m,i is G n,m,i-1 -measurable and E n,m,i-1 (d n,m,i ) = 0 a.s., we infer that

E m i=1 E i-1 (Y i ) 2 1+δ ≤ E m i=1 E i-1 (d i 1 |d i |>ε ) 2 m k=1 1 V k ≤M E k-1 (d 2 k ) δ ≤ M δ m i=1 E(d 2 i 1 |d i |>ε ) .
Hence by Lemma 6.3,

E m i=1 E i-1 (Y i ) 2 1+δ ≤ 12M δ m i=1 E(U 2 i 1 |U i |>ε/4 ) . (6.13)
On the other hand, using again the fact that V n,m,i is G n,m,i-1 -measurable and also that

V n,m,i ≤ V n,m,i+1 , we derive that E m i=1 E i-1 (Y 2 i ) -1 1+δ ≤ E 1 + m i=1 1 V i ≤M E i-1 (d 2 i ) δ 1 - m k=1 1 V k ≤M E k-1 (d 2 k 1 |d k |≤ε ) ≤ (M + 1) δ m i=1 E(d 2 i 1 |d i |>ε ) + (M + 1) δ E 1 - m k=1 1 V k ≤M E k-1 (d 2 k ) ≤ (M + 1) δ m i=1 E(d 2 i 1 |d i |>ε ) + (M + 1) δ E m k=1 E k-1 (d 2 k ) -1 +(M + 1) δ E 1 Vm>M m k=1 E k-1 (d 2 k ) .
Therefore,

E m i=1 E i-1 (Y 2 i ) -1 1+δ ≤ (M + 1) δ m i=1 E(d 2 i 1 |d i |>ε ) + 2(M + 1) δ E m k=1 E k-1 (d 2 k ) -1 + (M + 1) δ M 1 + E m k=1 E k-1 (d 2 k ) -1 ,
which together with Lemma 6.3 and the fact that

E k-1 (d 2 k ) = Var(U k |G k-1 ) imply that E m i=1 E i-1 (Y 2 i )-1 1+δ ≤ 12(M +1) δ m i=1 E(U 2 i 1 |U i |>ε/4 )+2(M +1) δ E m k=1 Var(U k |G k-1 )-1 + (M + 1) δ M 1 + E m k=1
Var(U k |G k-1 ) -1 . (6.14)

Starting from (6.12) and considering the bounds (6.13) and (6.14) together with the conditions (6.2) and (6.3), we then infer that (6.10) holds for any δ ∈]0, 1[ with v(M) = M -1 (M + 1) δ . This ends the proof of (6.7) and then of the theorem. With this notation and since E n,m,i-1 (d n,m,i ) = 0 almost surely,

m i=1 d n,m,i = m i=1 d ′ n,m,i -E n,m,i-1 (d ′ n,m,i ) + m i=1 d ′′ n,m,i -E n,m,i-1 (d ′′ n,m,i ) . Since E d ′′ n,m,i -E n,m,i-1 (d ′′ n,m,i ) ≤ 2E(|d n,m,i |1 |d n,m,i |>ε ) ,
by using the first part of (6.15), the lemma will follow if we can prove that lim ε→0 lim sup

m→∞ lim sup n→∞ E m i=1 d ′ n,m,i -E n,m,i-1 (d ′ n,m,i ) = 0 . (6.16)
With this aim, it suffices to notice that

E m i=1 d ′ n,m,i -E n,m,i-1 (d ′ n,m,i ) 2 ≤ m i=1 E(d ′ n,m,i ) 2 ≤ ε m i=1 E|d n,m,i | ,
showing that (6.16) holds under (6.15). Lemma 6.3. Let X be a real random variable and F a sigma-field. For any p ≥ 1 and any ε > 0,

E |X| p 1 |E(X|F )|>2ε ≤ 2 E |X| p 1 |X|>ε , (6.17)
and setting Y = X -E(X|F ),

E |X| p 1 |Y |>3ε ≤ 2 E |X| p 1 |X|>ε and E(|Y | p 1 |Y |>4ε ) ≤ 3 × 2 p E |X| p 1 |X|>ε ) . (6.18)
Proof of Lemma 6.3. We first write that

|X| p 1 |E(X|F )|>2ε ≤ |X| p 1 |X|>ε + ε p 1 |E(X|F )|>2ε . (6.19) Notice now that {|E(X|F )| > 2ε} ⊆ {|E(X1 |X|>ε )|F )| > ε}, implying that ε p 1 |E(X|F )|>2ε ≤ |E(X1 |X|>ε |F )| p ≤ E |X| p 1 |X|>ε |F , (6.20)
Starting from (6.19), using (6.20) and taking the expectation, (6.17) follows. To prove the first part of (6.18), we start by writing that

|X| p 1 |Y |>3ε ≤ |X| p 1 |X|>ε + ε p 1 |E(X|F )|>2ε ,
and we use (6.20). To prove the second part of (6.18), we first notice that for any positive reals a, b and ε, (a + b) p 1 a+b>4ε ≤ 2 p a p 1 a>2ε + 2 p b p 1 b>2ε . Therefore

E(|Y | p 1 |Y |>4ε ) ≤ 2 p E(|X| p 1 |X|>2ε ) + 2 p E(|X| p 1 |E(X|F )|>2ε ) .
The second part of (6.18) then follows by using (6.17).

Ergodic theorem

We gather below the ergodic theorems used in this paper. We keep the notations of Section 2 and we define

F ∞ = k∈Z F k . Lemma 7.1. Let Z be a F ∞ -measurable real-valued random variable in L 1 . Define Z k = Z • T k for any k in Z. Then 1 n n i=1 E 0 (Z i ) → E(Z|I) almost surely and in L 1 , (7.1
)

and 1 n E 0 max 1≤i≤n |Z i | → 0 almost surely and in L 1 . (7.2)
Proof. By definition of the operator K (see the beginning of Section 2),

1 n n i=1 E 0 (Z i ) = 1 n n i=1 K i (Z) .
Applying the Dunford-Schwartz ergodic theorem (see for instance [START_REF] Krengel | Ergodic theorems[END_REF]) we obtain that (K(Z) + • • • + K n (Z))/n converges almost surely and in L 1 to some g ∈ L 1 . We prove now that g = E(Z|I). Let N ∈ N. Define Z 0,N = E(Z|F N ) and Z k,N = Z 0,N • T k for any k in Z. From the stationarity of the sequence (Z k,N ) k∈Z and the invariance of E(Z 0,N |I), we have

E(Z 0,N |I) - 1 n n k=1 Z k,N 1 = E(Z 0,N |I) - 1 n -N k=1-(n+N ) Z k,N 1
.

Both this equality and the L 1 -ergodic theorem imply that E(Z 0,N |I) is the limit in L 1 of a sequence of F 0 -measurable random variables. Hence E(E(Z 0,N |I)|F 0 ) = E(Z 0,N |I) almost surely. Therefore, noticing that for any i ∈ N, E 0 (Z i ) = E 0 (Z i,N ) and using, once again, the L 1 -ergodic theorem, we derive that lim sup

n→∞ 1 n n i=1 E 0 (Z i ) -E(Z 0,N |I) 1 ≤ lim sup n→∞ 1 n n i=1 Z i,N -E(Z 0,N |I) 1 = 0 .
Hence the proof will be complete if we show that lim N →∞ E(Z 0,N |I)-E(Z|I) 1 = 0. Notice that

E(Z 0,N |I) -E(Z|I) 1 ≤ E(Z|F N ) -Z 1 .
Therefore since (E(Z|F N )) N ≥1 is an uniformly integrable martingale, and Z is F ∞ -measurable, the desired convergence follows by the martingale convergence theorem.

We turn now to the proof of (7.2). With this aim, we notice that for any N > 0,

1 n E 0 max 1≤i≤n |Z i | ≤ N n + 1 n n i=1 E 0 (|Z i |1 |Z i |>N ) .
By using (7.

Appendix

This section is devoted to the proof of Proposition 5.4. We shall see that it follows from a slight modification of the example given in Section 5.4 in [START_REF] Durieu | Comparison between criteria leading to the weak invariance principle[END_REF].

We consider the ergodic dynamical system (Ω, A, µ, T ) with positive entropy, the sequence (e i ) i∈Z of independent identically distributed (i.i.d.) Rademacher random variables with parameter 1/2, and the σ-algebra F 0 as described at the beginning of Section 4.1 in [START_REF] Durieu | Comparison between criteria leading to the weak invariance principle[END_REF]. Now, for any positive integer k, we define

N k = 4 k , ρ k = 1 4 k , θ k = 1 k2 k , ε k = 1 k 2 4 3k , (8.1)
and we consider mutually disjoint sets (A k ) k∈Z by using their Lemma 2 with 2N k instead of N k , and the sequences (ρ k ) and (ε k ) defined above. In addition to being disjoint, the sets

(A k ) k∈N * are such that (i) 2 3 ρ k ≤ µ(A k ) ≤ ρ k for all k ∈ N * ; (ii) for all k ∈ N * and all i, j ∈ {0, . . . , 2N k }, µ(T -i A k ∆T -j A k ) ≤ ε k ,
The function f is then defined as

f = k≥1 f k 1 A k with f k = θ k 2N k j=N k +1 e -j . (8.2)
The function f defined in (8.2) is centered, F 0 -measurable and, since k≥1 θ 2 k N k ρ k < ∞, it belongs to L 2 (see Proposition 7 in Durieu and Volný (2008)).

Let now X i = f • T i for any i ∈ Z. This sequence is adapted to the stationary and nondecreasing sequence of σ-algebras (F i ) i∈Z where F i = T -i (F 0 ). Let us first prove that the sequence (X i ) i∈Z satisfies the condition (2.1). With this aim, we first emphasize some additional important properties of (e i ) i∈Z and of (A k ) k∈Z . First, the sequence (e i ) i∈Z is adapted to (F i ) i∈Z and E(e i |F 0 ) = e i 1 i≤0 almost surely. Second, for all k and i, 1 A k • T i is F 0 -measurable. Finally, the e i 's and the 1 A k 's are independent for all i and k. As in relation (4) in [START_REF] Durieu | Comparison between criteria leading to the weak invariance principle[END_REF], we then write that for any i ∈ N,

E(X i |F 0 ) = k≥1 θ k 2N k j=N k +1 e i-j 1 T -i (A k ) 1 i≤j = k≥1 θ k 2N k j=N k +1 e i-j 1 A k 1 i≤j + k≥1 θ k 2N k j=N k +1 e i-j (1 T -i (A k )\A k -1 A k \T -i (A k ) )1 i≤j . (8.3)
Using Item (ii) above, and the fact that the e j 's are bounded by one, we obtain Since the e i 's are i.i.d., centered and with variance one, we have The second term in the right-hand side of (8.6) can be handled similarly. So overall, we infer that

i≥0 k≥1 θ k 2N k j=N k +1 e i-j 1 i≤j (1 T -i (A k )\A k -1 A k \T -i (A k ) ) 2 ≤ k≥1 N k i=0 θ k N k µ(T -i (A k )∆A k ) 1/2 + k≥1 θ k 2N k i=N k +1
i≥0 f k≥1 θ k 2N k j=N k +1 e i-j 1 A k 1 i≤j 1 ≤ 2 k≥1 θ 2 k N 2 k ρ k ,
which is finite according to (8.1). This ends the proof of (8.5) and then of the fact that the sequence (X i ) i∈Z satisfies the condition (2.1).

Let us prove now that the condition (2.4) fails for the sequence (X i ) i∈Z defined above. With this aim, we shall prove that sup K∈N E 0 (S N K ) 1 = ∞ .

(8.7)

Starting from (8.3) and using (8.4), it suffices to prove that sup

K∈N N K i=1 k≥1 θ k 2N k j=N k +1 e i-j 1 i≤j 1 A k 1 = ∞ . (8.8)
Let K ≥ 3. By disjointness of the A k 's and by independence between the e i 's and the 1 A k 's, it follows that

N K i=1 k≥1 θ k 2N k j=N k +1 e i-j 1 i≤j 1 A k 1 = k≥1 θ k µ(A k )E 2N k j=N k +1 N K i=1 e i-j 1 i≤j . Therefore N K i=1 k≥1 θ k 2N k j=N k +1 e i-j 1 i≤j 1 A k 1 ≥ K-1 k=1 θ k µ(A k )E 2N k -1 ℓ=(N k +1-N K )∨0 e -ℓ N K i=1 1 i≥N k +1-ℓ 1 i≤2N k -ℓ .
Notice now that for any k ∈ {1, . . . , K -1}, N k + 1 -N K ≤ N K-1 + 1 -N K ≤ 0 and for any ℓ ≥ 0, 2N k -ℓ ≤ N K (since 2N k -N K ≤ 2N K-1 -N K ≤ 0). Hence

N K i=1 k≥1 θ k 2N k j=N k +1 e i-j 1 i≤j 1 A k 1 ≥ K-1 k=1 θ k µ(A k )E N k N k ℓ=0 e -ℓ + 2N k -1 ℓ=N k +1 (2N k -ℓ)e -ℓ .
Next, by using the Marcinkiewicz-Zygmund's inequality together with Item (i) above, we get that there exists a positive constant A such that

N K i=1 k≥1 θ k 2N k j=N k +1 e i-j 1 i≤j 1 A k 1 ≥ A K-1 k=1 θ k ρ k E N 2 k N k ℓ=0 e 2 -ℓ + 2N k -1 ℓ=N k +1 (2N k -ℓ) 2 e 2 -ℓ 1/2 ≥ A K-1 k=1 θ k ρ k N 3/2 k ≥ A ln(K -1) ,
which proves (8.8) and therefore (8.7).

Let us prove now that the condition (5.19) fails for the sequence (X i ) i∈Z defined above. Following the computations page 339 in Durieu and Volný (2006), it suffices to prove that

n≥1 1 n 3/2 k≥1 1 2N k ≤n θ 2 k N 3 k ρ k 1/2 = ∞ . (8.9)
Since N k = 4 k , using (8.1), we get that where C is a positive constant. This shows (8.9) and then that (5.19) fails.

Let us prove now that the Hannan-Heyde condition (5.20) fails for the sequence (X i ) i∈Z defined above. With this aim, we first notice that

E(X i |F 0 ) -E(X i |F -1 ) = e 0 k≥1 θ k 2N k j=N k +1 1 T -i (A k ) 1 i=j .
Proceeding as in (8.3) and since k≥1 θ k N k √ ε k < ∞, it suffices to prove that

i≥1 k≥1 θ k 2N k j=N k +1 1 i=j 1 A k 2 = ∞ . (8.10)
But, by disjointness of the A k 's,

i≥1 k≥1 θ k 2N k j=N k +1 1 i=j 1 A k 2 = i≥1 k≥1 θ 2 k 2N k j=N k +1 1 i=j 2 µ(A k ) 1/2 . Therefore, since N k = 2 2k , i≥1 k≥1 θ k 2N k j=N k +1 1 i=j 1 A k 2 = ℓ≥0 2 2(ℓ+1) i=2 2ℓ +1 k≥1 θ 2 k 2 2k+1 j=2 2k +1 1 i=j 2 µ(A k ) 1/2 ≥ ℓ≥0 2 2ℓ+1 i=2 2ℓ +1 k≥1 θ 2 k 2 2k+1 j=2 2k +1 1 i=j 2 µ(A k ) 1/2 = ℓ≥0 2 2ℓ+1 i=2 ℓ +1 θ 2 ℓ µ(A ℓ ) 1/2 ≥ √ 2 √ 3 ℓ≥0 2 2ℓ θ ℓ √ ρ ℓ ,
which does not converge according to (8.1). This ends the proof of (8.10) and then of the fact that the sequence (X i ) i∈Z does not satisfy the Hannan-Heyde condition (5.20).

  ) then the conclusion of Theorem 2.1 holds. Hence the weak invariance principle of Doukhan et al. (1994) is also quenched. We refer to Theorem 2 in Doukhan et al. (1994) and to Bradley (1997) for a discussion on the optimality of the condition (3.3).

Proposition 4 . 1 .

 41 Assume that C 1 , C 2 and C 3 hold. Then, on a set of probability one, for any continuous and bounded function f ,

Before proving Proposition 4 . 2 ,

 42 let us show how it leads to Proposition 4.1. Let m be a fixed positive integer less than n. Set p = [n/m]. We apply Proposition 4.2 to the sequence Y n,m,i = S (i) p / √ mp and the filtration G n,m,i = F ip . We also replace the expectation E by the conditional expectation E 0 (recall that all the conditional expectations of functions of T with respect to F 0 are obtained through the regular conditional probability P T |F 0 ), and σ 2 by the non negative F 0 -measurable random variable η. With these notations, the conditions C 1 , C 2 and C 3 imply that (4.1), (4.2), (4.3) and (4.4) hold almost surely. It follows from Proposition 4.2 that, on a set of probability one, for any continuous and bounded function f , lim m→∞ lim sup n→∞

  d,k S (k) p , where λ m,d,k = d ℓ=1 a ℓ 1 mb ℓ-1 +1≤k≤mb ℓ . Hence to prove (4.16), it suffices to apply Proposition 4.2 to the random variables Y n,m,i = (mpc d ) -1/2 λ m,d,i S (i) p and the filtration G n,m,i = F ip , by replacing the expectation E by E 0 . The conditions (4.1) and (4.4) are verified by using respectively C 1 and C 3 . To verify (4.2) and (4.3) with σ

Proposition 4 . 4 .

 44 Assume that C 1 , C 2 and C 4 hold. Then, on a set of probability one, for any continuous and bounded function f from C([0, 1]) to R,

P

  S(i-1)p,ip > ε √ n F 0 = 0 a.s., which holds under C 4 . 5 Proof of Theorem 2.1 and additional comments 5.1 Proof of Theorem 2.1

  and note that, by assumption, Z j,N = Z 0,N • T j belongs to L 1 . Applying the ergodic theorem in relation (7.1) of Lemma 7.1 we obtain that lim

E 0 (E 0

 00 |X j X j+l |1 Γ (i-1)n,in (ε (Z j,N ) , where Z j,N = l>j+N |X j E j (X l )|. Since by (2.1), Z j,N = Z 0,N • T j belongs to L 1 , The ergodic theorem in relation (7.1) of Lemma 7.1 gives: for any positive integer i, lim n→∞ 1 n in j=(i-1)n+1 E 0 (Z j,N ) = E(Z 0,N |I) a.s.

. 18 )

 18 then the conditions C 1 , C 2 and C 4 are satisfied with η = E(d 2 0 |I).

Lemma 6 . 2 .

 62 Assume that (d n,m,i ) i≥1 is an array of random variables in L 2 adapted to an array (G n,m,i ) i≥1 of nested sigma fields, and such that for anyi ≥ 1, E n,m,i-1 (d n,m,i ) = 0 almost surely. Suppose that lim n,m,i |1 |d n,m,i |>ε ) = 0 and m i=1 E|d n,m,i | < K (6.15)for some positive constant K. Then lim m→∞ lim sup n→∞ E m i=1 d n,m,i = 0 . Proof of Lemma 6.2. Let ε > 0, and let for any i ≥ 1, d ′ n,m,i = d n,m,i 1 |d n,m,i |≤ε and d ′′ n,m,i = d n,m,i 1 |d n,m,i |>ε .

2 ≤ 2 k≥1θe -ℓ 1 A k 1 + i≥0 k≥1 1 N k +1≤i≤2N k θ 2 k

 22112 -i (A k )∆A k ) 1/k N k (N k + 1) √ ε k . (8.4) Since, by (8.1), k≥1 θ k N 2 k √ ε k < ∞, in order to prove that (2.1) holds, it is enough to show that 2N k j=i e i-j 2N k ℓ=N k +1 e -ℓ 1 A k 1 .(8.6)Now, by independence between the e i 's and the 1 A k 's,

2 ≤

 2 N k .

k≥1 1 2N k ≤n θ 2 k N 3 k

 13 ρ k = [(ln n-ln 2)/(2 ln 2)]

  a Markov chain with transition Kernel L γ and invariant measure ν γ .

	Definition 3.6. A function H from R + to [0, 1] is a tail function if it is non-increasing,
	right continuous, converges to zero at infinity, and x → xH(x) is integrable. If µ is a probability measure on R and H is a tail function, let Mon * (H, µ) denote the set of functions
	f : R → R which are monotonic on some open interval and null elsewhere and such that µ(|f | > t) ≤ H(t). Let F * (H, µ) be the closure in L 1 (µ) of the set of functions which can be written as L ℓ=1 a ℓ f ℓ , where L ℓ=1 |a ℓ | ≤ 1 and f ℓ ∈ Mon * (H, µ).
	Corollary 3.7. Let γ ∈ (0, 1/2) and (Y i ) i≥1 be a stationary Markov chain with transition kernel L γ and invariant measure ν γ . Let H be a tail function such that
	∞
	0

  >= d i=1 a i x i .

Proposition 4.3. Assume that C 1 , C 2 and C 3 hold. Then, on a set of probability one, for any continuous and bounded function h, for any a ∈ Q d and any t 1 , t 2 , . . . , t d rational numbers such that

  2 max 1≤i≤n |X i |, by using (7.2) of Lemma 7.1, it

	follows that (4.18) is equivalent to
			m
	lim m→∞	lim sup n→∞	i=1

  Proposition 5.1. If (2.1) holds, then C 1 ,C 2 and C 4 hold, with η defined in (2.2). In addition the conclusion of Proposition 4.4 also holds for f in H * . Proof of Proposition 5.1. We first prove that the following reinforced version of C 2 holds:

	C * 2

  1), n -1 n i=1 E 0 (|Z i |1 |Z i |>N ) converges to E(|Z|1 |Z|>N |I) almost surely and in L 1 , as n tends to infinity. Therefore (|Z i |1 |Z i |>N ) = 0 almost surely and in L 1 , which ends the proof of (7.2).

	lim N →∞	lim sup n→∞	1 n	n i=1	E 0

Acknowledgements. The authors would like to thank the referee for carefully reading the manuscript and for numerous suggestions that improved the presentation of this paper. The authors are also indebted to Christophe Cuny for helpful discussions.

* Supported in part by a Charles Phelps Taft Memorial Fund grant, the NSA grant H98230-11-1-0135, and the NSF grant DMS-1208237.


Raphaël Lachièze-Rey

Realisability conditions for second order marginals of biphased media

This paper concerns the second order marginals of biphased random media. We give discriminating necessary conditions for a bivariate function to be such a valid marginal, and illustrate our study with two practical applications: (1) the spherical variograms are valid indicator variograms if and only if they are multiplied by a sufficiently small constant, which upper bound is estimated, and (2) not every covariance/indicator variogram can be obtained with a Gaussian level set. The theoretical results backing this study are contained in a companion paper.

Introduction

A random geometric structure is sometimes modelled as a bi-phased medium in the euclidean space. Its low order characteristics, such as fraction volume, variogram, or covariance, although far from containing a total description of the model, provide interesting features on the microscopic regularity of the set boundary, as well as on its long range dependency. Characterising the class of second order marginals is an old problem going back to the 50's, with applications in telecommunications, materials sciences, geostatistics, and marginal problems in general are present under different occurrences in fields as various as quantum mechanics, computer science, game theory; see the surveys [START_REF] Torquato | Random Heterogeneous Materials[END_REF] or [START_REF] Fritz | Entropic inequalities and marginal problems[END_REF] . Experts from these fields have contributed valuable necessary or sufficient conditions for a function to be admissible, still a deep understanding of the problem is unavailable. Our purpose is to provide improvements to the existing methods for checking the admissibility of a second-order characteristic, and illustrate it with practical examples. Related theoretical results are proved in [START_REF] Lachièze-Rey | The convex class of realisable unit covariances[END_REF].

The two point covering function of a random set X contained in an ambient space E is defined as p X

x,y = P(x, y ∈ X), x, y ∈ E. If E = R d and X is stationary, meaning its law is invariant under the action of translations, the covariance can be factorized p X

x,y =: pX x-y = P(0, x -y ∈ X), x, y ∈ E.

1

The precise notion of a random set, evasive in most of the applied literature, is of no importance in this paper because we are only interested in combinatorial properties of the marginals, and leave aside regularity issues. Thus we call random set here a Borel random element of the subsets of E, endowed with the pointwise convergence topology. Most of the study is conducted in a discrete, or even finite, ambient set E. The indicator function of a random set is a {0, 1}-valued random field, sometimes referred to as a binary field. Any other sufficiently rich framework (e.g. random closed sets, see [START_REF] Molchanov | Theory of random sets[END_REF], or random measurable sets, see for instance [START_REF] Galerne | Computation of the perimeter of measurable sets via their covariogram. Applications to random sets[END_REF]), would do.

The central question here is the inverse realisability problem, given a bivariate function {p x,y } in the class F E of symmetric functions on E, whether it can be realised by some random set X ( i.e. p = p X ), or the equivalent problem if one is interested in another second order marginal, such as the covariance, the geometric variogram, the indicator variogram, or the unit covariance (see below); it is for instance necessary that p is semi-definite positive, but insufficient. Since p X

x,y = E1 {x∈X} 1 {y∈X} , the problems of characterising covariances and two point covering functions are closely related, therefore both these problems are referred to as the covariance realisability problem. The problem is also called the S 2 problem in materials science. This study can serve many purposes, especially in modelisation; one needs to know admissibility conditions to propose and use new models of covariances. In reconstruction and estimation, one should test whether the estimated/reconstructed covariance indeed corresponds to a random structure (see [START_REF] Jiao | Modeling heterogeneous materials via twopoint correlation functions: Basic principles[END_REF]).

The realisability problem seems to have deep combinatorial roots, which substance remains elusive after several decades. In a recent paper [START_REF] Lachièze-Rey | Regularity conditions in the realisability problem in applications to point processes and random closed sets[END_REF] and the forthcoming paper [START_REF] Galerne | Realisability of random sets of finite perimeter via grid approximation[END_REF], it is made clear that this problem can be uncoupled in two independent problems, referred to as the positivity problem, and the regularity problem. The regularity problem, treated for random sets in [START_REF] Galerne | Realisability of random sets of finite perimeter via grid approximation[END_REF], is not addressed here. The positivity problem, central in this paper, is about the compatibility of a candidate p with the algebraic properties of a set covariance, and is of combinatorial nature.

This problem has been posed by McMillan [START_REF] Mcmillan | History of a problem[END_REF] in the field of telecommunications. It is more or less implicit in many articles, and has been to the author's knowledge first addressed directly by Shepp [START_REF] Shepp | On positive definite functions associated with certain stochastic processes[END_REF], and more recently by Quintanilla [START_REF] Quintanilla | Necessary and sufficient conditions for the two-point probability function of two-phase random media[END_REF]. A series of works by Torquato and his coauthors (see [START_REF] Jiao | Modeling heterogeneous materials via twopoint correlation functions: Basic principles[END_REF] and [20, Sec. 2.2] and references therein), in the field of materials science, gather known necessary conditions and illustrate them in many 2D and 3D theoretical models. This question was developed alongside in the field of geostatistics; Matheron [START_REF] Matheron | Une conjecture sur la covariance d'un ensemble aléatoire[END_REF] has found via arithmetic considerations a wide class of necessary conditions, that he has proven to be sufficient if E has cardinality less or equal to 5, and he has conjectured these conditions to be sufficient for any finite E. This conjecture is disproved in the companion paper [START_REF] Lachièze-Rey | The convex class of realisable unit covariances[END_REF]. Other authors do not attack frontally this question, but address the realisability problem within some particular classes of models, e.g. Gaussian, mosaic, or boolean model (see [START_REF] Chilès | Modelling spatial uncertainty[END_REF][START_REF] Emery | On the existence of mosaic and indicator random fields with spherical, circular, and triangular variograms[END_REF][START_REF] Lantuéjoul | Geostatistical simulation: models and algorithms[END_REF][START_REF] Masry | On covariance functions of unit processes[END_REF]).

Depending on the community, the problem has different formulations, in function of the second order marginal of interest. Section 2 recalls the relations between those marginals. Section 3 focuses on the mathematical problem of testing a candidate second order marginal, giving an algorithm able to efficiently discard non-valid marginals. The method is illustrated in Section 4 with the Gaussian model and the spherical variograms.

Formulations of the problem

Let E be a set and endow the class of its subsets with the σ-algebra generated by the mappings A ⊂ E → 1 {x∈A} , x ∈ E. An important part of the paper focuses on finite E = [N] = {1, . . . , N} for some N ≥ 1. Denote by X : x → 1 {x∈X} the random indicator function of X. The two point covering function p X of X is also defined as p X

x,y = E Xx Xy .

Function p X lives in the space F E of symmetric functions on E. Call C E = {p X ; X ⊆ E random set} ⊂ F E the class of realisable functions.

Remark 2.1. It is clear, as p X is the two point covering function of a random process, that C E ⊂ P E ⊂ F E , where P E is the convex cone of semi-definite positive functions, meaning q i,j=1

p X x i ,x j h i h j ≥ 0 for every q ≥ 1, q-tuple of points (x 1 , . . . , x q ) of E, and vector h ∈ R q . Thus an example of a necessary condition for a function p x,y to be realisable is its semi-definite positiveness.

Second order marginals

Other second order characteristics of random sets can be found in the literature. With the same notation define for x, y

∈ E γ X x,y = 1 2 E( Xx -Xy ) 2 = 1 2 P( Xx = Xy ) the indicator variogram of X, ρ X x,y = 1 -4γ X x,y
the unit covariance of X.

(2.1)

The function ρ X takes on a special combinatorial interest because, if one defines the random field

Y X x = 1 if x ∈ X, -1 if x / ∈ X, = 2 Xx -1, ρ X x,y = EY X x Y X y . Defining F c E = {f ∈ F : f (x, x) = c; x ∈ E} for c = 0 or 1, the class V E of realisable variograms is contained in F 0 E , the class U E of realisable unit covari- ances is contained in F 1
E . An alternative definition of U E is as the class of the covariances of random functions taking values in {-1, 1}; it implies that U E , like C E , is contained in P E .

The two problems are fundamentally the same, and some geostatisticians seek to determine which functions {γ x } are valid models of indicator variograms. The "unit field formulation" is preferred in [START_REF] Matheron | Une conjecture sur la covariance d'un ensemble aléatoire[END_REF][START_REF] Quintanilla | Necessary and sufficient conditions for the two-point probability function of two-phase random media[END_REF][START_REF] Shepp | On positive definite functions associated with certain stochastic processes[END_REF]. The natural symmetry of the set {-1, 1} confers to this version a more handy combinatorial structure, and is exploited in the companion paper [START_REF] Lachièze-Rey | The convex class of realisable unit covariances[END_REF] to give theoretical realisability results, transferred in this paper to other marginals. Remark 2.2 (centred marginals). Other versions of these functionals where the first order marginal is subtracted are also used, such as the proper covariance E( Xx -E Xx )( Xy -E Xy ), but they only complicate the already difficult realisability problem, and our findings can be passed on to centred marginals.

In the light of (2.1), the class of indicator variograms can be paired up with that of unit covariances, while two point covering functions, covariances and geometric variograms (see Sec. 4) form another group in the literature treating of second order marginals. The relations between these two families are precised below. There is a one-to-one mapping

Φ V →U : F 0 E → F 1 E γ → ρ = 1 -4γ
that maps indicator variograms to unit covariances. The relation with the covariances is more subtle. Since for a random set X, ρ X x,y = EY x Y y where Y = 2 X -1, it is clear that the two point covering function of X and its unit covariance are related through ρ X

x,y = 4p X x,y -2(p X x,x + p X yy ) + 1. It is on the converse not possible to pass directly from ρ X to p X . This can be understood by noting that ρ X = ρ X c but p X = p X c where X c is the complementary of X, thus a given unit covariance might correspond to different covariances. There is no canonical way to pass from two point covering functions to unit covariances, so we give below the method proposed in [START_REF] Matheron | Une conjecture sur la covariance d'un ensemble aléatoire[END_REF]. We introduce an exterior point x 0 / ∈ E and put

E ′ = E ∪ {x 0 }.
Let X be a random set, and

p X ∈ C E its two point covering function. Define X ′ = {x 0 } ∪ X. The unit covariance ρ X ′ of X ′ is ρ X ′ x,y = Φ C →U (p X ) x,y :=      4p X x,y -2(p X x,x + p X y,y ) + 1 if x, y = x 0 2p X yy -1, if x = x 0 , y ∈ E 1 if x = y = x 0 . (2.2)
Let reciprocally ρ X ′ be the unit covariance of a random set X ′ ⊆ E ′ . Defining

X ′′ = X ′ if x 0 ∈ X ′ (X ′ ) c if x 0 / ∈ X ′ yields ρ X ′′ = ρ X ′ , thus we can assume without loss of generality that x 0 ∈ X ′ a.s. Define X = X ′ ∩ E = X ′ \ {x 0 }.
Then the two point covering function of X can be obtained by reversing (2.2), explicitly for all x, y ∈ E,

p X x,y = 1 4 (ρ X ′ x,y + 1 + ρ X ′ x 0 ,y + ρ X ′ x 0 ,x ) = (Φ -1 C →U (ρ X ′ )) xy .
Remark 2.3. The bijection between U E ′ and C E corresponds to the transformation X ′ = X ∪ {x 0 }. There are many other ways to pass from covariances to unit covariances, one can for instance impose that the law of X′ x 0 is that of a Bernoulli variable with parameter p for every p ∈ [0, 1], but the construction is slightly more complicated.

Proposition 2.4. We have the following relations between C E , U E and V E . Assume that x 0 is a fixed point external to E, and E ′ = E ∪ {x 0 }. We have

U E = Φ V →U (V E ), V E = Φ U →V (U E ), U E ′ = Φ C →U (C E ), C E = Φ U →C (U E ′ )
where

Φ V →U : F 1 E → F 0 E and Φ C →U : F E → F 1
E ′ are the one-to-one affine mappings described above, and

Φ U →V = Φ -1 V →U , Φ U →C = Φ -1 C →U . If E = [N] for N ≥ 1, identify E ′ = [N + 1].
For u, v two real functions on E, denote by (u ⊗ v) x,y = u x v y , x, y ∈ E, their tensor product. Marginals of central theoretic importance are the deterministic ones, namely the p A = Ã ⊗ Ã for A ⊆ E deterministic, and the corresponding deterministic unit covariances ρ A = (2 Ã -1) ⊗ (2 Ã -1). The deterministic variograms are the

γ A = Φ U →V (ρ A ) = 1 4 (1 -ρ A ) ∈ V E , A ⊂ E.
The following theorem unveils the convex structure of the class of valid second order marginals.

Theorem 2.5. (i)C E (resp. V E ) is a convex subset of F E which extreme points are the deterministic two point covering functions p A for A ⊂ E (resp. the deterministic variograms γ Proof. It is proved in the companion paper [START_REF] Lachièze-Rey | The convex class of realisable unit covariances[END_REF] that U E is convex, that its extreme points are the u ⊗ u for u ∈ {-1, 1} E , and that furthermore if card(E) = N ≥ 1 there are exactly 2 N -1 extreme points and U N has inner dimension d N . The isomorphisms Φ U →V and Φ U →C yield the announced conclusions for V E and C E .

A , A ⊂ E). (ii) If E = [N], N ≥ 1, call d N = N (N +1)
Similarly, the last statement is a direct consequence of Prop. 1.4 in [START_REF] Lachièze-Rey | The convex class of realisable unit covariances[END_REF], itself an application of the Minkowski-Carathéodory Theorem.

Remark 2.6. Algorithms yielding a simplex of d N + 1 points containing a realisable unit covariance ρ provide a reconstruction algorithm for a structure with given unit covariance ρ, see [START_REF] Ziegler | Lectures on polytopes[END_REF] Sec. 1.6 and references therein. Still the obtained model completely lacks structure and geometrical meaning, and such a naive approach is probably useless for applications.

Still it is proved in the forthcoming paper [START_REF] Galerne | Realisability of random sets of finite perimeter via grid approximation[END_REF] that one can control the mean perimeter of the obtained set via its second order marginals, giving a geometric substance to the model, therefore such a constrained reconstruction algorithm could be fruitful.

If E is finite with cardinal N ≥ 1, a function p x,y is sometimes identified with the N × N matrix (p x i ,x j ) 1≤i,j≤N , {x i ; 1 ≤ i ≤ N} being an enumeration of E. The following theorem enlightens the way the realisability problem should be posed in a discrete setting.

Theorem 2.7. Let N ≥ 1. Given a N × N matrix Σ with only 0's on the diagonal, for s > 0 sufficiently small, I + sΣ ∈ U N is realisable in E = [N] as a unit covariance, and -sΣ ∈ V N is realisable as a variogram. Calling s c (Σ) > 0 the critical value such that I + sΣ ∈ U N and

-s 4 Σ ∈ V N if and only if s ≤ s c (Σ), we have s c (Σ) ≥ 2 π Σ -1 2
, where • 2 is the Euclidean norm for matrices.

Proof. It is proved in Prop. 4.2 that if Λ is a semi-definite positive matrix with 1's on the diagonal and entries in [-1, 1],

ρ x,y = 2 π arcsin(Λ x,y ), x, y ∈ E
is a realisable unit covariance (realised by the 0 level set of a centred Gaussian process on E with covariance Λ). Given Σ semi-definite positive, the strategy here is to write

I + sΣ = 2 π arcsin(Λ)
for some such Λ, hence verifying for x, y ∈ E, Λ x,y = 1 if x = y sin( π 2 sΣ x,y ) otherwise. Let us prove that for s small, Λ so defined is semi-definite positive. We have for any

vector h = (h x ) x∈E x,y∈E Λ x,y h x h y = x∈E h 2 x + x =y Λ x,y h x h y ≥ x∈E h 2 x - x =y Λ 2 x,y x =y h 2 x h 2 y ≥ x∈E h 2 x - x =y sin π 2 sΣ x,y 2 x∈E h 2 x ≥ x∈E h 2 x   1 -s π 2 x =y Σ 2
x,y   by the Cauchy-Schwarz inequality. Hence if s ≤ 2 π Σ -1 2 , Λ is semi-definite positive, and therefore I + sΣ is a valid unit covariance, and in virtue of (2.1), 1 4 (I -

(I + sΣ)) = -s 4 Σ is a valid indicator variogram.
Finding precisely this value is not possible in general because one cannot characterise completely U N , but we give in Section 3 a heuristic procedure to estimate it, and illustrate it in Section 4 on the spherical variograms.

Stationarity and Isotropy

In most of the literature about random structures, models are assumed to be stationary and/or isotropic. If X is a random subset of R d , X is said to be stationary -or homogeneous -if its law is invariant under translations, and isotropic it its law is invariant under rotations. A second order marginal {α X

x,y } of X (covariance, indicator variogram, or two point covering function) then factorizes to a simpler form denoted ᾱX on a smaller space H :

α X x = α X 0,x , x ∈ H = R d , if X is stationary, with α X x,y = α X x-y , (2.3)
α X r = α X 0,ru , r ∈ H = R + , if X is stationary and isotropic, with α x,y = α x-y ,
where u is some unit vector of R d . The realisability problem is therefore posed in this terms for ᾱ.

Theorem 2.8. A function α on H is the reduced covariance of a stationary (and/or isotropic) random set if and only if α defined by (2.3) is realisable.

The proof of Th. 2.8, as well as more general statements, can be derived from Th. 2.13 in [START_REF] Lachièze-Rey | Regularity conditions in the realisability problem in applications to point processes and random closed sets[END_REF]. Under this form, the problem of characterising numerically stationary covariances is the same without stationarity. Nevertheless it might be possible to take advantage of the special Toeplitz form of stationary field covariances to make efficient computations, for instance by imbedding it in a circulant matrix, as it has been made in [START_REF] Wood | Simulation of stationary gaussian processes in [0, 1] d[END_REF][START_REF] Dembo | Embedding nonnegative definite toeplitz matrices in nonnegative definite circulant matrices, with application to covariance estimation[END_REF]. For the same reasons, the numerical complexity of the problem reduces since the dimension of the space where live stationary covariances is smaller, but the combinatorial symmetry is lost, thus there is no gain on the theoretical point of view.

Checking realisability numerically and Matheron's conjecture 3.1 A heuristic for checking Matheron's conditions

Matheron formulated necessary conditions for an indicator variogram γ on E to be admissible, namely γ must satisfy q i,j=1

ε i ε j γ x i ,x j ≤ 0 (3.1)
for every ε ∈ {-1, 0, 1} N such that i ε i = 1, and x 1 , . . . , x q ∈ E. It is acknowledged in the literature that they are hard to check in practice (see [START_REF] Jiao | Modeling heterogeneous materials via twopoint correlation functions: Basic principles[END_REF] for a heuristic approach). The unique condition used in practice for proving the non-validity of some indicator variogram is the so-called triangular inequality γ x,z ≤ γ x,y + γ y,z , x, y, z ∈ E, that arises from (3.1) by specialising for q = 3, (x 1 , x 2 , x 3 ) = (x, y, z), ε = (1, -1, 1). See Markov [START_REF] Markov | On the triangular inequality in the theory of two-phase random media[END_REF] for a discussion of this inequality and its consequences. This condition implies for instance that a valid variogram in a continuous medium must have a cusp at the origin (see [START_REF] Lantuéjoul | Geostatistical simulation: models and algorithms[END_REF][START_REF] Jiao | Modeling heterogeneous materials via twopoint correlation functions: Basic principles[END_REF]). Authors have been able to discard for instance the Gaussian variogram γ x,y = 1 -exp(-x -y 2 ) ( [START_REF] Lantuéjoul | Geostatistical simulation: models and algorithms[END_REF] p. 27) as a valid indicator variogram. We claim here that in a discrete setting it is more convenient to characterise realisability in terms of the unit covariance ρ = 1 -4γ. Assume that N = card(E) < ∞ and call unitary vector some e ∈ Z N such that there exists u ∈ {-1, 1} N satisfying i e i u i = 1. Call E N the class of unitary vectors. It is proved in the companion paper (with numeric computations) that the conditions i,j

ρ x i ,x j e i e j ≥ 1 e ∈ E N , (3.2)
are necessary for the realisability of ρ (and hence of γ), and sufficient if N ≤ 6. The advantage of using instead the conditions (3.2) is twofold:

1. Due to spectral considerations, it is easier to check than (3.1) (see Th. 3.1 and the heuristic below).

2. Conditions (3.2) are more discriminant. Indeed, conditions (3.1) only characterise the positive convex cone generated by 0 in V N , while (3.1) delimit a compact convex set. The new conditions involve the e such that i u i e i = 1 for unitary e such that i e i = 1.

Theorem 3.1. Take ρ ∈ F 1 N a non-singular symmetric matrix. If ρ is not definite positive, then it is not realisable. Otherwise, call λ > 0 its smallest eigenvalue. Then ρ verifies (3.2) for every e ∈ E N if and only if it satisfies them only for e ∈ E N such that

i e 2 i < λ -1 . (3.3) Proof.
For any e ∈ Z N , we have ij ρ ij e i e j ≥ λ e 2 2 .

Thus (3.2) is automatically fulfilled if e 2 ≥ λ -1 .
At this stage the number of conditions to effectively check in (3.2) for a non-singular symmetric matrix is finite, even though the number depends on the smallest eigenvalue λ of ρ. Still if λ is very small, this number might explode. The heuristic procedure below can help find e more likely to fail (3.2). Real data more often concerns non-singular matrices; still the singular case is treated in Th. 3.2 . Theorem 3.2. Assume ρ ∈ F 1 N has rank r < N, and let σ ∈ F 1 r and I ⊆ [N] with card(I) = r be such that σ = (ρ ij ) ij∈I is non-singular. Then for k / ∈ I, the k-th line L k of ρ can be written as a linear combination

L k = i∈I α k i L i
of the lines L i , i ∈ I. Then ρ satisfies conditions (3.2) if and only if σ satisfies them with N = r and for i, j

∈ [N] ρ ij = k,q∈I α i k α j q ρ kq . (3.4)
where

α i k := δ k=i if k ∈ I. Proof. Assume that ρ is realisable by a random field Y ∈ {-1, 1} N . Then σ is realisable as the covariance of the unit field Y I = {Y i ; i ∈ I} that furthermore satisfies for k / ∈ I, j ∈ [N], EY j (Y k - i∈I α k i Y i ) = ρ kj - i∈I α k i ρ ij = 0,
summing up with the proper coefficients yields

E(Y k - i∈I α k i Y i ) 2 = 0 and Y k = N i=1 α k i Y i a.
Y k = i∈I α i k Y i for k / ∈ I, ρ kj defined by (3.4) satisfies ρ kj = EY k Y j for k, j ∈ [N].

Heuristic algorithm

The following algorithm can be used to apply (3.2) to a potential unit covariance ρ. Its complexity explodes quickly with N, the strategy used in Section 4 is to test the realisability of a bidimensional data by sampling a low dimensional restriction ρ and applying the following procedure.

1. If ρ is not singular, extract a singular matrix σ of ρ with maximal rank r and check that ρ and σ satisfy the assumptions of Th. 3.2. Then continue the algorithm with N = r, ρ = σ.

2. Define a cut-off value µ > 0 and compute the q smallest eigenvalues λ = λ 1 ≤ • • • ≤ λ q ≤ µ.

3. Introduce the q-dimensional space V µ spanned by eigenvectors associated to λ 1 , . . . , λ q . Find by optimisation procedures the unitary vectors e ∈ E N as close of V λ as possible, and such that i e 2 i is small (at least smaller than λ -1 ). 4. If one of those e fails (3.2), then ρ is not realisable.

We know that for v ∈ V µ , we have

λ 1 ≤ vρv T v 2 2 ≤ µ.
In view of (3.3), the cut-off value µ need not be taken larger than 1/4 because i e 2 i < 4 implies that e ∈ E N has at most three non-zero components; in this case (3.2) is reduced to the triangular inequality, which can be checked by other means (see for instance [START_REF] Markov | On the triangular inequality in the theory of two-phase random media[END_REF]). This heuristic is illustrated in Section 4 to study the realisability of the circular variogram.

Spherical variograms and Gaussian level sets

This section focuses on two particular models widely used in the literature; the Gaussian level sets and the spherical variograms. We apply the work of the previous sections to establish two facts: (1) the circular and spherical variograms are not valid indicator variograms if multiplied by a too large constant, for which we give an upper bound, and (2) some admissible covariances and indicator variograms cannot be realised by Gaussian models.

Spherical variograms

Spherical variograms are among the most popular models used to fit experimental samples of geostatistical data. It is a recurrent question to know wether such a variogram can be used as a valid indicator variogram for a stationary model (see [START_REF] Lantuéjoul | Geostatistical simulation: models and algorithms[END_REF], p. 28). Emery [START_REF] Emery | On the existence of mosaic and indicator random fields with spherical, circular, and triangular variograms[END_REF] proved that indicator variograms of the most popular families of geometric models (boolean models, Gaussian models, Poisson mosaics) do not in general provide an indicator variogram under the spherical form. It does not discard a priori the spherical variograms as admissible variograms. Our study is not model-dependant as we estimate the critical value of the constant by which one can multiply a spherical variogram so that it is realisable at all. Our computations rely on the realisability conditions of Th. 3.1 and the subsequent heuristic.

For d ≥ 1, denote by K d (r) the geometric spherical covariogram,

K d (r) = ℓ(B d ∩ (B d + ru)), 0 ≤ r ≤ 1.
where B d is a ball of R d with diameter 1, and u is some unitary vector of R d . For instance for d = 2,

K 2 (r) = 1 2 (arccos(r) -r √ 1 -r 2 ).
Define the spherical variogram by A recurring question in geostatistics [START_REF] Lantuéjoul | Geostatistical simulation: models and algorithms[END_REF][START_REF] Chilès | Modelling spatial uncertainty[END_REF][START_REF] Emery | On the existence of mosaic and indicator random fields with spherical, circular, and triangular variograms[END_REF] is the admissibility of γd , up to some multiplicative constant, as the (reduced) indicator variogramof a (stationary) random set. Does there exists a random binary field X ⊆ R d such that

γd r = K d (0) -K d (r) K d (0) .
sγ d r = P ( X0 = Xru ), r ∈ [0, 1]
for every unitary u in R d , and some constant s ∈ [0, 1] called the sill ? As is usual in this paper we prefer to study the corresponding unit covariance

ρd r = 1 -2sγ d r ∈ [-1, 1].
We call s c,d ≥ 0 the critical sill such that ρd is realisable if and only if s ≤ s c,d . We have for r ∈ [0, 1]

ρ1 r = 1 -2sr ρ2 r = 1 - 4s π π 2 -arccos(r) + r √ 1 -r 2 ρ3 r = 1 -s 3r -r 3 .
The case of the triangular variogram is quickly settled as for s = 1, sin(πρ 1 (r)/2) = cos(πr) is a semi-definite positive function. It follows by (4.4) that the triangular variogram is the indicator variogram of the centred stationary Gaussian field W with covariance EW x W y = cos(π|x -y|) thresholded at the level 0. Therefore s c,1 = 1.

For higher dimensions, to study numerically the realisability of ρ d , we study for N ≥ 1 the realisability of its N d × N d restriction matrix

ρ d,N i,j = ρd x i -x j , i ∈ [N] d , j ∈ [N] d
where

x i = ((i 1 -1)/N, . . . , (i N -1)/N), i ∈ [N] d .
If d = 2 the heuristic described at Section 3 gave the following results.

1. For s = 0.72, with N = 3, and

v = [1 0 1 0 -1 0 1 0 1],
we have Remark 4.1. The uniform sampling scheme {x i ; i ∈ [N] d } is arbitrary. Even though the efficiency of the method should increase as the smallest eigenvalue of ρ N decreases (letting theoretically the possibility for more vectors e ∈ E N to be tested), choosing the x i,j more concentrated instead of equally spaced did not give good results.

vρ d,N v T = N i,j=1 ρ i,j v i v j < 0,
For d = 3, ρ 3,8 is not of positive type for s > 0.58, thus Theorem (3.1) enables us to say that s c,3 ≤ 0.58.

Gaussian level sets covariances

When confronted to a symmetric function p that might be the second order characteristic of some random set, the default strategy is sometimes to associate it with a Gaussian structure, in general a Gaussian process thresholded at a given value (see [START_REF] Emery | On the existence of mosaic and indicator random fields with spherical, circular, and triangular variograms[END_REF][START_REF] Lantuéjoul | Geostatistical simulation: models and algorithms[END_REF][START_REF] Torquato | Random Heterogeneous Materials[END_REF]). This approach might not be successful, therefore a legitimate question is whether it is realisable at all. In other words, are there realisable second-order characteristics that cannot be realised by a Gaussian level set? The aim of this section is to prove that the answer is yes unless E is pathologically small.

A standard Gaussian field on E is a collection of random variables {W x ; x ∈ E} any linear combination of which is Gaussian and such that EW x = 0 and EW 2 x = 1 for x ∈ E (see [START_REF] Lantuéjoul | Geostatistical simulation: models and algorithms[END_REF] or [START_REF] Lifshits | Gaussian random functions[END_REF] for more details on Gaussian fields) . The covariance function of W Σ x,y = EW x W y , x, y ∈ E is semi-definite positive. Bochner's theorem states that conversely, given Σ ∈ F 1 E ∩ P of positive type, there exists a unique (in law) standard Gaussian field W Σ with covariance Σ. In this section we consider the random set obtained by thresholding W Σ at some level z ∈ R

X Σ,z = {x ∈ E : W Σ x ≥ z}. (4.1)
The second order marginals of X Σ,z can be found in the literature, see for instance [START_REF] Lantuéjoul | Geostatistical simulation: models and algorithms[END_REF].

Proposition 4.2. Let Σ ∈ P ∩ F 1 E , z ∈ R. The indicator variogram of X Σ,z is γ Σ,z x,y := 1 2π 1 Σx,y 1 √ 1 -r 2 exp - z 2 1 + r dr, x, y ∈ E, (4.2)
The two point covering function of X Σ,z is

p Σ,z x,y := Φ(z) 2 + 1 2π Σx,y 0 1 √ 1 -r 2 exp - z 2 1 + r dr, x, y ∈ E, (4.3)
where

Φ(z) = -∞ z e -r 2 /2 dr √ 2π
is the Gaussian tail function.

We have in particular a compact expression of the unit covariance if z = 0:

ρ Σ,0 x,y := 1 -4γ Σ,0 x,y = 2 π arcsin(Σ x,y ), x, y ∈ E. (4.4) Proof. Let us call f r (u, v), u, v ∈ R 2 the density of a Gaussian vector of R 2 with covariance r ∈ [-1, 1]. Direct computations yield ∂f r (u, v) ∂r = ∂ 2 f r (u, v) ∂u∂v .
We have

γ Σ,z x,y = P(W (x) ≥ z, W (y) ≤ z) = ∞ z z -∞ f Σx,y (u, v)dudv = ∞ z z -∞ f 1 (u, v) + 1 Σx,y ∂f r (u, v) ∂r dr dudv = 0 + 1 Σx,y ∞ z z -∞ ∂ 2 f r (u, v) ∂u∂v dudvdr = 1 Σx,y f r (z, z)dr,
which proves relation (4.2). (4.3) is obtained in an exactly similar fashion and can also be found as Prop. 16.1.1 in [START_REF] Lantuéjoul | Geostatistical simulation: models and algorithms[END_REF]. One can also note that p Σ,z x,y = P(x, y ∈ X Σ,z ) -γ Σ,z x,y = Φ(z) -γ Σ,z

x,y .

Theorem 4.3. If card(E) ≥ 4, there exists p ∈ C N that cannot be obtained by thresholding a standard Gaussian field, i.e. that is not under the form (4.3) for some definite positive function {Σ x,y } and some level z ∈ R.

Proof. The strategy is to prove that the class of all Gaussian two point covering functions for card(E) = N = 4 is not convex, and therefore does not coincide with C E . The result follows with a restriction argument for E with a larger cardinality. Take Σ 0 = Σ 1 ∈ F 1 E ∩ P the covariances of two standard Gaussian fields, resp. W 0 , W 1 , and consider the random fields X0 := XΣ 0 ,0 and X1 := XΣ 1 ,0 obtained by thresholding resp. W 0 and W 1 at the level 0. Call p i the two point covering function of the binary field Xi , for i = 1, 2. We claim that if Σ 0 and Σ 1 have been chosen judiciously, the two point covering function p := 1 2 (p 1 + p 2 ) (4.5)

is not the two point covering function of a Gaussian level set of the form (4.1). On the other hand, it is the two point covering function of a random set because U E is convex. If X is a random set such that p = p X , then p x,x = p 0 x,x = p 1 x,x = 1/2. Thus p x,y = 1/4+ 1 2π arcsin(Σ x,y ) for some semi-definite positive matrix Σ. (4.5) imposes also arcsin(Σ x,y ) = 1 2 arcsin(Σ 0 x,y ) + arcsin(Σ 1

x,y ) .

The conclusion follows directly from the following lemma:

Lemma 4.4. It is possible to choose Σ 0 , Σ 1 ∈ P ∩ F 1 4 such that Σ x,y = sin 1 2 (arcsin(Σ 0 x,y ) + arcsin(Σ 1 x,y )) , x, y ∈ E, is not semi-definite positive.

Proof. Defining the Toeplitz matrices

Σ 0 =     1 0 -1 0 0 1 0 -1 -1 0 1 0 0 -1 0 1     Σ 1 =      1 √ 3 2 1 2 0 √ 3 2 1 √ 3 2 1 2 1 2 √ 3 2 1 √ 3 2 0 1 2 √ 3 2 1      gives Σ =     1 1 2 -1 2 0 1 2 1 1 2 -1 2 -1 2 1 2 1 1 2 0 -1 2 1 2 1     .
which is not of positive type (counter example provided by the user Robert Israel on the mathematical forum mathoverflow.com ).

Thus Σ cannot be the covariance of a Gaussian field.

For variograms and unit covariances, one does not have direct access to the first order marginal through the diagonal elements, thus the situation is more subtle but the result still holds if we enlarge E with two elements. Theorem 4.5. For N ≥ 6, there exists γ ∈ V N such that γ is not the indicator variogram of any Gaussian field thresholded at some level z ∈ R.

Proof. Consider two block-matrices covariances of the form

Λ 0 =   1 -1 0 -1 1 0 0 0 Σ 1   , Λ 1 =   1 -1 0 -1 1 0 0 0 Σ 2  
where Σ i are given by lemma 4.4, and put γ i = γ Λ i ,0 . Then 1 2 (γ 0 + γ 1 ) is the variogram of a binary field X such that X1 = X2 a.s.. If X was the level set of a standard Gaussian field W = W Σ at a level z ∈ R, we would have W x > z if and only if W y < z with probability 1, whence necessarily Σ 1,2 = -1 and z = 0. Then Λ x,y = sin((1/2)(arcsin(Λ 1

x,y ) + arcsin(Λ 2

x,y ))), x, y ∈ E, is not semi-definite positive but still it is the covariance matrix of Σ. Contradiction.

A similar result holds of course for unit covariances.

2 .

 2 Then V E has inner dimension d N , and C E has inner dimension d N +1 . Every valid variogram (resp. valid two point covering function) can be realised by a random set taking at most d N + 1 (resp. d N +1 + 1) different values.

  s., k / ∈ I. Condition (3.4) follows easily. If conversely σ is realisable by a unit field Y I ∈ {-1, 1} I , then if one defines

For d = 1 ,

 1 γd is called the triangular variogram, for d = 2 the circular variogram, and for d = 3 the spherical variogram. Their expressions can be found in [1] pp. 81-82.

  987 < 1, contradicting(3.2). It implies that ρ 2,N , and thus ρ2 , is not a valid indicator variogram. 2. for s = 0.7 and N = 3, the smallest eigenvalue is λ ∼ 0.2370, meaning finding e ∈ Z N with odd sum such that i e 2 i ≤ [1/λ] = 4 not satisfying (3.2) might turn up difficult. Thus we have s c,2 < 0.72 and point 2. might lead us to think that s c,2 should not be far from 0.7. At this stage, the only possible way to ensure realisability is to provide directly a model realising ρ.