Nicolas Chenavier
email: nicolas.chenavier@univ-littoral.fr

Ahmad Darwiche
email: darwich.ahmad.92@gmail.com

Some properties on extremes for transient random walks in random sceneries

Keywords: extreme values, random walks, point processes. Mathematics Subject Classification: 60G70, 60G50, 60G55

Let (S n ) n≥0 be a transient random walk in the domain of attraction of a stable law and let (ξ(s)) s∈Z be a stationary sequence of random variables. In a previous work, under conditions of type D(u n ) and D (u n ), we established a limit theorem for the maximum of the first n terms of the sequence (ξ(S n )) n≥0 as n goes to infinity. In this paper we show that, under the same conditions and under a suitable scaling, the point process of exceedances converges to a Poisson point process. We also give some properties of (ξ(S n )) n≥0 .

Introduction

In 2009, Franke and Saigo [START_REF] Franke | The extremes of a random scenery as seen by a random walk in a random environment[END_REF][START_REF] Franke | The extremes of random walks in random sceneries[END_REF] considered the following problem. Let (X k ) k≥1 be a sequence of centered, integer-valued i.i.d. random variables and let S 0 = 0 a.s. and S n = X 1 + • • • + X n , n ≥ 1. Assume that, for any x ∈ R,

P S n n 1/α ≤ x -→ n→∞ F α (x),
where F α is the distribution function of a stable law with characteristic function given by

φ(θ) = exp(-|θ| α (C 1 + iC 2 sgn θ)), α ∈ (0, 2].
Let (ξ(s)) s∈Z be a stationary sequence of R-valued random variables which are independent of the sequence (X k ) k≥1 . The sequence (ξ(S n )) n≥0 is referred to as a random walk in a random scenery. In [START_REF] Franke | The extremes of random walks in random sceneries[END_REF], Franke and Saigo derive limit theorems for the random variable max i≤n ξ(S i ) as n goes to infinity when the ξ(s)'s are i.i.d.. The statements of their theorems depend on the value of α. When α < 1 (resp. α > 1), it is known that the random walk (S n ) n≥0 is transient (resp. recurrent) [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF][START_REF] Gall | The range of stable random walks[END_REF]. An important concept concerning random walks is the 1 range. The latter is defined as the number of sites visited by the first n terms of the random walk, namely R n := #{S 1 , . . . , S n }. The following result, due to Le Gall and Rosen [START_REF] Gall | The range of stable random walks[END_REF], deals with its asymptotic behavior.

Theorem 1 (LeGall and Rosen). (i) If α < 1, then

R [nt] n -→
n→∞ qt P -a.s.

with q := P (S k = 0, ∀k ≥ 1).

(ii) If α = 1, then

h(n)R [nt] n -→ n→∞ t in L p (P),
where h(n) := 1 + n k=1 P (S k = 0).

(iii) If 1 < α ≤ 2, then for any L ∈ N and any

t 1 < • • • < t L , 1 n 1/α R nt 1 , . . . , R nt L -→ n→∞ (m(Y (0, t 1 )), . . . , m(Y (0, t L ))) , in distribution.
In the above result, {Y (t), t ∈ R} denotes the right-continuous α-stable Lévy process with characteristic function given by φ(tθ) and m is the Lebesgue measure on R. One of the results of [START_REF] Franke | The extremes of random walks in random sceneries[END_REF] is the following. If u n is a threshold such that nP (ξ > u n ) -→ n→∞ τ for some τ > 0, with ξ = ξ [START_REF] Chenavier | Extremes for transient random walks in random sceneries under weak independence conditions[END_REF], and if the ξ(s)'s are i.i.d. then

P max i≤n ξ(S i ) ≤ u n -→ n→∞ e -τ q
for α < 1. Such a result was generalized in [START_REF] Chenavier | Extremes for transient random walks in random sceneries under weak independence conditions[END_REF] for sequences (ξ(s)) s∈Z which are not necessarily i.i.d., but which satisfy a slight modification of the classical D(u n ) and D (u n ) conditions of Leadbetter (see [START_REF] Leadbetter | On extreme values in stationary sequences[END_REF][START_REF] Leadbetter | Extremes and local dependence in stationary sequences[END_REF] for a statement of these conditions).

In this paper, we give a more precise treatment of the extremes of (ξ(S n )) n≥0 . To do it, we assume that the threshold is of the form

u n = u n (x) = a n x + b n (a n ∈ R, b n > 0 and x ∈ R)
and that, for any x ∈ R, the following term exists and is finite:

ν(x, ∞) := lim n→∞ nP (ξ > u n (x)) . ( 1
)
The quantity ν defines a measure on some topological space E. According to the Gnedenko's theorem [START_REF] Gnedenko | Sur la distribution limite du terme d'une série aléatoire[END_REF], if ξ is in the domain of attraction of an extreme value distribution G, then ν is of the form:

ν(x, ∞) =        x -β , E = (0, ∞] if G is a Fréchet distribution; (-x) -δ , E = (-∞, 0] if G is a Weibull distribution; e -x , E = (-∞, ∞] if G is a Gumbel distribution;
for some β, δ > 0. Notice that if P n denotes the distribution of ξ-an bn , then (1) can be rephrased as

nP n (A) -→ n→∞ ν(A), (2)
for any Borel subset A ⊂ R. Secondly, we assume that the (stationary) sequence (ξ(s)) s∈Z satisfies conditions of type D(u n ) and D (u n ) in the same spirit as in [START_REF] Chenavier | Extremes for transient random walks in random sceneries under weak independence conditions[END_REF]. To introduce the first one, we write for each i 1 < • • • < i p and for each u ∈ R,

F i 1 ,...,ip (u) = P (ξ(i 1 ) ≤ u, . . . , ξ(i p ) ≤ u) .

D(u n ) condition

We say that (ξ(s)) s∈Z satisfies the D(u n ) condition if there exist a sequence (α n, ) (n, )∈N 2 and a sequence ( n ) of positive integers such that α n, n -→ n→∞ 0, n = o(n), and

|F i 1 ,...,ip,j 1 ,...,j p (u n ) -F i 1 ,...,ip (u n )F j 1 ,...,j p (u n )| ≤ α n, for any integers i 1 < • • • < i p < j 1 < • • • < j p such that j 1 -i p ≥ .
Notice that the bound holds uniformly in p and p . Roughly, the D(u n ) condition (see e.g. p29 in [START_REF] Lucarini | Extremes and Recurrence in Dynamical Systems[END_REF]) is a weak mixing property for the tails of the joint distributions.

The D (u n ) condition (see e.g. p29 in [START_REF] Lucarini | Extremes and Recurrence in Dynamical Systems[END_REF]) is a local type property and precludes the existence of clusters of exceedances. To introduce it, we consider a sequence (k n ) such that

k n -→ n→∞ ∞, n 2 k n α n, n -→ n→∞ 0, k n n = o(n), (3)
where ( n ) and (α n,l ) (n,l)∈N 2 are the same as in the D(u n ) condition.

D (u n ) condition

In conjunction with the D(u n ) condition, we say that (ξ(s)) s∈Z satisfies the D (u n ) condition if there exists a sequence of integers (k n ) satisfying (3) such that

lim n→∞ n n/kn s=1 P (ξ(0) > u n , ξ(s) > u n ) = 0.
In the classical literature, the sequences (α n,l ) (n,l)∈N 2 and (k n ) only satisfy k n α n, n -→ n→∞ 0 (see e.g. (3.2.1) in [START_REF] Lucarini | Extremes and Recurrence in Dynamical Systems[END_REF]) whereas in (3) we have assumed that n 2 kn α n, n -→ n→∞ 0. In this sense, the D (u n ) condition as written above is slightly more restrictive than the usual D (u n ) condition.

Our paper is organized as follows. In Section 2, we prove that under suitable scaling the socalled point process of exceedances converges to a Poisson point process in the transient case.

In Section 3, we give some properties of the random walk in random scenery. More precisely, we show that the (stationary) sequence (ξ(S n )) n≥0 satisfies the classical D(u n ) condition of Leadbetter, but does not satisfy the D (u n ) condition. Our results generalize [START_REF] Franke | The extremes of random walks in random sceneries[END_REF] for sequences (ξ(s)) s∈Z which are not i.i.d. but which only satisfy the D(u n ) and D (u n ) conditions. We also give some remarks on the so-called extremal index and on the D (k) (u n ) condition.

2 Point process of exceedances

Poisson approximation

The main result of this section claims that the point process of exceedances converges to a Poisson point process in the transient case, i.e. α < 1. To introduce it, we denote for any k ≥ 1 by

τ k = inf{m ≥ 0 : #{S 1 , . . . , S m } ≥ k}
the time at which the random walk visits its k-th site. The point process of exceedances is defined as

Φ n = τ k n , ξ(S τ k ) -b m(n) a m(n) : τ k ≤ n k≥1 ⊂ [0, 1] × R, (4)
where m(n) = qn .

Proposition 2. Let α < 1. Assume that the sequence (ξ(s)) s∈Z satisfies the D(u n ) and D (u n ) conditions for any threshold

u n = u n (x) = a n x + b n , x ∈ R,
⊂ [0, 1] × R with m [0,1] ⊗ ν(∂B i ) = 0, 1 ≤ i ≤ K, (#Φ n ∩ B 1 , . . . , #Φ n ∩ B K ) D -→ n→∞ (#Φ ∩ B 1 , . . . , #Φ ∩ B K ) .
By using the Laplace functional, Franke and Saigo (Theorem 3 in [START_REF] Franke | The extremes of random walks in random sceneries[END_REF]) obtained a similar result when the ξ(s)'s are i.i.d. Proposition 2 extends it and is based on Kallenberg's theorem. Our result is stated only in the transient case, i.e. for α < 1. However, it remains true for α = 1 by taking m(n) = n h(n) . When α > 1, the point process of exceedances is defined in the same spirit as (4) by taking this time m(n) = n 1/α . In this case, similarly to Theorem 4 in [START_REF] Franke | The extremes of random walks in random sceneries[END_REF], we can show by adapting the proof of Proposition 2 that Φ n converges weakly to a Cox point process Φ Y , i.e. a Poisson point process in

[0, 1] × R with random intensity measure µ(dt, dx) = m Y (dt)ν(dx), where m Y (t) = m(Y (0, t)).

Technical results

The proof of Proposition 2 is mainly based on Kallenberg's theorem (see e.g. Proposition 3.22 in [START_REF] Resnick | Extreme Values, Regular Variation and Point Processes[END_REF]) and on two technical lemmas which are stated below.

Theorem 3 (Kallenberg). Suppose Φ is a simple point process on E and I is a basis of relatively compact open sets such that I is closed under finite unions and intersections and, for I ∈ I,

P (#Φ ∩ ∂I = 0) = 1,
where ∂I is the boundary of I. Let (Φ n ) be a sequence of point processes on E such that, for all I ∈ I,

lim n→+∞ E (#Φ n ∩ I) = E (#Φ ∩ I) and lim n→+∞ P (#Φ n ∩ I = 0) = P (#Φ ∩ I = 0) . Then Φ n converges weakly to Φ in distribution.
The following lemma is a direct adaptation of Lemma 1 in [START_REF] Franke | The extremes of random walks in random sceneries[END_REF] and deals with the independence between the sequence (ξ(S n )) n≥0 and the sequence (τ k ) k≥1 .

Lemma 1. For all measurable sets B ⊂ N + and A ⊂ R, we have

P (τ k ∈ B, ξ(S τ k ) ∈ A) = P (τ k ∈ B) P (ξ ∈ A) .
The second lemma is an extension of [START_REF] Chenavier | Extremes for transient random walks in random sceneries under weak independence conditions[END_REF]. More precisely, under the assumptions that the D(u n ) and D (u n ) conditions hold for the sequence (ξ(s)) s∈Z , we have shown in [START_REF] Chenavier | Extremes for transient random walks in random sceneries under weak independence conditions[END_REF] that, for almost all realization of (S n ) n≥0 ,

P (Sn)    k≥1: τ k n ∈(0,1] ξ(S τ k ) -b m(n) a m(n) / ∈ (x, ∞)    -exp - R n m(n) ν(x, ∞) -→ n→∞ 0
when (1) holds for any threshold u n = u n (x), x ∈ R. In the above equation, P (Sn) stands for the probability conditional on (S n ) n≥0 . The following lemma deals with the case where the interval (0, 1] (resp. (x, ∞)) is replaced by (a, b] (resp. A ⊂ R) in the above equation.

Lemma 2.

Let A be a Borel subset in R and let 0 ≤ a < b ≤ 1. Under the same assumptions as Proposition 2, for almost all realization of (S n ) n≥0 , we have

lim n→∞ P (Sn)    k≥1: τ k n ∈(a,b] ξ(S τ k ) -b m(n) a m(n) / ∈ A    -exp - R nb -R na m(n) ν(A) = 0.

Proofs

Proof of Lemma 1. Since the random walk and the random scenery are independent, we have

P (τ k ∈ B, ξ(S τ k ) ∈ A) = m∈B P (τ k = m, ξ(S m ) ∈ A) = m∈B s∈Z P (τ k = m, S m = s, ξ(s) ∈ A) = m∈B s∈Z P (τ k = m, S m = s) P (ξ(s) ∈ A) = P (τ k ∈ B) P (ξ ∈ A) .
Proof of Lemma 2. The proof will be sketched since it relies on a simple adaptation of the proof of Theorem 1 in [START_REF] Chenavier | Extremes for transient random walks in random sceneries under weak independence conditions[END_REF]. Let (k n ), ( n ) be as in (3) and let

r n = n k n -1 + 1, (5)
for n large enough. Given a realization of (S n ) n≥0 , we write

S (na,nb] = S τ k : k ≥ 1, τ k n ∈ (a, b] and R nb -R na = #S (na,nb] .
To capture the fact that (ξ(s)) s∈Z satisfies the condition D(u n ), we construct blocks and stripes as follows. Let

K n = R nb -R na r n + 1.
We subdivide the set S (na,nb] into subsets

B i ⊂ S (na,nb] , 1 ≤ i ≤ K n , referred to as blocks, in such a way that #B i = r n and max B i < min B i+1 for all i ≤ K n -1. Notice that K n ≤ k n and #B Kn = R nb -R na -(K n -1)
• r n a.s.. For each j ≤ K n , we denote by L j the family consisting of the n largest terms of B j (e.g. if

B j = {x 1 , . . . , x rn }, with x 1 < • • • < x rn , j ≤ K n -1, then L j = {x rn-n+1 , . . . , x rn }). When j = K n , we take the convention L Kn = ∅ if #B Kn < n .
The set L j is referred to as a stripe, and the union of the stripes is denoted by L n = j≤Kn L j . Proceeding in the same spirit as in the proofs of Lemmas 1 and 2 of [START_REF] Chenavier | Extremes for transient random walks in random sceneries under weak independence conditions[END_REF], we can easily prove that for almost all realization of (S n ) n≥0 ,

• P (Sn)

s∈S (na,nb] ξ(s)-b m(n) a m(n) / ∈ A -P (Sn) s∈S (na,nb] \Ln ξ(s)-b m(n) a m(n) / ∈ A -→ n→∞ 0; • P (Sn) s∈S (na,nb] \Ln ξ(s)-b m(n) a m(n) / ∈ A -i≤Kn P (Sn) s∈B i \Ln ξ(s)-b m(n) a m(n) / ∈ A -→ n→∞ 0;
• i≤Kn P (Sn)

s∈B i \Ln ξ(s)-b m(n) a m(n) / ∈ A -i≤Kn P (Sn) s∈B i ξ(s)-b m(n) a m(n) / ∈ A -→ n→∞ 0;
• i≤Kn P (Sn)

s∈B i ξ(s)-b m(n) a m(n) / ∈ A -exp - R nb -R na m(n) ν(A) -→ n→∞ 0.
The first and the third assertions come from the fact that the size of the stripes is negligible compared to the size of the blocks, i.e. n = o(r n ). The second assertion is a consequence of the fact that the sequence (ξ(s)) s∈Z satisfies the D(u n ) condition and the last one is obtained by using the D(u n ) and D (u n ) conditions. Lemma 2 follows directly from the four assertions.

Proof of Proposition 2. According to Kallenberg's theorem, it is sufficient to show that

(i) lim n→∞ E (#Φ n ∩ I) = m [0,1] ⊗ ν(I), (ii) lim n→∞ P (#Φ n ∩ I = 0) = e -m [0,1] ⊗ν(I) ,
for all set I of the form I = (a, b] × A, where 0 ≤ a < b ≤ 1 and where A is an open subset of E.

To deal with (i), we write

E (#Φ n ∩ I) = k≥1 P τ k n , ξ(S τ k ) -b qn a qn ∈ I = k≥1 P τ k n ∈ (a, b] P ξ -b qn a qn ∈ A = k≥1 P τ k n ∈ (a, b] P qn (A),
where the second line comes from Lemma 1. Using the fact that k≥1

1 τ k n ∈(a,b] = R nb -R na , we have E (#Φ n ∩ I) = E   k≥1 1 τ k n ∈(a,b]   P qn (A) = E R nb -R na P qn (A).
Moreover, according to Theorem 1 and to the Lebesgue's dominated convergence theorem, we know that E(R nb -R na ) ∼ n→∞ nq(b -a). This, together with (2) implies

E (#Φ n ∩ I) -→ n→∞ (b -a) × ν(A) = m [0,1] ⊗ ν(I).
To deal with (ii), we observe that

P (#Φ n ∩ I = 0) = P    k≥1: τ k n ∈(a,b] ξ(S τ k ) -b qn a qn / ∈ A    .
According to Lemma 2, Theorem 1 and the Lebesgue's dominated convergence theorem, we have

P (#Φ n ∩ I = 0) = E exp - R nb -R na qn ν(A) + o(1) -→ n→∞ exp (-(b -a)ν(A)) .
This, together with the fact that (b-a)ν(A) = m [0,1] ⊗ν(I), concludes the proof of Proposition 2.

3 Properties of (ξ(S n )) n≥0

In this section, we give some properties of (ξ(S n )) n≥0 . More precisely, we show that the latter satisfies the D(u n ) condition and an extension of the so-called D (k) (u n ) condition, but does not satisfy the D (u n ) condition.

Distributional mixing property

For technical reasons, we assume only in this section that the random walk satisfies the following property

1 n 2 sup 0≤i 1 <•••<ip≤n V R i 1 ,...,ip -→ n→∞ 0, (6)
where V [•] denotes the variance and R i 1 ,...,ip = #{S i 1 , . . . , S ip }. We hope that the above condition is not very restrictive since R i 1 ,...,ip ≤ n a.s.. The following result extends Proposition 2 in [START_REF] Franke | The extremes of random walks in random sceneries[END_REF], which deals with the case where the ξ(s)'s are i.i.d., to sequences which only satisfy the D(u n ) and D (u n ) conditions.

Proposition 4. Let α < 1. Assume that the sequence (ξ(s)) s∈Z satisfies the D(u n ) and D (u n ) conditions for a threshold u n such that nP (ξ > u n ) -→ n→∞ τ , with τ > 0. Then (ξ(S n )) n≥0 satisfies the D(u n ) condition.
Proof of Proposition 4. We adapt several arguments of [START_REF] Franke | The extremes of random walks in random sceneries[END_REF]

in our context. Let 0 ≤ i 1 < • • • < i p < j 1 < • • • < j p ≤
n be a family of integers, with j 1 -i p > n and n = o(n). To prove that (ξ(S n )) n≥0 satisfies the D(u n ), we have to show that

|F i 1 ,...,ip,j 1 ,...,j p (u n ) -F i 1 ,...,ip (u n )F j 1 ,...,j p (u n )| ≤ αn, n ,
for some sequence (α n, ) (n, )∈N 2 such that αn, n -→ n→∞ 0, with

F i 1 ,...,ip (u n ) = P ξ(S i 1 ) ≤ u n , . . . , ξ(S ip ) ≤ u n .
We have

|F i 1 ,...,ip,j 1 ,...,j p (u n ) -F i 1 ,...,ip (u n )F j 1 ,...,j p (u n )| ≤ F i 1 ,...,ip,j 1 ,...,j p (u n ) -E exp - R i 1 ,...,ip,j 1 ,...,j p n τ + E exp - R i 1 ,...,ip,j 1 ,...,j p n τ -E exp - R i 1 ,...,ip + R j 1 ,...,j p n τ + E exp - R i 1 ,...,ip + R j 1 ,...,j p n τ -F i 1 ,...,ip (u n )F j 1 ,...,j p (u n ) . ( 7
)
To deal with the first and the third terms of the right-hand side of (7), we will use the following lemma.

Lemma 3. We have

sup 0≤i 1 <•••<ip≤n E F i 1 ,...,ip (u n ) -exp - R i 1 ,...,ip n τ -→ n→∞ 0.
Proof of Lemma 3. Similarly to Lemma 2, the main idea is to adapt several arguments appearing in the proofs of Lemmas 1 and 2 in [START_REF] Chenavier | Extremes for transient random walks in random sceneries under weak independence conditions[END_REF] in our context. Let (k n ) and (r n ) be as in ( 3) and [START_REF] Franke | The extremes of random walks in random sceneries[END_REF]. Given 1

≤ i 1 < i 2 < • • • < i p ≤ n, we subdivide the random set {S i 1 , . . . , S ip } into K n blocks, with K n = R i 1 ,...,ip rn
+ 1, in the same spirit as we did in the proof of Lemma 2. More precisely, there exists a unique K n -tuple of subsets B i ⊂ S n , i ≤ K n , such that the following properties hold: j≤Kn B j = {S i 1 , . . . , S ip }, #B i = r n and max B i < min B i+1 for all i ≤ K n -1. In particular, we have K n ≤ k n and #B Kn = R n -(K n -1) • r n a.s.. Without loss of generality, we assume that #B Kn = #B i = r n for all i ≤ K n -1, so that R i 1 ,...,ip = K n r n . For each j ≤ K n , we also denote by L j the family consisting of the n largest terms of B j and we let L n = j≤Kn L j . In the rest of the paper, we write M B = max s∈B ξ(s) for all subset B ⊂ Z.

Adapting the proof of Lemma 1 in [START_REF] Chenavier | Extremes for transient random walks in random sceneries under weak independence conditions[END_REF], we can show that the following inequalities hold for almost all realization of (S n ) n≥0 and for n larger than some deterministic integer n 0 :

P (Sn) M {S i 1 ,...,S ip } ≤ u n -P (Sn) M {S i 1 ,...,S ip }\Ln ≤ u n ≤ k n n P (ξ > u n ) ; P (Sn) M {S i 1 ,...,S ip }\Ln ≤ u n - j≤Kn P (Sn) M B j \Ln ≤ u n ≤ k n α n, n ; j≤Kn P (Sn) M B j \Ln ≤ u n - j≤Kn P (Sn) M B j ≤ u n ≤ 2 τ k n n n . Since P (ξ > u n ) ∼ n→∞ τ n and F i 1 ,...,ip (u n ) = E P (Sn) M {S i 1 ,...,S ip } ≤ u n , we get sup 0≤i 1 <•••<ip≤n F i 1 ,...,ip (u n ) -E   j≤Kn P (Sn) M B j ≤ u n   -→ n→∞ 0.
Without loss of generality, we assume from now on that P (ξ > u n ) = τ n . We show below that sup

0≤i 1 <•••<ip≤n E   j≤Kn P (Sn) M B j ≤ u n   -E exp - R i 1 ,...,ip n τ -→ n→∞ 0. ( 8
)
To do it, we adapt several arguments of Lemma 2 in [START_REF] Chenavier | Extremes for transient random walks in random sceneries under weak independence conditions[END_REF]. First, we notice that for n large enough, j≤Kn

P (Sn) M B j ≤ u n -exp - R i 1 ,...,ip n τ ≥ exp (K n log(1 -r n P (ξ > u n ))) -exp - R i 1 ,...,ip n τ ≥ exp -K n r n P (ξ > u n ) -K n (r n P (ξ > u n )) 2 -exp - R i 1 ,...,ip n τ ,
where the last line comes from the facts that log(1 -x) ≥ -x -x 2 for |x| small enough and that

r n P (ξ > u n ) -→ n→∞ 0. Because K n r n = R i 1 ,...,ip and P (ξ > u n ) = τ n , we have j≤Kn P (Sn) M B j ≤ u n -exp - R i 1 ,...,ip n τ ≥ exp - R i 1 ,...,ip n τ exp -K n (r n P (ξ > u n )) 2 -1 ≥ exp(-k n (r n P (ξ > u n )) 2 ) -1,
where the last line comes from the fact that

K n ≤ k n a.s.. Since k n r n ∼ n→∞ n, we have j≤Kn P (Sn) M B j ≤ u n -exp - R i 1 ,...,ip n τ ≥ c • 1 k n .
Moreover, because j≤Kn P (Sn) M B j ≤ u n ≤ exp -j≤Kn P (Sn) M B j > u n , it follows from the Bonferroni inequalities (see e.g. p110 in Feller [START_REF] Feller | An introduction to probability theory and its applications[END_REF]) that j≤Kn

P (Sn) M B j ≤ u n ≤ exp   -(K n -1)r n P (ξ > u n ) + j≤Kn α<β;α,β∈B j P (ξ(α) > u n , ξ(β) > u n )   . Since K n r n = R i 1 ,...,ip and P (ξ > u n ) = τ n , we have j≤Kn P (Sn) M B j ≤ u n -exp - R i 1 ,...,ip n τ = exp - R i 1 ,...,ip n τ ×   exp   r n P (ξ > u n ) + j≤Kn α<β;α,β∈B j P (ξ(α) > u n , ξ(β) > u n )   -1  
and therefore j≤Kn

P (Sn) M B j ≤ u n -exp - R i 1 ,...,ip n τ ≤ exp   r n P (ξ > u n ) + j≤Kn α<β;α,β∈B j P (ξ(α) > u n , ξ(β) > u n )   -1.
Proceeding along the same lines as in the proof of Lemma 2 in [START_REF] Chenavier | Extremes for transient random walks in random sceneries under weak independence conditions[END_REF], we can show that exp

  r n P (ξ > u n ) + j≤Kn α<β;α,β∈B j P (ξ(α) > u n , ξ(β) > u n )   -1 ≤ c   1 k n + n n/kn s=1 P (ξ(0) > u n , ξ(s) > u n )   .
Thus, for almost all realization of (S n ) n≥0 , j≤Kn

P (Sn) M B j ≤ u n -exp - R i 1 ,...,ip n τ ≤ c   1 k n + n n/kn s=1 P (ξ(0) > u n , ξ(s) > u n )   ,
This shows (8) by taking the expectations and the triangular inequality. It remains to prove that sup

0≤i 1 <•••<ip≤n E exp - R i 1 ,...,ip τ n -E exp - R i 1 ,...,ip τ n -→ n→∞ 0.
To do it, we write for any ε > 0,

E exp - R i 1 ,...,ip τ n -E exp - R i 1 ,...,ip τ n = E exp - R i 1 ,...,ip τ n -E exp - R i 1 ,...,ip τ n 1 |Ri 1 ,...,ip /n-E[R i 1 ,...,ip /n]|≤ε + E exp - R i 1 ,...,ip τ n -E exp - R i 1 ,...,ip τ n 1 |Ri 1 ,...,ip /n-E[R i 1 ,...,ip /n]|>ε .
The first term of the right-hand of the equality is lower than some function f (ε), with f (ε) -→ ε→0 0 whereas the second one is lower than

2 n -2 ε -2 sup 0≤i 1 <•••<ip≤n V R i 1 ,...,ip
, which converges to 0 as n goes to infinity according to [START_REF] Gnedenko | Sur la distribution limite du terme d'une série aléatoire[END_REF]. This concludes the proof of Lemma 3 by taking first the limit over n → ∞ and then the limit over ε → 0.

As a consequence of Lemma 3, the first and the third terms of the right-hand side of (7) converge to 0 as n goes to infinity. To deal with the second one, we write

exp - R i 1 ,...,ip,j 1 ,...,j p n τ -exp - R i 1 ,...,ip + R j 1 ,...,j p n τ = exp - R i 1 ,...,ip + R j 1 ,...,j p n τ   exp   R j 1 ,...,j p i 1 ,...,ip n τ   -1   ≤ exp   R ip+ n+1,...,n 1,...,ip n τ   -1,
where the last line comes from the fact that j 1 -i p > n , with

R j 1 ,...,j p i 1 ,...,ip = #{S i 1 , . . . , S ip } ∩ {S j 1 , . . . , S j p } = R i 1 ,...,ip + R j 1 ,...,j p -R i 1 ,...,ip,j 1 ,...,j p .
Since n ≥ 0, we get

sup exp - R i 1 ,...,ip,j 1 ,...,j p n τ -exp - R i 1 ,...,ip + R j 1 ,...,j p n τ ≤ sup i≤n exp   R i+1,...,n 1,...,i n τ   -1, (9)
where the supremum in the left-hand side is taken over all integers 0

≤ i 1 < • • • < i p < j 1 < • • • < j p ≤ n, with j 1 -i p > n .
Moreover, using the fact that R i+1,...,n 1,...,i = R 1,...,i + R i+1,...,n -R 1,...,n and following [START_REF] Gall | The range of stable random walks[END_REF], we have sup i≤n R i+1,...,n 1,...,i n -→ n→∞ 0 a.s.. This, together with (9) and the Lebesgue's dominated convergence theorem implies

sup E exp - R i 1 ,...,ip,j 1 ,...,j p n τ -E exp - R i 1 ,...,ip + R j 1 ,...,j p n τ -→ n→∞ 0
and consequently concludes the proof of Proposition 4.

The

D (k) (u n ) as k → ∞
In [START_REF] Chernick | Calculating the extremal index for a class of stationary sequences[END_REF], the authors introduce a local mixing condition, referred to as the D (k) (u n ) condition, which allows to express the extremal index in terms of joint distribution. We recall the latter below.

Condition D (k) (u n ) Let (ξ(s)) s∈Z be a sequence of random variables and let u n be a threshold such that nP (ξ > u n ) -→ n→∞ τ , for some τ > 0. In conjunction with the D(u n ) condition, we say that the D (k) (u n ) condition, k ≥ 1, holds if there exist two sequences of integers (k n ) and ( n ) such that

k n → ∞, k n α n, n → 0, k n n = o(n) and lim n→∞ nP (ξ(1) > u n ≥ M 2,k , M k+1,rn > u n ) = 0, ( 10
)
where r n is as in ( 5) and where M i,j = max{ξ(i), ξ(i + 1), . . . , ξ(j)} for all i ≤ j, with the convention M i,j = -∞ if i > j. As mentioned in [START_REF] Chernick | Calculating the extremal index for a class of stationary sequences[END_REF], Equation ( 10) is implied by the condition

lim n→∞ n rn s=k+1 P (ξ(1) > u n ≥ M 2,k , ξ(s) > u n ) = 0.
Observe that the last line is the

D (u n ) condition if k = 1.
Roughly, the following proposition states that the sequence (ξ(S n )) n≥0 satisfies the D (k) (u n ) condition as k goes to infinity.

Proposition 5. Under the same assumptions as Proposition 4, we have

lim k→∞ lim n→∞ n rn j=k+1 P ξ(S 1 ) > u n ≥ M 2,k , ξ(S j ) > u n = 0,
where M i,j = max i≤t≤j ξ(S t ) if i ≤ j and M i,j = -∞ if i > j.

Proof of Proposition 5. For all k ≥ 1, we have

n rn j=k+1 P ξ(S 1 ) > u n ≥ M 2,k , ξ(S j ) > u n = n rn j=k+1 P ξ(S 1 ) > u n ≥ M 2,k , ξ(S j ) > u n |S j = S 1 P (S j = S 1 ) + n rn j=k+1 P ξ(S 1 ) > u n ≥ M 2,k , ξ(S j ) > u n |S j = S 1 P (S j = S 1 ) . (11)
The first term of the right-hand side of [START_REF] Lucarini | Extremes and Recurrence in Dynamical Systems[END_REF] tends to zero as k, n → ∞. Indeed, To prove that the second term of the right-hand side of [START_REF] Lucarini | Extremes and Recurrence in Dynamical Systems[END_REF] goes to 0, we write

P ξ(S 1 ) > u n ≥ M 2,k , ξ(S j ) > u n |S j = S 1 P (S j = S 1 ) ≤ P (ξ(S 1 ) > u n ) P (S j = S 1 ) .
n rn j=k+1 P ξ(S 1 ) > u n ≥ M 2,k , ξ(S j ) > u n |S j = S 1 P (S j = S 1 ) = n rn j=k+1 P ξ(S 1 ) > u n ≥ M 2,k , ξ(S j ) > u n |S j ∈ B * (S 1 , r n ) P (S j ∈ B * (S 1 , r n )) + n rn j=k+1 P ξ(S 1 ) > u n ≥ M 2,k , ξ(S j ) > u n |S j / ∈ B(S 1 , r n ) P (S j / ∈ B(S 1 , r n )) , ( 12
)
where

B(S 1 , r n ) := {S ∈ S n : |S -S 1 | ≤ r n } and B * (S 1 , r n ) = B(S 1 , r n ) \ {S 1 }.
We prove below that the last two terms in ( 12) converge to 0. For the first one, we write

n rn j=k+1 P ξ(S 1 ) > u n ≥ M 2,k , ξ(S j ) > u n |S j ∈ B * (S 1 , r n ) P (S j ∈ B * (S 1 , r n )) ≤ n rn j=2 P (ξ(0) > u n , ξ(S j -S 1 ) > u n |S j ∈ B * (S 1 , r n )) .
The last quantity converges to 0 as n goes to infinity since the sequence (ξ(s)) s∈Z satisfies the D (u n ) condition. To deal with the second term of (12), we write

n rn j=k+1 P ξ(S 1 ) > u n ≥ M 2,k , ξ(S j ) > u n |S j / ∈ B(S 1 , r n ) P (S j / ∈ B(S 1 , r n )) ≤ n rn j=k+1 P (ξ(S 1 ) > u n , ξ(S j ) > u n |S j / ∈ B(S 1 , r n )) ≤ n rn j=k+1 P (ξ > u n ) 2 + n rn j=k+1 P (ξ(S 1 ) > u n , ξ(S j ) > u n |S j / ∈ B(S 1 , r n )) -P (ξ > u n ) 2 .
The first series tends to 0 as n goes to infinity because

n rn j=k+1 P (ξ > u n ) 2 ≤ nr n P (ξ > u n ) 2 ∼ n→∞ τ 2 r n n ,
and r n = o(n). To deal with the second series, we use the D(u n ) condition. This gives

n rn j=k+1 |P (ξ(S 1 ) > u n , ξ(S j ) > u n |S j / ∈ B(S 1 , r n )) -P (ξ > u n ) 2 | ≤ nr n α n,rn ≤ n 2 k n α n,rn ,
which converges to 0 as n goes to infinity according to (3). This concludes the proof of Proposition 5.

The extremal index

Let (k n ) and (r n ) be as in ( 3) and ( 5). Let us denote by R n = #S n and K n = Rn rn + 1. The following proposition deals with M Sn under the D(u n ) condition. Proposition 6. Let α < 1. Assume that the sequence (ξ(s)) s∈Z satisfies the D(u n ) conditions for a threshold u n such that nP (ξ > u n ) -→ n→∞ τ , with τ > 0. Then for almost all realization of (S n ) n≥0 ,

P (Sn) (M Sn ≤ u n )-exp   - Kn j=1 rn i=1 P (Sn) ξ(S ((j-1)rn+i) ) > u n ≥ M ((j-1)rn+i+1, jrn)   -→ n→∞ 0,
where

M (i,j) :=    max i≤t≤j ξ(S (t) ), i ≤ j -∞, i > j
and where S (t) is the t-th largest value of the ξ(S i ) s, i ≤ n.

A similar result was obtained by O'Brien (Theorem 2.1. in [START_REF] O'brien | Extreme values for stationary and Markov sequences[END_REF]). However, the above proposition is not a consequence of the latter. Proposition 6 remains true if the sequence (ξ(s)) s∈Z only satisfies the D(u n ) condition (i.e. when k n α n, n -→ for some θ ∈ [0, 1], then P (M Sn ≤ u n ) -→ n→∞ e -θτ . In this case, the term θ is referred to as the extremal index (see e.g. [START_REF] Leadbetter | Extremes and local dependence in stationary sequences[END_REF]) and can be interpreted as the reciprocal of the mean size of a cluster of exceedances. As stated in Theorem 1 in [START_REF] Chenavier | Extremes for transient random walks in random sceneries under weak independence conditions[END_REF], when the sequence (ξ(s)) s∈Z satisfies the D(u n ) and D (u n ) conditions,

P (M Sn ≤ u n ) -→ n→∞ e -qτ . ( 13
)
In other words, under these conditions, the extremal index θ exists and θ = q.

Proof of Proposition 6. Let us write S n = {S (1) , . . . , S (Rn) } with S (1) < S (2) < • • • < S (Rn) , and partitition S n into K n blocks as in Lemma 2. Without loss of generality, assume that the last block has the same size as the others, so that Rn Kn is an integer. Let B j = {S ((j-1)rn+1) , . . . , S (jrn) } be the j-th block of size r n . According to Lemma 1 in [START_REF] Chenavier | Extremes for transient random walks in random sceneries under weak independence conditions[END_REF], for almost all realization of (S n ) n≥0 , we have This together with (14) concludes the proof of Proposition 6.

The D (u n ) condition

Recall that, in the classical literature (see e. The following result is an extension of Proposition 3 in [START_REF] Franke | The extremes of random walks in random sceneries[END_REF]. However, we give a simpler proof which is based on [START_REF] Leadbetter | Extremes and local dependence in stationary sequences[END_REF].

P

  Moreover, because (S n ) n≥1 is a transient random walk, we have∞ j=2 P (S j = S 1 ) < ∞, which implies lim k→∞ lim n→∞ nP (ξ(S 1 ) > u n ) rn j=k+1 P (S j = S 1 ) = 0, ξ(S 1 ) > u n ≥ M 2,k , ξ(S j ) > u n |S j = S 1 P (S j = S 1 ) = 0.

n→∞ 0 instead of n 2 P

 2 kn α n, n -→ n→∞ 0). As a direct consequence of such a result, if for almost all realization of (S n ) n≥0 , (Sn) M ((j-1)rn+i+1, jrn) ≤ u n |ξ(S ((j-1)rn+i) ) > u n -→ n→∞ θ,

P 2 ≤

 2 (Sn) (M Sn ≤ u n ) -exp   j≤Kn log 1 -P (Sn) M B j > u n | log(1-x)+x| ≤ Cx2 for |x| small enough and because P (Sn) M B j > u n ≤ r n P (ξ > u n ) converges to 0 as n goes to infinity, we havej≤Kn log 1 -P (Sn) M B j > u n + j≤Kn P (Sn) M B j > u n ≤ j≤Kn log 1 -P (Sn) M B j > u n + P (Sn) M B j > u n ≤ C j≤Kn P (Sn) M B j > u n Ck n r 2 n P (ξ > u n ) 2 .The last term converges to 0 as n goes to infinity sincek n r n ∼ n→∞ n, nP (ξ > u n ) -→ n→∞ τ and r n P (ξ > u n ) -→ n→∞ 0. This shows that for almost all realization of (S n ) n≥0 P (Sn) (M Sn ≤ u n ) -exp   -j≤Kn P (Sn) M B j > u n the same lines as[START_REF] O'brien | Extreme values for stationary and Markov sequences[END_REF], we haveP (Sn) M B j ≤ u n = 1 -P (Sn) M B j > u n = 1 -rn i=1P (Sn) ξ(S ((j-1)rn+i) ) > u n ≥ M ((j-1)rn+i+1, jrn)

P

  g. (3.2.1) in[START_REF] Lucarini | Extremes and Recurrence in Dynamical Systems[END_REF]), the D (u n ) condition holds for the sequence(Z n ) if, in conjunction with the D(u n ) condition, (Z 1 > u n , Z i > u n ) = 0, for some sequence of integers (k n ) such that k n -→ n→∞ ∞, k n α n, n -→ n→∞ 0 and k n n = o(n).

  satisfying Equation[START_REF] Chenavier | Extremes for transient random walks in random sceneries under weak independence conditions[END_REF]. Then Φ n converges weakly to a Poisson point process Φ with intensity measure m [0,1] ⊗ ν, where m [0,1] denotes the Lebesgue measure in [0, 1], i.e. for any Borel subsets B 1 , . . . , B K

Proposition 7. Under the same assumptions as Proposition 4, the sequence (ξ(S n )) n≥0 does not satisfy the D (u n ) condition.

Proof of Proposition 7. On the opposite, if (ξ(S n )) n≥0 satisfies the D (u n ) condition, then P (M Sn ≤ u n ) -→ n→∞ e -τ according to Theorem 1.2 in [START_REF] Leadbetter | Extremes and local dependence in stationary sequences[END_REF]. This contradicts (13) since q = 1.


Nicolas Chenavier
email: nicolas.chenavier@univ-littoral.fr

Christian Y Robert
email: chrobert@ensae.fr

Fixed-domain asymptotic properties of maximum composite likelihood estimators for max-stable Brown-Resnick random fields

Keywords: Brown-Resnick random fields, Composite likelihood estimators, Fixed-domain asymptotics, Gaussian random fields, Poisson random sampling, Delaunay triangulation 62G32, 62M30, 60F05, 62H11

Likelihood inference for max-stable random fields is in general impossible because their finite-dimensional probability density functions are unknown or cannot be computed efficiently. The weighted composite likelihood approach that utilizes lower dimensional marginal likelihoods (typically pairs or triples of sites that are not too distant) is rather favored. In this paper, we consider the family of spatial max-stable Brown-Resnick random fields associated with isotropic fractional Brownian fields. We assume that the sites are given by only one realization of a homogeneous Poisson point process restricted to C = (-1/2, 1/2] 2 and that the random field is observed at these sites. As the intensity increases, we study the asymptotic properties of the composite likelihood estimators of the scale and Hurst parameters of the fractional Brownian fields using different weighting strategies: we exclude either pairs that are not edges of the Delaunay triangulation or triples that are not vertices of triangles.

Introduction

Gaussian random fields are widely used to model spatial data because their finite-dimensional distributions are only characterized by the mean and covariance functions. In general it is assumed that these functions 1 belong to some parametric models which leads to a parametric estimation problem. When extreme value phenomena are of interest and meaningful spatial patterns can be discerned, max-stable random field models are preferred to describe such phenomena. However, likelihood inference is challenging for such models because their corresponding finite-dimensional probability density functions are unknown or cannot be computed efficiently. In this paper we study composite likelihood estimators in a fixed-domain asymptotic framework for a widely used class of stationary max-stable random fields: the Brown-Resnick random fields.

As a preliminary, we provide brief reviews of work on maximum likelihood estimators and on composite likelihood estimators for Gaussian random fields under fixed-domain asymptotics and present max-stable random fields with their canonical random tessellations.

Maximum likelihood estimators for Gaussian random fields under fixed-domain asymptotics

The fixed-domain asymptotic framework is sometimes called infill asymptotics [START_REF] Stein | Interpolation of Spatial Data[END_REF], [START_REF] Cressie | Statistics for Spatial Data[END_REF])

and corresponds to the case where more and more data are observed in some fixed bounded sampling domain (usually a region of R d , d ∈ N * ). Within this framework, the maximum likelihood estimators (MLE) of the covariance parameters of Gaussian random fields have been deeply studied in the last three decades.

It is noteworthy that two types of covariance parameters have to be distinguished: microergodic and non-microergodic parameters. A parameter is said to be microergodic if, for two different values of it, the two corresponding Gaussian measures are orthogonal [START_REF] Ibragimov | Gaussian Random Processes[END_REF], [START_REF] Stein | Interpolation of Spatial Data[END_REF]). It is non-microergodic if, even for two different values of it, the two corresponding Gaussian measures are equivalent. Non-microergodic parameters cannot be estimated consistently under fixed-domain asymptotics. No general results are available for the asymptotic properties of microergodic MLE. Most available results are specific to particular covariance models.

The initial covariance model that has been studied is the exponential model with its variance and scale parameters. When d = 1, only a reparameterized quantity obtained from the variance and scale parameters is microergodic [START_REF] Ying | Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process[END_REF]). It is shown that the MLE of this microergodic parameter is consistent and asymptotically normal. When d > 1 and for a separable exponential covariance function, all the covariance parameters are microergodic, and the asymptotic normality of the MLE is proved in [START_REF] Ying | Maximum likelihood estimation of parameters under a spatial sampling scheme[END_REF]. Other results are also given in van der [START_REF] Van Der Vaart | Maximum likelihood estimation under a spatial sampling scheme[END_REF] and in [START_REF] Abt | Fisher information and maximum-likelihood estimation of covariance parameters in Gaussian stochastic processes[END_REF].

The Matern covariance model (Matern (1960)) is very popular in spatial statistics for its flexibility with respect to the parameterization of smoothness (in the mean square sense) of the underlying Gaussian field. This model has three parameters: the variance, the scale and the smoothness parameters. [START_REF] Zhang | Inconsistent estimation and asymptotically equivalent interpolations in model-based geostatistics[END_REF] showed that when the smoothness parameter is known and fixed, not all parameters can be estimated consistently when d = 1, 2, 3; only the ratio of variance and scale parameters (to the power of the smoothing parameter) is microergodic. [START_REF] Kaufman | The role of the range parameter for estimation and prediction in geostatistics[END_REF] proved strong consistency and provided the asymptotic distributions of the microergodic parameters when estimating jointly the scale and variance parameters (see also [START_REF] Du | Fixed-domain asymptotic properties of tapered maximum likelihood estimators[END_REF] and [START_REF] Wang | On fixed-domain asymptotics and covariance tapering in Gaussian random field models[END_REF]

Maximum composite likelihood estimators for Gaussian random fields under fixeddomain asymptotics

From a theoretical point of view, the maximum likelihood method is the best approach for estimating the covariance parameters of a Gaussian random field. Nevertheless, the evaluation of the likelihood function under the Gaussian assumption requires a computational burden of order O(n 3 ) for n observations (because of the inversion of the n × n covariance matrix), making this method computationally impractical for large datasets. The composite likelihood (CL) methods rather use objective functions based on the likelihood of lower dimensional marginal or conditional events (Varin et al. ( 2011)). These methods are generally appealing when dealing with large data sets or when it is difficult to specify the full likelihood, and provide estimation methods with a good balance between computational complexity and statistical efficiency.

There is not a lot of results under fixed domain asymptotics for maximum CL estimators (MCLE).

However, [START_REF] Bachoc | Composite likelihood estimation for a Gaussian process under fixed domain asymptotics[END_REF] studied the problem of estimating the covariance parameters of a Gaussian process (d = 1) with exponential covariance function. They showed that the weighted pairwise maximum likelihood estimator of the microergodic parameters can be consistent, but also inconsistent, according to the objective function; e.g. the weighted pairwise conditional maximum likelihood estimator is always consistent (and also asymptotically Gaussian). Bachoc and Lagnoux (2020) considered a Gaussian process (d = 1) whose covariance function is parametrized by variance, scale and smoothness parameters. They focused on CL objective functions based on the conditional log likelihood of the observations given the K (resp. L) observations corresponding to the left (resp. right) nearest neighbor observation points. They examined the case where only the variance parameter is unknown and the case where the variance and the spatial scale are jointly estimated. In the first case they proved that for small values of the smoothness parameter, the composite likelihood estimator converges at a sub-optimal rate and they showed that the asymptotic distribution is not Gaussian. For large values of the smoothness parameter, they proved that the estimator converges at the optimal rate.

Fixed-domain asymptotics for non-Gaussian random fields

To the best of our knowledge, there is a few papers that study MLE or MCLE for non-Gaussian random fields under fixed-domain asymptotics. For example, [START_REF] Li | Maximum-likelihood estimation for diffusion processes via closed-form density expansions[END_REF] proposed approximate maximum-likelihood estimation for diffusion processes (d = 1) and provided closed-form asymptotic expansion for transition density. But diffusion processes may not be generalized for d ≥ 2.

Other papers rather considered variogram-based or power variation-based estimators. Chan and Wood (2004) considered a random field of the form g (X), where g : R→R is an unknown smooth function and X is a real-valued stationary Gaussian field on R d (d = 1 or 2) whose covariance function obeys a power law at the origin. The authors addressed the question of the asymptotic properties of variogram-based estimators when g (X) is observed instead of X under a fixed-domain framework. They established that the asymptotic distribution theory for nonaffine g is somewhat richer than in the Gaussian case (i.e. when g is an affine transformation). Although the variogram-based estimators are not MLE or MCLE, this study shows that their asymptotic properties can differ significantly from the Gaussian random field case. Robert (2020) considered a particular class of max-stable processes (d = 1), the class of simple Brown-Resnick max-stable processes whose spectral processes are continuous exponential martingales. He developed the asymptotic theory for the realized power variations of these max-stable processes, that is, sums of powers of absolute increments. He considered a fixed-domain asymptotic setting and obtained a biased central limit theorem whose bias depends on the local times of the differences between the logarithms of the underlying spectral processes.

Max-stable random fields

Max-stable random fields appear as the only possible non-degenerate limits for normalized pointwise maxima of independent and identically distributed (i.i.d.) random fields with continuous sample paths (see e.g.

de [START_REF] De Haan | Extreme Value Theory: An Introduction[END_REF]). The one-dimensional marginal distributions of max-stable fields belong to the parametric class of Generalized Extreme Value distributions. Since we are interested in the estimation of parameters characterizing the dependence structure, we restrict our attention to max-stable random fields η = (η(x)) x∈X on X ⊂R d with standard unit Fréchet margins, that is, satisfying

P [η(x) ≤ z] = exp -z -1
, for all x ∈ X and z > 0.

The max-stability property has then the simple form

n -1 n i=1 η i d = η
where (η i ) 1≤i≤n are i.i.d. copies of η, is the pointwise maximum, and

η(x) = i≥1 U i Y i (x), x ∈ X , (1)
where (U i ) i≥1 is the decreasing enumeration of the points of a Poisson point process on (0, +∞) with intensity measure u -2 du, (Y i ) i≥1 are i.i.d. copies of a non-negative stochastic random field Y on X such that E[Y (x)] = 1 for all x ∈ X , the sequences (U i ) i≥1 and (Y i ) i≥1 are independent.

The spectral representation (1) makes it possible to construct a canonical tessellation of X as in Dombry and [START_REF] Dombry | Random tessellations associated with max-stable random fields[END_REF]. We define the cell associated with each index i ≥ 1 by

C i = {x ∈ X : U i Y i (x) = η(x)}.
It is a (possibly empty) random closed subset of X and each point x ∈ X belongs almost surely (a.s.) to a unique cell (the point process {U i Y i (x)} i≥1 is a Poisson point process with intensity u -2 du so that the maximum η(x) is almost surely attained for a unique i). It is noteworthy that the terms cell and tessellation are meant in a broader sense than in Stochastic Geometry where they originated. Here, a cell is a general (not necessarily convex or connected) random closed set and a tessellation is a random covering of X by closed sets with pairwise disjoint interiors.

Likelihood inference is challenging for max-stable random fields because their finite-dimensional probability density functions are unknown or cannot be computed efficiently. [START_REF] Padoan | Likelihood-based inference for max-stable processes[END_REF] proposed to use a composite-likelihood approach but only discussed asymptotic properties of the estimators when the data-sites are fixed and when there is a large number of i.i.d. data replications.

Contributions of the paper

In this paper, we consider the class of spatial max-stable Brown-Resnick random fields (d = 2) associated with isotropic fractional Brownian random fields as defined in [START_REF] Kabluchko | Stationary max-stable fields associated to negative definite functions[END_REF]. We assume a Poisson stochastic spatial sampling scheme and use the Poisson-Delaunay triangulation to select the pairs and triples of sites with their associated marginal distributions that will be integrated into the CL objective functions (we exclude pairs that are not edges of the Delaunay triangulation or triples that are not vertices of triangles of this triangulation). Note that using the Delaunay triangulation is relatively natural here since we only use the distributions of pairs and triples. Moreover, the Delaunay triangulation appears to be the most "regular" triangulation in the sense that it is the one that maximises the minimum of the angles of the triangles.

We In this paper we consider spatial max-stable random fields (d = 2) and assume that the random field Y introduced in the spectral representation (1) has the following form

Y (x) = exp (W (x) -γ (x)) , x ∈ R 2 .
With this choice, η is a stationary random field while W is not stationary but has (linear) stationary increments (see [START_REF] Kabluchko | Stationary max-stable fields associated to negative definite functions[END_REF]).

Pairwise joint distributions and asymptotic score contributions

Let us consider two sites x 1 , x 2 ∈ R 2 and denote by d = x 2 -x 1 the distance between these sites. Let

z 1 , z 2 ∈ R + , a = σd α/2 , u = log(z 2 /z 1 )/a and v (u) = a/2 + u.
It is well known that the joint probability distribution function of (η (x 1 ) , η (x 2 )) is given by (see e.g. Huser and Davison (2013))

F x 1 ,x 2 (z 1 , z 2 ) = P [η (x 1 ) ≤ z 1 , η (x 2 ) ≤ z 2 ] = exp (-V x 1 ,x 2 (z 1 , z 2 )) ,
where

V x 1 ,x 2 (z 1 , z 2 ) = 1 z 1 Φ (v (u)) + 1 z 2 Φ (v (-u)) , z 1 , z 2 > 0.
Here Φ denotes the cumulative distribution function of the standard Gaussian distribution. The term V x 1 ,x 2 is referred to as the pairwise exponent function. Let us now consider the "normalized" (linear) increments of the logarithm of the Brown-Resnick random field

U = d -α/2 σ -1 log (η (x 2 ) /η (x 1 )) .
The following proposition provides the conditional and marginal distributions of U and allows us to deduce that it has asymptotically a standard Gaussian distribution as the distance d tends to 0. Such a result generalizes Proposition 3 in Robert (2020).

Proposition 1 The conditional distribution of U given η (x 1 ) = η > 0 is characterized by

P [ U ≤ u| η (x 1 ) = η] = exp - 1 η V x 1 ,x 2 (1, e σd α/2 u ) -1 Φ (v (u)) , u ∈ R,
and its marginal distribution by

P [U ≤ u] = Φ (v (u)) V x 1 ,x 2 (1, e σd α/2 u ) , u ∈ R.
It follows that

lim d→0 P [U ≤ u] = Φ (u) , u ∈ R.
The fact that the asymptotic distribution of U (as d tends to 0) is a standard Gaussian distribution is not a surprise since the probability that x 1 and x 2 belong to the same cell of the canonical tessellation of the max-stable random field tends to 1. Indeed, in a common cell, the values of the max-stable random field are generated by the same isotropic fractional Brownian random field. It is natural to first study the asymptotic behaviors of the increment sums for an isotropic fractional Brownian random field before considering a Brown-Resnick random field (see Section 4.2).

The distribution of (η (x 1 ) , η (x 2 )) is absolutely continuous with respect to the Lebesgue measure on (R + ) 2 . Its density function satisfies

f x 1 ,x 2 (z 1 , z 2 ) = ∂ ∂z 1 ∂z 2 F x 1 ,x 2 (z 1 , z 2 ) , z 1 , z 2 ∈ R + ,
and will be used for the contribution of the pair (η (x 1 ) , η (x 2 )) to the pairwise CL function. For any α ∈ (0, 2), σ 2 > 0 and z 1 , z 2 ∈ R + , this joint density function is a differentiable function with respect to (α, σ). The following proposition provides the asymptotic contributions of the pair to the pairwise score functions.

Proposition 2 Let u ∈ R be fixed. Let x 1 , x 2 ∈ R 2 and z 1 , z 2 ∈ R + be such that d -α/2 σ -1 log (z 2 /z 1 ) = u, where d = x 2 -x 1 > 0. Then lim d→0 ∂ ∂σ log f x 1 ,x 2 (z 1 , z 2 ) = 1 σ u 2 -1 , lim d→0 1 log d ∂ ∂α log f x 1 ,x 2 (z 1 , z 2 ) = 1 2 u 2 -1 .
The asymptotic score contributions of a pair are therefore proportional to u 2 -1 . Further, u will be replaced by the normalized increment of log (η) which has asymptotically a standard Gaussian distribution as stated in Proposition 1. This fact ensures that the asymptotic score contributions are asymptotically unbiased.

Triplewise joint distributions and asymptotic score contributions

Let us now consider three sites x 1 , x 2 , x 3 ∈ R 2 and denote by

d 1,2 = x 2 -x 1 , d 1,3 = x 3 -x 1 , d 2,3 =
x 3 -x 2 the distances between two different sites. Let z 1 , z 2 , z 3 ∈ R + and, for i, j = 1, 2, 3 such that i = j, let a i,j = σd α/2 i,j , u i,j = log(z j /z i )/a i,j and v i,j (u) = a i,j /2 + u i,j . The joint probability distribution function of (η (x 1 ) , η (x 2 ) , η (x 3 )) is given by (see e.g. Huser and Davison (2013))

F x 1 ,x 2 ,x 3 (z 1 , z 2 , z 3 ) = P [η (x 1 ) ≤ z 1 , η (x 2 ) ≤ z 2 , η (x 3 ) ≤ z 3 ] = exp (-V x 1 ,x 2 ,x 3 (z 1 , z 2 , z 3 )) , where V x 1 ,x 2 ,x 3 (z 1 , z 2 , z 3 ) = 1 z 1 Φ 2 v 1,2 (u 1,2 ) v 1,3 (u 1,3 ) ; 1 R 1 R 1 1 + 1 z 2 Φ 2 v 1,2 (-u 1,2 ) v 2,3 (u 2,3 ) ; 1 R 2 R 2 1 + 1 z 3 Φ 2 v 1,3 (-u 1,3 ) v 2,3 (-u 2,3 ) ; 1 R 3 R 3 1 (2)
with

R 1 = d α 1,2 + d α 1,3 -d α 2,3 2 (d 1,2 d 1,3 ) α/2 , R 2 = d α 1,2 + d α 2,3 -d α 1,3 2 (d 1,2 d 2,3 ) α/2 , R 3 = d α 1,3 + d α 2,3 -d α 1,2 2 (d 1,3 d 2,3 ) α/2 .
Here Φ 2 (•, Σ) denotes the bivariate cumulative distribution function of the centered Gaussian distribution with covariance matrix Σ. As for the pairs, let us also consider the "normalized" (linear) increments of the logarithm of the Brown-Resnick random field

U 1,2 = d -α/2 1,2 σ -1 log (η (x 2 ) /η (x 1 )) , U 1,3 = d -α/2 1,3 σ -1 log (η (x 3 ) /η (x 1 )) .
The following proposition provides the conditional and marginal distributions of the vector (U 1,2 , U 1,3 ) and allows us to deduce that it has asymptotically a bivariate Gaussian distribution as the distances d 1,2 and d 1,3 tend to 0 proportionally.

Proposition 3

The conditional distribution of (U 1,2 , U 1,3 ) given η (x 1 ) = η > 0 is characterized by

P [ U 1,2 ≤ u 2 , U 1,3 ≤ u 3 |η (x 1 ) = η] = exp - 1 η V x 1 ,x 2 ,x 3 (1, e σd α/2 12 u 2 , e σd α/2 13 u 3 ) -1 × Φ 2 v 1,2 (u 2 ) v 1,3 (u 3 ) ; 1 R 1 R 1 1 , u 1 , u 2 ∈ R,
and its marginal distribution by

P [U 1,2 ≤ u 2 , U 1,3 ≤ u 3 ] = Φ 2 v 1,2 (u 2 ) v 1,3 (u 3 ) ; 1 R 1 R 1 1 V x 1 ,x 2 ,x 3 (1, e σd α/2 12 u 2 , e σd α/2 13 u 3 ) , u 1 , u 2 ∈ R. It follows that, if x 2 -x 1 = δd 1,2 , x 3 -x 1 = δd 1,3 , x 3 -x 2 = δd 2,3
, where d i,j , i = j, is fixed, then

lim δ→0 P [U 1,2 ≤ u 2 , U 1,3 ≤ u 3 ] = Φ 2 u 2 u 3 ; 1 R 1 R 1 1 , u 1 , u 2 ∈ R.
The comment concerning the asymptotic distribution of U also holds for (U 1,2 , U 1,3 ). The probability that x 1 , x 2 and x 3 belong to the same cell of the canonical tessellation of the max-stable random field tends to 1 as δ tends to 0. Therefore the vector (U 1,2 , U 1,3 ) tends to have the same distribution as the vector of normalized linear increments of an isotropic fractional Brownian random field.

The distribution of (η (x 1 ) , η (x 2 ) , η (x 3 )) is absolutely continuous with respect to the Lebesgue measure on (R + ) 3 . Its density function satisfies

f x 1 ,x 2 ,x 3 (z 1 , z 2 , z 3 ) = ∂ ∂z 1 ∂z 2 ∂z 3 F x 1 ,x 2 ,x 3 (z 1 , z 2 , z 3 ) , z 1 , z 2 , z 3 ∈ R + ,
and will be used for the contribution of the triple (η (x 1 ) , η (x 2 ) , η (x 3 )) to the triplewise CL function.

For any α ∈ (0, 2), σ 2 > 0 and z 1 , z 2 , z 3 ∈ R + , this joint density function is a differentiable function with respect to (α, σ). The following proposition provides the asymptotic contributions of the triple (η (x 1 ) , η (x 2 ) , η (x 3 )) to the triplewise score functions.

Proposition 4 Let u 2 , u 3 ∈ R be fixed. Let x 1 , x 2 , x 3 ∈ R 2 and z 1 , z 2 , z 3 ∈ R + be such that δ -α/2 d -α/2 1,2 σ -1 log (z 2 /z 1 ) = u 2 and δ -α/2 d -α/2 1,3 σ -1 log (z 3 /z 1 ) = u 3 , where δd 1,2 = x 2 -x 1 , δd 1,3 = x 3 -x 1 , δd 2,3 = x 3 -x 2 . Then lim δ→0 ∂ ∂σ log f x 1 ,x 2 ,x 3 (z 1 , z 2 , z 3 ) = 1 σ   u 2 u 3 1 R 1 R 1 1 -1 u 2 u 3 -2   , lim δ→0 1 log δ ∂ ∂α log f x 1 ,x 2 ,x 3 (z 1 , z 2 , z 3 ) = 1 2   u 2 u 3 1 R 1 R 1 1 -1 u 2 u 3 -2   .
The asymptotic contributions of a triple are therefore proportional to a quadratic function of (u 2 , u 3 ).

Further, u 2 and u 3 will be replaced by normalized increments of log (η) over a triangle of the Delaunay triangulation (see Section 3.1) which have asymptotically a bivariate standard Gaussian distribution with correlation coefficient R 1 , as stated in Proposition 3. We can also conclude that the asymptotic score contributions are asymptotically unbiased.

If we let δ -α/2 d -α/2 1,2 σ -1 log (z 1 /z 2 ) = ũ1 and δ -α/2 d -α/2 2,3 σ -1 log (z 3 /z 2 ) = ũ3 with fixed ũ1 , ũ3 ∈ R, we also get lim δ→0 ∂ ∂σ log f x 1 ,x 2 ,x 3 (z 1 , z 2 , z 3 ) = 1 σ   ũ1 ũ3 1 R 2 R 2 1 -1 ũ1 ũ3 -2   .
In particular, an invariance property with respect to the choice of the order of the points x 1 , x 2 and x 3 holds for the triplewise score functions. However, it will be necessary to order these points later.

3 The weighted CL approach

The randomized sampling scheme

We assume that the data-sites are given by a realization of a homogeneous Poisson point process of intensity N in R 2 , denoted by P N , which is independent of the Brown-Resnick random field. Let us denote by

C = (-1/2, 1/2]
2 the square where we will consider the sites for the observations of the max-stable random field.

The Delaunay graph Del(P N ) based on P N is our connection scheme and is defined as the unique triangulation with vertices in P N such that the circumball of each triangle contains no point of P N in its interior. With a slight abuse of notation, we identify Del(P N ) to its skeleton. When x 1 , x 2 ∈ P N are Delaunay neighbors, we write x 1 ∼ x 2 in Del(P N ).

For a Borel subset B in R 2 , let E N,B be the set of couples (x 1 , x 2 ) such that the following conditions hold:

x 1 ∼ x 2 in Del(P N ), x 1 ∈ B, and x 1 x 2 ,
where denotes the lexicographic order. When B = C, we only write

E N = E N,C .
For a Borel subset B in R 2 , let DT N,B be the set of triples (x 1 , x 2 , x 3 ) satisfying the following properties ∆(x 1 , x 2 , x 3 ) ∈ Del(P N ), x 1 ∈ B, and

x 1 x 2 x 3 ,
where ∆(x 1 , x 2 , x 3 ) is the convex hull of (x 1 , x 2 , x 3 ). When B = C, we only write DT N = DT N,C .

The weighted CL objective functions and the CL estimators

The (tapered) pairwise CL objective function is defined as

2,N (σ, α) = (x 1 ,x 2 )∈E N log f x 1 ,x 2 (η (x 1 ) , η (x 2 )) ,
while the (tapered) triplewise CL objective function is defined as

3,N (σ, α) = (x 1 ,x 2 ,x 3 )∈DT N log f x 1 ,x 2 ,x 3 (η (x 1 ) , η (x 2 ) , η (x 3 )) .
Thereby, in the CL objective functions, we exclude pairs that are not edges of the Delaunay triangulation or triples that are not vertices of triangles of this triangulation. Restricting the CL objective functions to the most informative pairs and triples for the estimation of the parameters does not modify the approach that follows, but allows us to simplify the presentation and the proofs.

From Section 4.4 of Dombry et al. (2018), we know that there exist families of positive functions (l x 1 ,x 2 ) x 1 ,x 2 ∈R 2 and (l x 1 ,x 2 ,x 3 ) x 1 ,x 2 ,x 3 ∈R 2 with l x 1 ,x 2 : R 2 → R and l x 1 ,x 2 ,x 3 : R 3 → R such that the following Lipschitz conditions hold: for any σ 1 , σ 2 > 0 and α 1 , α 2 ∈ (0, 2)

log f x 1 ,x 2 (z 1 , z 2 ; (σ 2 , α 2 )) f x 1 ,x 2 (z 1 , z 2 ; (σ 1 , α 1 )) ≤ l x 1 ,x 2 (z 1 , z 2 ) (|σ 2 -σ 1 | + |α 2 -α 1 |) and log f x 1 ,x 2 ,x 3 (z 1 , z 2 , z 3 ; (σ 2 , α 2 )) f x 1 ,x 2 ,x 3 (z 1 , z 2 , z 3 ; (σ 1 , α 1 )) ≤ l x 1 ,x 2 ,x 3 (z 1 , z 2,z 3 ) (|σ 2 -σ 1 | + |α 2 -α 1 |) .
Let us denote by (σ 0 , α 0 ) the true parameters. We assume that σ 0 belongs to a compact set S σ of R + and that α 0 belongs to a compact set S α of (0, 2). We can now define the MCLEs of σ and α.

When α 0 is assumed to be known, the pairwise and triplewise maximum (tapered) CL estimators of σ 0 , σj,N , are respectively defined as a solution of the maximization problems max σ∈Sσ j,N (σ, α 0 ) , j = 2, 3.

When σ 0 is assumed to be known, the pairwise and triplewise maximum (tapered) CL estimators of α 0 , αj,N , are respectively defined as a solution of the maximization problems max α∈Sα j,N (σ 0 , α) , j = 2, 3.

Note that the solutions of these maximization problems become unique as N → ∞. This can be viewed from the first-order optimality conditions and the asymptotic approximations of the score functions obtained in Propositions 2 and 4.

Main results

Our aim is to characterize the asymptotic distributions of the MCLEs. We provide intermediate results

for different random fields in order to understand how we obtained the different families of asymptotic distributions of our estimators. We first provide some definitions and notations related to the Poisson-Delaunay triangulation. Then we consider sums of square increments of an isotropic fractional Brownian field on the edges of the Delaunay triangles and provide Central Limit Theorems using Malliavin calculus.

We only consider the case α ∈ (0, 1) for which the asymptotic distributions are Gaussian. This is not a 2018)). Third we consider sums of square increments of the pointwise maximum of two independent isotropic fractional Brownian fields and show that the asymptotic behaviors of the sums now depend on the local time at the level 0 of the difference between the two fractional Brownian fields. Fourth we generalize these results to the max-stable Brown-Resnick random field and, using approximation of the pairwise and triplewise CL objective functions, we derive the asymptotic properties of the MCLEs. (0, ∞). The cell intensity β 2 of Del(P 1 ) is defined as the mean number of cells per unit area, i.e.

Definitions and notations

β 2 = 1 a(B) E [|{C ∈ Del(P 1 ) : z(C) ∈ B}|] .
It is known that β 2 = 2, see e.g. Theorem 10.2.9. in [START_REF] Schneider | Stochastic and Integral Geometry[END_REF]. Then, we define the typical cell as a random triangle C with distribution given as follows: for any positive measurable and translation invariant function g : K 2 → R, we have

E [g(C)] = 1 β 2 a(B) E   C∈Del(P 1 ):z(C)∈B g(C)   ,
where K 2 denotes the set of convex compact subsets in R 2 , endowed with the Fell topology (see Section

E [g(C)] = 1 6 ∞ 0 (S 1 ) 3 r 3 e -πr 2 a(∆(u 1 , u 2 , u 3 ))g(∆(ru 1 , ru 2 , ru 3 ))σ(du 1 )σ(du 2 )σ(du 3 )dr, (3)
where S 1 is the unit sphere of R 2 and σ is the spherical Lebesgue measure on

S 1 with normalization σ S 1 = 2π. It means that C is equal in distribution to R∆(U 1 , U 2 , U 3 )
, where R and (U 1 , U 2 , U 3 ) are independent with probability density functions given respectively by 2π 2 r 3 e -πr 2 and a(∆(u 1 , u 2 , u 3 ))/(12π 2 ).

In a similar way, we can define the notion of typical edge. The edge intensity β 1 of Del(P 1 ) is defined as the mean number of edges per unit area and is equal to β 1 = 3 (see e.g. Theorem 10.2.9. in Schneider and Weil (2008)). The distribution of the length of the typical edge is the same as the distribution of

D = R||U 1 -U 2 ||.
Its probability density function f D satisfies the following equality

P [D ≤ ] = 0 f D (d)dd = π 3 ∞ 0 (S 1 ) 2 r 3 e -πr 2 a(∆(u 1 , u 2 , e 1 ))I [r u 1 -u 2 ≤ ] σ(du 1 )σ(du 2 )dr, (4)
where e 1 = (1, 0) and > 0. Following Eq. ( 3), a typical couple of (distinct) Delaunay edges with a common vertex can be defined as a 3-tuple of random variables (D 1 , D 2 , Θ), where

D 1 , D 2 ≥ 0 and Θ ∈ [-π 2 , π 2
), with distribution given by

P[(D 1 , D 2 , Θ) ∈ B] = 1 6 ∞ 0 (S 1 ) 3 r 3 e -πr 2 a(∆(u 1 , u 2 , u 3 )) × I[(r||u 3 -u 2 ||, r||u 2 -u 1 ||, arcsin (cos(ζ u 1 ,u 2 /2))) ∈ B]σ(du 1 )σ(du 2 )σ(du 3 )dr,
where ζ u 1 ,u 2 is the measure of the angle (u 1 , u 2 ) and where

B is any Borel subset in R 2 + × [-π 2 , π 2
). The random variables D 1 , D 2 (resp. Θ) can be interpreted as the lengths of the two typical edges (resp. as the angle between the edges). In particular, the length of a typical edge is equal in distribution to D = R||U 2 -U 1 || with distribution given in Eq. (4).

Asymptotic distributions of squared increment sums for an isotropic fractional Brownian field

Let (W (x)) x∈R 2 be an isotropic fractional Brownian field where W (0) = 0 a.s. and var (W (x)) = σ 2 x α for some α ∈ (0, 1) and σ 2 > 0. For two sites x 1 , x 2 ∈ R 2 , let us define the normalized increment between

x 1 and x 2 as

U (W ) x 1 ,x 2 = σ -1 d -α/2 1,2 (W (x 2 ) -W (x 1 )) with d 1,2 = x 2 -x 1 .
The (normalized) squared increment sum for the edges of the Delaunay triangulation is given by

V (W ) 2,N = 1 |E N | (x 1 ,x 2 )∈E N (U (W ) x 1 ,x 2 ) 2 -1 ,
while the (normalized) squared increment sum for the pairs of edges of Delaunay triangles is defined as

V (W ) 3,N = 1 |DT N | (x 1 ,x 2 ,x 3 )∈DT N   U (W ) x 1 ,x 2 U (W ) x 1 ,x 3 1 R x 1 ,x 2 ,x 3 R x 1 ,x 2 ,x 3 1 -1 U (W ) x 1 ,x 2 U (W ) x 1 ,x 3 -2   , where R x 1 ,x 2 ,x 3 = corr(U (W ) x 1 ,x 2 , U (W ) x 1 ,x 3 ) = d α 1,2 + d α 1,3 -d α 2,3 2 (d 1,2 d 1,3 ) α/2 , (5)
with

d 1,3 = x 3 -x 1 > 0 and d 2,3 = x 3 -x 2 > 0. Let Ũ (W ) x 1 ,x 2 ,x 3 = (1 -R 2 x 1 ,x 2 ,x 3 ) -1/2 U (W ) x 1 ,x 2 -R x 1 ,x 2 ,x 3 U (W )
x 1 ,x 3 and Ũ (W )

x 1 ,x 3 = U (W ) x 1 ,x 3 .

Note that Ũ (W )

x 1 ,x 2 ,x 3 is a normalized increment based on the three points x 1 , x 2 , x 3 (see e.g. Chan and Wood (2002)) and that corr( Ũ (W )

x 1 ,x 2 ,x 3 , Ũ (W ) x 1 ,x 3 ) = 0. The sum V (W )
3,N may be rewritten as

V (W ) 3,N = 1 |DT N | (x 1 ,x 2 ,x 3 )∈DT N [( Ũ (W ) x 1 ,x 2 ,x 3 ) 2 -1] + [( Ũ (W ) x 1 ,x 3 ) 2 -1] .
The following theorem states that the asymptotic distributions of

V (W ) 2,N and V (W )
3,N are Gaussian. Their asymptotic variances are known, but quite intricate. We provide their integral representations in Section 1 in the Supplementary Material.

Theorem 1 Let α ∈ (0, 1). Then there exist constants σ 2

V 2 > 0 and σ 2 V 3 > 0 such that, as N → ∞,

V (W ) 2,N D → N 0, σ 2 V 2 , V (W ) 3,N D → N 0, σ 2 V 3 .
We note that the rates of convergences of both sums are the same as in Theorem 3.2 of [START_REF] Chan | Increment-based estimators of fractal dimension for two-dimensional surface data[END_REF] or in Theorem 1 of [START_REF] Zhu | Parameter estimation for fractional Brownian surfaces[END_REF] where statistics based on square increments on regular grids have been considered.

Asymptotic distributions of squared increment sums for the (pointwise) maximum of two independent fractional Brownian fields

Let W (1) (x) x∈R 2 and W (2) (x) x∈R 2 be two independent isotropic fractional Brownian fields, where W (1) (0) = W (2) (0) = 0 a.s. and var W (1) (x) = var W (2) (x) = σ 2 x α for some α ∈ (0, 1) and σ 2 > 0.

We denote by W ∨ the pointwise maximum of the two isotropic fractional Brownian fields, i.e.

W ∨ (x) = W (1) (x) ∨ W (2) (x), x ∈ R 2 .
For two distinct sites x 1 , x 2 ∈ R 2 , let

U (W∨) x 1 ,x 2 = σ -1 d -α/2 1,2 (W ∨ (x 2 ) -W ∨ (x 1 )) .
Then we define

V (W∨) 2,N = 1 |E N | (x 1 ,x 2 )∈E N (U (W∨) x 1 ,x 2 ) 2 -1 V (W∨) 3,N = 1 |DT N | (x 1 ,x 2 ,x 3 )∈DT N   U (W∨) x 1 ,x 2 U (W∨) x 1 ,x 3 1 R x 1 ,x 2 ,x 3 R x 1 ,x 2 ,x 3 1 -1 U (W∨) x 1 ,x 2 U (W∨) x 1 ,x 3 -2   ,
where R x 1 ,x 2 ,x 3 is given in Eq. ( 5) .

The main result of this section concerns the asymptotic behaviors of

V (W∨) 2,N and
V (W∨)
3,N . To state it, let us denote the difference between both fractional Brownian fields as W (2\1) (x) = W (2) (x) -W (1) (x) for any x ∈ R 2 . Similarly to Section 5.1 in Robert (2020), we observe that, for any real measurable function f : R → R and for any (

x 1 , x 2 ) ∈ E N , f (U (W∨) x 1 ,x 2 ) = f (U (1) x 1 ,x 2 )I[W (2\1) (x 1 ) < 0] + f (U (2) x 1 ,x 2 )I[W (2\1) (x 1 ) > 0] + Ψ f U (1) x 1 ,x 2 , U (2) x 1 ,x 2 , W (2\1) (x 1 )/(σd α/2 1,2 ) , (6)
where

U (1) x 1 ,x 2 = 1 σd α/2 1,2 W (1) (x 2 ) -W (1) (x 1 ) , U (2) x 1 ,x 2 = 1 σd α/2 1,2 W (2) (x 2 ) -W (2) (x 1 )
and

Ψ f (x, y, w) = (f (y + w) -f (x))I [x -y ≤ w ≤ 0] + (f (x -w) -f (y))I [0 ≤ w ≤ x -y] .
In particular, taking f (u) = H 2 (u) = u 2 -1, for all u ∈ R, and Ψ = Ψ H 2 , the above decomposition implies that

V (W∨) 2,N = V (1) 2,N + V (2) 2,N + V (2/1) 2,N , (7)
where

V (1) 2,N = 1 |E N | (x 1 ,x 2 )∈E N ,W (2\1) (x 1 )<0 (U (1) x 1 ,x 2 ) 2 -1 V (2) 2,N = 1 |E N | (x 1 ,x 2 )∈E N ,W (2\1) (x 1 )>0 (U (2) x 1 ,x 2 ) 2 -1 V (2/1) 2,N = 1 |E N | (x 1 ,x 2 )∈E N Ψ(U (1) x 1 ,x 2 , U (2) x 1 ,x 2 , W (2\1) (x 1 ) /(σd α/2 1,2 )).
To obtain a similar decomposition for the triples, let us denote, for -1 < R < 1, by Ω the following function

Ω(u 1 , v 1 , u 2 , v 2 , w 1 , w 2 ; R) = 1 1 -R 2 [Ψ H 2 (u 1 , v 1 , w 1 ) + Ψ H 2 (u 2 , v 2 , w 2 )] -2 R 1 -R 2 Ψ I (u 1 , v 1 , w 1 ) Ψ I (u 2 , v 2 , w 2 ) -2 R 1 -R 2 [u 1 Ψ I (u 2 , v 2 , w 2 ) + u 2 Ψ I (u 1 , v 1 , w 1 )] I[w 1 < 0] -2 R 1 -R 2 [v 1 Ψ I (u 2 , v 2 , w 2 ) + v 2 Ψ I (u 1 , v 1 , w 1 )] I[w 1 > 0] (8)
with I (u) = u for all u ∈ R. Then we have (see Section 3.3.2 in the Supplementary Material)

V (W∨) 3,N = V (1) 3,N + V (2) 3,N + V (2/1) 3,N , (9)
where

V (1) 3,N = 1 |DT N | (x 1 ,x 2 ,x 3 )∈DT N , W (2\1) (x 1 )<0   U (1)
x 1 ,x 2 U

(1)

x 1 ,x 3 1 R x 1 ,x 2 ,x 3 R x 1 ,x 2 ,x 3 1 -1 U (1) x 1 ,x 2 U (1) x 1 ,x 3 -2   V (2) 3,N = 1 |DT N | (x 1 ,x 2 ,x 3 )∈DT N , W (2\1) (x 1 )>0   U (2) x 1 ,x 2 U (2) x 1 ,x 3 1 R x 1 ,x 2 ,x 3 R x 1 ,x 2 ,x 3 1 -1 U (2) x 1 ,x 2 U (2) x 1 ,x 3 -2   V (2/1) 3,N = 1 |DT N | (x 1 ,x 2 ,x 3 )∈DT N Ω U (1) x 1 ,x 2 , U (1) x 1 ,x 3 , U (2) x 1 ,x 2 , U (2) x 1 ,x 3 , W (2\1) (x 1 ) σd α/2 1,2 , W (2\1) (x 1 ) σd α/2 1,3 ; R x 1 ,x 2 ,x 3 .
An adaptation of the proof of Theorem 1 shows that, for α ∈ (0, 1), as N → ∞, V

,N + V (2) 2,N D → N 0, σ 2 V 2 (1) 2
and

V (1) 3,N + V (2) 3,N D → N 0, σ 2 V 3 . (11)
To obtain the asymptotic behaviors of V The local time of W (2\1) . Let ν (2\1) be the occupation measure of W (2\1) over C defined by

ν (2\1) (A) = C I W (2\1) (x) ∈ A dx, for any Borel measurable set A ⊂ R. Observe that, for any s, t ∈ [0, 1] 2 , ∆(s, t) := E (W (2\1) (s) -W (2\1) (t)) 2 = 2σ 2 s -t α . Because C (∆(s, t)) -1/2 ds is finite for all t ∈ C,
L W (2\1) ( ) = lim ε→0 C 1 √ 2πε exp - 1 2ε W (2\1) (x) - 2 dx or L W (2\1) ( ) = 1 2π lim M →∞ -[M,M ] R e iξ(W (2\1) (x)-) dxdξ, (12)
where the limits hold in L 2 .

The asymptotic behaviors of V (W∨) 2,N and V (W∨) 3,N . Let F 2 be the function defined, for any z ∈ R, by

F 2 (z) = R 2 ×R + Ψ H 2 (x, y, z/d α/2 ) 1 2π e -(x 2 +y 2 )/2 f D (d) dxdydd,
where f D is the density function of the length of the typical edge defined in Eq. ( 4). Let us also define

F 3 (z) = R 4 ×(R + ) 3 Ω(x 1 , y 1 , x 2 , y 2 , z/d α/2 1 , z/d α/2 3 ; R(d 1 , d 2 , d 3 )) × ϕ 2 (x 1 , y 1 ; R(d 1 , d 2 , d 3 )) ϕ 2 (x 2 , y 2 ; R(d 1 , d 2 , d 3 )) × f D 1 ,D 2 ,D 3 (d 1 , d 2 , d 3 ) dx 1 dy 1 dx 2 dy 2 dd 1 dd 2 dd 3 ,
where

ϕ 2 (x, y; R) = 1 2π 1 (1 -R 2 ) exp   - 1 2 x y 1 R R 1 -1 x y   , R(d 1 , d 2 , d 3 ) = d α 1 + d α 3 -d α 2 2 (d 1 d 3 ) α/2 ,
and where f D 1 ,D 2 ,D 3 is the density function of the edge lengths of the typical Delaunay triangle C.

Moreover let

c V 2 = R F 2 (z)dz and c V 3 = R F 3 (z)dz.
The following proposition provides the asymptotic behaviors of V

(2/1) 2,N and V

(2/1) 3,N .

Proposition 5 Let α ∈ (0, 1). Then, as N → ∞, √ → 2 as N → ∞, respectively. As a consequence of the above proposition, we obtain the following result.

3 N -(2-α)/4 V (2/1) 2,N P → c V 2 L W (2\1) (0) √ 2 2 N -(2-α)/4 V (2/1) 3,N P → c V 3 L W (2\1) 3
Theorem 2 Let α ∈ (0, 1). Then, as N → ∞, √ 3 3 N -(2-α)/4 V (W∨) 2,N P → c V 2 L W (2\1) (0) √ 2 2 N -(2-α)/4 V (W∨) 3,N P → c V 3 L W (2\1) (0).
An important observation is that the rates of convergence of V

Y (x) = exp (W (x) -γ (x)) , x ∈ R 2 ,
where (W (x)) x∈R 2 is an isotropic fractional Brownian field satisfying W (0) = 0 a.s. and γ (x) = var (W (x)) /2 = σ 2 x α /2 for some α ∈ (0, 1) and σ 2 > 0.

Let us define, for k = j ≥ 1,

Z k\j (x) = Z k (x) -Z j (x) , x ∈ R 2 ,
where

Z i (x) = log U i + log Y i (x), x ∈ R 2 .
In the same spirit as Dombry and Kabluchko (2018), we build a random tessellation of C, (C k,j ) k =j≥1

where

C k,j =    x ∈ C :Z k (x) Z j (x) > i =j,k Z i (x)    . (13)
If C k,j = ∅, we define for any Borel subset A of R the occupation measure of Z k\j over C k,j by

ν (k\j) (A) = C k,j I Z k\j (x) ∈ A dx.
The associated local time at level 0 is given by L Z k\j (0) := dν (k\j) d (0). If C k,j = ∅, we let L Z k\j (0) := 0.

Let U (η)

x 1 ,x 2 be the (normalized) increment of log (η) defined as

U (η) x 1 ,x 2 = 1 σ x 2 -x 1 α/2 log η(x 2 ) η(x 1
) .

The square increment sums are given respectively by

V (η) 2,N = 1 |E N | (x 1 ,x 2 )∈E N (U (η) x 1 ,x 2 ) 2 -1 V (η) 3,N = 1 |DT N | (x 1 ,x 2 ,x 3 )∈DT N   U (η) x 1 ,x 2 U (η) x 1 ,x 3 1 R x 1 ,x 2 ,x 3 R x 1 ,x 2 ,x 3 1 -1 U (η) x 1 ,x 2 U (η) x 1 ,x 3 -2   ,
where R x 1 ,x 2 ,x 3 is given in Eq. ( 5). As a consequence we deduce that c V 2 is negative.

Asymptotic properties of the MCLEs

We are now able to present the asymptotic properties of σ2 j,N and αj,N for j = 2, 3. Let us recall that the sums of the contributions of the observations to the composite likelihood are proportional to the square increment statistics (see Propositions 2 and 4). Moreover the asymptotic behaviors of these statistics are characterized in Theorem 3.

Theorem 4 Assume that σ 0 belongs to the interior of a compact set of R + , and that α 0 belongs to the interior of a compact set of (0, 1). Then, as N → ∞, √ Several important points have to be highlighted. First the MCLEs of σ 2 0 and α 0 (when the other parameter is known) are consistent in our infill asymptotic setup. They have rates of convergence proportional to N α 0 /4 for σ2 2,N and log (N ) N α 0 /4 for α2,N that differ from the expected rates of convergence N 1/2 and log (N ) N 1/2 as in [START_REF] Zhu | Parameter estimation for fractional Brownian surfaces[END_REF] for the isotropic fractional Brownian field. Second the type of convergence is in probability. The random variables appearing in the limits in Theorem 4 are proportional to a sum of local times. However these local times have unknown distributions and they cannot be estimated from the data since the underlying random fields (Y i ) i≥1 and the point process (U i ) i≥1 are not observed.

In particular, if the spatial data are only observed for a single date, the Gaussian approximation for the MCLEs given in Padoan et al. (2010) (when several independent replications over time of the spatial data are available) should not be used.

The problem of joint parameter estimation of σ 2 0 , α 0 is left for future work, but it is expected that the respective rates of convergence will be modified into N α 0 /4 / log (N ) and N α 0 /4 as suggested by Brouste and Fukasawa (2018) in the case of a fractional Gaussian process (d = 1) observed on a regular grid.

  for tapered MLE as well as Loh et al. (2021) for quadratic variation estimators). For d = 5, Anderes (2010) proved the orthogonality of two Gaussian measures with different Matern covariance functions. In this case, all the parameters are microergodic. The case d = 4 is still open. More recently Bevilacqua et al. (2019) considered the generalized Wendland (GW) covariance model. They characterized conditions for equivalence of two Gaussian measures and they established strong consistency and asymptotic normality of the MLE for the microergodic parameters associated with the GW covariance model. Bevilacqua and Faouzi (2019) considered the generalized Cauchy covariance model that is able to separate the characterizations of the fractal dimension and the long range dependence of the associated Gaussian random fields. They also characterized conditions for the equivalence of two Gaussian measures, and established strong consistency and asymptotic normality of the MLE of the microergodic parameters.

d=

  denotes the equality of finitedimensional distributions. Max-stable random fields are characterized by their spectral representation (see e.g., de Haan (1984), Giné et al. (1990)): any stochastically continuous max-stable process η can be written as

  study for the first time the asymptotic properties of the MCLE of the scale and Hurst parameters of the max-stable Brown-Resnick random fields under fixed domain asymptotics (for only one realization of a Poisson point process). Pairwise and triplewise CL objective functions (considering all pairs and triples) have been proposed for inference for max-stable processes, but the properties of the MCLE have only been studied when the sites are fixed and when there is a large number of independent observations over time of the max-stable random field (see, e.g.,[START_REF] Blanchet | Spatial modeling of extreme snow depth[END_REF],[START_REF] Davison | Statistical modeling of spatial extremes[END_REF] or Huser and Davison (2013)). Note that the tapered CL estimators for max-stable random field excluding pairs that are at a too large distance apart have also been studied in Sang and Genton (2014) (here again with independent observations), but this is the first time that a Delaunay triangulation is used to select the pairs and triples.To obtain the asymptotic distributions of the MCLEs, we proceed in several steps. First we consider sums of square increments of an isotropic fractional Brownian field on the edges of the Delaunay triangles and provide asymptotic results using Malliavin calculus (see Theorem 1).[START_REF] Zhu | Parameter estimation for fractional Brownian surfaces[END_REF] also studied sums of generalized variations for this random field but assumed data-sites on a regular grid. Second we consider sums of square increments of the pointwise maximum of two independent isotropic fractional Brownian fields and show that the asymptotic behaviors of the sums now depend on the local time at the level 0 of the difference between the two fractional Brownian fields (see Theorem 2). Third we generalize these results to the max-stable Brown-Resnick random field which is built as the pointwise maximum of an infinite number of isotropic fractional Brownian fields (see Theorem 3). Using approximations of the pairwise and triplewise CL objective functions, we derive the asymptotic properties of the MCLEs (seeTheorem 4).The family of stationary Brown-Resnick random fields defined in Kabluchko et al. (2019) is presented in Section 2. We also provide the asymptotic distributions of pairs and triples as the distances between sites tend to zero. In Section 3, we introduce the randomized sampling scheme and define the CL estimators of the scale and Hurst parameters. Our main results are stated in Section 4. The proofs and some intermediate results are deferred into a Supplementary Material.

  very restrictive constraint since almost all empirical studies that use the spatial Brown-Resnick random field obtain values for α in this interval (see e.g. Davison et al. (2012), Engelke et al. (2014), Einmahl et al. (2015) or de Fondeville and Davison (

A

  classical object in Stochastic Geometry is the typical cell. To define it, let us consider a Delaunay triangulation Del(P 1 ) based on a homogeneous Poisson point process of intensity 1. With each cell C ∈ Del(P 1 ), we associate the circumcenter z(C) of C. Now, let B be a Borel subset in R 2 with area a(B) ∈ 13

12. 2 in

 2 [START_REF] Schneider | Stochastic and Integral Geometry[END_REF] for the definition). The distribution of C has the following integral representation (see e.g. Theorem 10.4.4. in[START_REF] Schneider | Stochastic and Integral Geometry[END_REF]):

  are investigated. This requires to introduce the notion of local time of W (2\1) .

  it follows from Section 22 in[START_REF] Geman | Occupation densities[END_REF] that the occupation measure ν (2\1) admits a Lebesgue density, referred to as the local time, that we denote byL W (2\1) ( ) := dν (2\1) d ( ) .An immediate consequence of the existence of the local time is the occupation time formula, which states thatC g(W (2\1) (x))dx = R g ( ) L W (2\1) ( ) dfor any Borel function g on R. Adapting the proof of Lemma 1.1 in Jaramillo et al. (2021), we can easily show that, for any ∈ R,

  from the facts that |E N |/N a.s. → 3 and |DT N |/N a.s.

Theorem 3 L(x 1 1 

 311 Let α ∈ (0, 1). Then, as N → ∞, Z k\j (0) .The results in Theorem 3 are quite similar with those in Theorem 2. It can be noted that there is an a.s. finite number of local times L Z k\j (0), j ≥ 1 and k > j, which are positive. This is related to the fact that there is an a.s. finite number of non-empty cells of the canonical tessellation in C.Using theSlivnyak-Mecke formula (see e.g. Theorem 3.2.5 in Schneider-Weil (2008)) and the same arguments as in the proof of Proposition 3 in Robert (2020), we can state that lim ,x 2 )∈E N (U (η) x 1 ,x 2 ) 2 -= 4σE D α/2 ψ with ψ = ∞ 0 uϕ(u) 1/2 -Φ (u) -u Φ (u) Φ (u) /ϕ(u) du -0.094.

3 |EN

 3 |N -(2-α 0 )/4 σ2 2,N --(2-α 0 )/4 log(N ) (α 2,N -α 0 ) -(2-α 0 )/4 log(N ) (α 3,N -α 0 ) P → -c V 3 j≥1 k>j L Z k\j (0) .

2

  The max-stable Brown-Resnick random fields2.1 Definition of the max-stable Brown-Resnick random fieldsThis paper concerns the class of max-stable random fields known as Brown-Resnick random fields. This class of random fields is based on Gaussian random fields with stationary increments and was introduced in Kabluchko et al.(2009). Recall that a random process (W (x)) x∈R d is said to have stationary increments if the law of (W (x + x 0 ) -W (x 0 )) x∈R d does not depend on the choice of x 0 ∈ R d . A prominent example is the isotropic fractional Brownian field where W (0) = 0 a.s. and semi-variogram given by γ (x) = var (W (x)) /2 = σ 2 x α /2 for some α ∈ (0, 2) and σ 2 > 0, where x is the Euclidean norm of x. The parameter σ is called the scale parameter while α is called the range parameter (H = α/2 is also known

as the Hurst parameter and relates to the Hölder continuity exponent of W ). It is noteworthy that W is a self-similar random field with linear stationary increments as presented in Definition 3.3.1 of Cohen and Istas (2013) and it differs from the fractional Brownian sheet which is a self-similar random field with stationary rectangular increments (see e.g. Section 3.3.2 of the same book). Functional limit theorems for generalized variations of this fractional Brownian sheet have been studied in

[START_REF] Pakkanen | Functional limit theorems for generalized variations of the fractional Brownian sheet[END_REF]

, but these theorems cannot be extended to the isotropic fractional Brownian field whose rectangular increments are not stationary.

  are actually the dominant terms. These increments depend on both isotropic fractional Brownian fields and they reveal the local time of W (2\1) at level 0 in the limits. It is also noteworthy that the convergence is now in probability.Let (η (x)) x∈R 2 be a max-stable Brown-Resnick random field such that η(x)= i≥1 U i Y i (x) for any x ∈ R 2 ,where (U i ) i≥1 is a decreasing enumeration of the points of a Poisson point process on (0, +∞) with intensity measure u -2 du, and (Y i ) i≥1 are i.i.d. copies of

	4.4 Asymptotic distributions of squared increment sums for the max-stable Brown-
		Resnick random field
				(W∨) 2,N	and V	(W∨) 3,N	differ from those of V	(W ) 2,N
	and V	(W ) 3,N . The sums of square increments in V 2,N (2/1)	and V	(2/1) 3,N


Victoria Kamp

Maud Pélissier

Dominance shift during the stay-abroad experience: a tip-of-the-tongue experiment

Keywords: attrition, tip-of-the-tongue experiment, lexical retrieval, language dominance

Previous research found that changes in the linguistic and socio-linguistic environment of bilinguals can provoke language dominance shifts. Investigating vocabulary loss permits to find differences in the quality and quantity of linguistic knowledge that deteriorates through attrition processes. This longitudinal study evaluated the effects of study abroad programs on L1 cognate and non-cognate word retrieval. Subjects were French-English bilinguals spending one to two semesters in an English-speaking country. Results of a picture-naming task show that the participants experienced less tip-of-the-tongue states in their L1 than in their L2 and retrieved cognates more easily than non-cognates. Profiling questionnaires helped to evaluate the correlation between those results and their monthly exposure to English.

Introduction

The phenomenon of non-pathological loss of a native or first language (L1) is called attrition and can occur in a bilingual's (or multilingual's) life when he or she acquires and uses an additional language (L2), leading to an impoverishment of the first language [START_REF] Schmid | L1 attrition and the mental lexicon[END_REF]. This provokes changes in the structural aspects of the language and the speaker's proficiency in the first language. Previous research indicates that vocabulary is one of the more vulnerable linguistic abilities [START_REF] Köpke | L'attrition de la première langue en tant que phénomène psycholinguistique[END_REF]. The attrition of the L1 is not only a consequence of the lack of use of it, but also a consequence of translinguistic influences that are entailed by the contact with the second language [START_REF] Kroll | Category interference in translation and picture naming: Evidence for asymmetric connections between bilingual memory representations[END_REF].

During studies abroad, a shift of dominance can ensue: the L1 is used less frequently, and lexical access becomes harder, while L2 proficiency increases. The competition between both languages as well as a dominance switch may lead to L1 attrition. Lexical access has been found to be one of the most vulnerable linguistic skills; and tip-of-the-tongue occurrences hint at a deterred access to the phonological form of the word, while the concept is activated. To account for TOTs in L2, two main frameworks have been used: the "weaker link" hypothesis [START_REF] Gollan | More use almost always means a smaller frequency effect: Aging, bilingualism, and the weaker links hypothesis[END_REF], which postulates that the more restricted use of L2 leads to weakened connections between form and concepts, and language competition or cross-language interference, which assumes that impaired retrieval comes from competition from the other language. This study focused on investigating the effects of studies abroad on L1 vocabulary loss. As it becomes harder to retrieve words in the neglected language, more tip-of-the-tongue states should occur.

Methods

Participants 19 native speakers of French aged 19 to 25 took part in this study. All were university students completing a bachelor or master's degree and had intermediate to advanced proficiency in English. All participants were spending one (n=3) or two semesters (n=16) abroad for studies in an English-speaking country. All participants filled out a questionnaire investigating their background as bilinguals. This was then followed by monthly forms to record the frequency of use for both languages.

Materials and procedure

The participants were tested on L1 word retrieval on three occasions during their stay. A list of 420 stimuli were selected, comprising 180 English noncognate words, 180 French non-cognate words and 180 French-English cognates. All stimuli were low-frequency words (frequency inferior to 50 per million occurrences) and were associated with black line drawings taken from the IPNP [START_REF] Szekely | A new on-line resource for psycholinguistic studies[END_REF] or drawn in the same style by the authors. Six balanced sets of 120 words were created, three in each language. Cognates were the same in English and French equivalent sets. The pictures were shown one by one on the screen and the participant had to name them as fast as possible while the experimenter recorded possible TOTs. We predicted that participants would experience more TOTs in French by the end of the year, fewer TOTs in English, and name cognates more accurately than non-cognates.

Analysis and results

Analyses were run in R (R Core Team, 2022) with linear mixed-effect models and generalised linear mixed-effect. Accuracy data was analysed with a binomial linear mixed-effect model with a logit link with Accuracy as a dependent variable and Language, Cognate status, Session and their interaction as fixed effects. Random effects included an intercept by Participant and an intercept by Item.

Analyses on the TOT accuracy data showed a significant effect of Language (β =-0,58, SE = 0.07, z = 7.69, p < .001), as participants experienced more TOTs in English than in French (English: M = 9.31 %; French: M = 5.67 %). The effect of Cognate status (cognate or non-cognate) was also significant (β = 0.86, SE = 0.12, z = 6.95, p < .001) as TOTs were experienced more frequently when naming non-cognate pictures than cognate ones (cognates: M = 5.31 %; non-cognates: M = 9.70 %). Finally, there was also a significant effect of Session: performance in session 1 was significantly different from session 2 and session 3 (β = -0,43, SE = 0.06, z = 7.31, p < .001). The analysis showed that there was a significant interaction between Session and Cognate status (β = 0,41, SE = 0.12, z = 3.49, p < .001; see Figure 1). Pairwise comparisons showed that the cognate effect was present in each session (all ps < .001), but that the effect was twice as big in session 3 (β = 1.269) than in session 1 (β = 0.594) (see Figure 1).

Discussion

This study investigated the L1 vocabulary loss of French-English bilinguals spending a year abroad.

Results of this exploratory experiment are in line with previous studies on two points: there was a facilitation effect linked to the cognate status of the stimuli and participants experienced more TOTs in their L2.

However, the results were inconclusive in relation to the starting hypothesis, given that the number of TOTs decreased over the course of the year more for cognates than for non-cognates, no matter the language. Several reasons can explain the results that contradict the starting hypothesis. For one, the pool of participants was not ideal in the sense that three of them only stayed for one semester abroad and that three participants were French Language teachers in their host country. Additionally, two semesters have proven to be too short a time to have any conclusive results regarding loss of lexical access in L1, especially when the reality of study abroad programs is that students tend to speak more of their L1 than anticipated.

As for the difference between the first session and the two that followed, one explanation can be that the experimenter was not as precise in the way she registered the answers as she was getting used to the task and the process and tended to confuse real tip-of-the-tongues with a "feeling of knowing". If the experiment was to be redone, it would also be important to have the option to record quickly resolved TOTs so as to have more accurate data.

The fact that the TOT rate diminished more for cognates than for noncognates could point to habituation to the task. It could also suggest that there is increased non-selective activation [START_REF] Marian | Competing activation in bilingual language processing: Within-and between-language competition[END_REF], due to more balanced bilingualism as the participants' proficiency in French does not decrease while their proficiency in English increases. Additional research is needed to investigate this possibility.

Figure 1 .

 1 Figure 1. TOT rate as a function of Cognate status and Session (in both languages)