Anne Sabourin

Johan Segers

MARGINAL STANDARDIZATION OF UPPER SEMICONTINUOUS PROCESSES. WITH APPLICATION TO MAX-STABLE PROCESSES

Keywords: extreme-value theory, max-stable processes, semicontinuous processes, copulas

Extreme-value theory for random vectors and stochastic processes with continuous trajectories is usually formulated for random objects all of whose univariate marginal distributions are identical. In the spirit of Sklar's theorem from copula theory, such marginal standardization is carried out by the pointwise probability integral transform. Certain situations, however, call for stochastic models whose trajectories are not continuous but merely upper semicontinuous (usc). Unfortunately, the pointwise application of the probability integral transform to a usc process does in general not preserve the upper semicontinuity of the trajectories. In the present work, we give sufficient conditions for marginal standardization of usc processes to be possible, and we state a partial extension of Sklar's theorem for usc processes. We specialize the results to max-stable processes whose marginal distributions and normalizing sequences are allowed to vary with the coordinate.

Introduction

A common way to describe multivariate max-stable distributions is as follows: their margins are univariate max-stable distributions; after standardization of the marginal distributions to a common one, the joint distribution has a specific representation, describing the dependence structure. The separation into margins and dependence is in line with Sklar's theorem [START_REF] Sklar | Fonctions de répartition à n dimensions et leurs marges[END_REF], which provides a decomposition of a multivariate distribution into its margins and a copula, that is, a multivariate distribution with standard uniform margins. If the margins of the original distribution are continuous, the copula is unique and can be found by applying the probability integral transform to each variable. Conversely, to recover the original distribution, it suffices to apply the quantile transformation to each copula variable. Although it is more common in extreme-value theory to standardize to the Gumbel or the unit-Fréchet distribution rather than to the uniform distribution, the principle is the same. The advantage of breaking up a distribution into its margins and a copula is that both components can be modelled separately.

Date: December 21, 2016.

Applications in spatial statistics have spurred the development of extreme-value theory for stochastic processes. If the trajectories of the process are continuous almost surely, then the process can be reduced to a process with standardized margins and continuous trajectories via the probability integral transform applied to each individual variable. Conversely, the original process can be recovered from the standardized one by applying the quantile transform to each standardized variable. Moreover, these maps, sending one continuous function to another one, are measurable with respect to the sigma-field on the space of continuous functions that is generated by the finite-dimensional cylinders. These results hinge on the following two properties. First, the marginal distributions of a stochastic process with continuous trajectories depend continuously on the index variable. Second, the distribution of the random continuous function associated to the process is determined by the finite-dimensional distributions.

For stochastic processes with upper semicontinuous (usc) trajectories, however, the two properties mentioned above do not hold: the marginal distributions need not depend in a continuous way on the index variable, and the distribution of path functionals such as the supremum of the process is not determined by the finite-dimensional distributions of the process. Still, max-stable processes with usc trajectories have been proposed as models for spatial extremes of environmental variables [START_REF] Davison | Geostatistics of extremes[END_REF]; [START_REF] Huser | Space-time modelling of extreme events[END_REF]; [START_REF] Schlather | Models for stationary max-stable random fields[END_REF]. As in the continuous case, construction of and inference on such models is carried out by a separation of concerns regarding the margins and the dependence structure. However, up to date, there is no theoretical foundation for such an approach. Another possible application of max-stable usc process is random utility maximization when the alternatives range over a compact metric space rather than a finite set [START_REF] Mcfadden | Econometric models of probabilistic choice[END_REF][START_REF] Mcfadden | Econometric modeling of locational behavior[END_REF]; [START_REF] Resnick | Random usc functions, max-stable processes and continuous choice[END_REF].

The present paper aims to fill the gap in theory and develop a framework for marginal standardization for stochastic processes with usc trajectories. For the mathematical framework, we follow [START_REF] Norberg | Semicontinuous processes in multi-dimensional extreme value theory[END_REF] and [START_REF] Resnick | Random usc functions, max-stable processes and continuous choice[END_REF] and we work within the space USC(D) of usc functions on a locally compact subset D of some Euclidean space. The space USC(D) is equipped with the hypo-topology: a usc function is identified with its hypograph, a closed subset of D × R; the hypotopology on USC(D) is then defined as the trace topology inherited from the Fell hit-and-miss topology on the space F of closed subsets of D×R [START_REF] Salinetti | On the convergence in distribution of measurable multifunctions (random sets) normal integrands, stochastic processes and stochastic infima[END_REF]; [START_REF] Vervaat | Stationary self-similar extremal processes and random semicontinuous functions[END_REF].

The theory is specialized to max-stable usc processes. Our definition of maxstability allows the shape parameter of the marginal distributions to vary with the index variable of the stochastic process. As a consequence, the stabilizing affine transformations in the definition of max-stability may depend on the index variable too. However, the coordinatewise affine transformation of a usc function does not necessarily produce a usc function, so that care is needed in the formulation of the definition and the results. In de [START_REF] De Haan | A spectral representation for max-stable processes[END_REF]; [START_REF] Resnick | Random usc functions, max-stable processes and continuous choice[END_REF], in contrast, the marginal distributions are assumed to be Fréchet with unit shape parameter, so that the stabilizing sequences in the definition of max-stability are the same for all margins. The possibility to reduce the general case to this simpler case is taken for granted, but, as argued in the paper, for usc processes, it is not.

Section 2 sets up the necessary background about usc processes. A general class of measurable transformations on the space of usc functions is introduced in Section 3. Under regularity conditions on the marginal distributions, this class includes the pointwise probability integral transform and its inverse. This property allows us to state a partial generalization of Sklar's theorem for usc processes in general (Section 4) and for max-stable ones in particular (Section 5). Section 6 concludes. Some additional results are deferred to the appendices.

Random usc processes

We review some essential definitions and properties of random usc processes, or usc processes for short. The material in this section may for instance be found in [START_REF] Salinetti | On the convergence in distribution of measurable multifunctions (random sets) normal integrands, stochastic processes and stochastic infima[END_REF], [START_REF] Beer | Topologies on Closed and Closed Convex Sets[END_REF] (Chapter 5), Vervaat (1988a), Vervaat and Holwerda (1997) and [START_REF] Molchanov | Theory of Random Sets[END_REF](Chapter 1.1 and Appendix B).

Let D be a non-empty, locally compact subset of some finite-dimensional Euclidean space. A function

x : D → [-∞, ∞] is upper semicontinuous (usc) if and only lim sup n→∞ x(s n ) ≤ x(s) whenever s n → s ∈ D. An equivalent property is that the set {s ∈ D : α ≤ x(s)} is closed for each α ∈ R.
The definition of lower semicontinuous (lsc) functions is similar, with a lim inf rather than a lim sup and with the inequalities being reversed.

For x : D → [-∞, +∞], the epigraph and hypograph are commonly defined by

epi x = {(s, α) ∈ D × R : x(s) ≤ α}, hypo x = {(s, α) ∈ D × R : α ≤ x(s)}.
Observe that epi x and hypo x are subsets of D × R, even if the range of x contains -∞ or +∞. A function is upper or lower semicontinuous if and only if its hypograph or epigraph, respectively, is closed. Let USC(D) be the collection of all upper semicontinuous functions from D into [-∞, ∞]. By identifying the function z ∈ USC(D) with the set hypo z ⊂ D × R, any topology on the space F = F (D × R) of closed subsets of D × R results in a trace topology on the space of usc functions.

The Fell hit-and-miss topology on F is defined as follows. Let K and G denote the families of compact and open subsets of D × R, respectively. A base for the Fell topology on F is the family of sets of the form

F K G 1 ,...,Gn = {F ∈ F : F ∩ K = ∅, F ∩ G 1 = ∅, . . . , F ∩ G n = ∅} for K ∈ K and G 1 , . . . , G n ∈ G.
A net of closed sets converges to a limit set F if and only if every compact K missed by F is eventually also missed by the net and if every open G hit by F is eventually also hit by the net.

The Fell topology on F (D × R) induces a trace topology on USC(D), the hypotopology. Subsets of USC(D) which are open or closed with respect to the hypotopology will be referred to as hypo-open or hypo-closed, respectively. A sequence x n in USC(D) is said to hypo-converge to x in USC(D) if and only if hypo x n converges to hypo x in the Fell topology. Since the underlying space D × R is locally compact, Hausdorff, and second countable (LCHS), the space USC(D) thus becomes a compact, Hausdorff, second-countable space (see e.g. [START_REF] Molchanov | Theory of Random Sets[END_REF], Appendix, Theorem B.2). A convenient pointwise criterion for hypo-convergence is the following: a sequence x n ∈ USC(D) hypo-converges to x ∈ USC(D) if, and only if, ∀s ∈ D :

∀s n ∈ D, s n → s : lim sup n→∞ x n (s n ) ≤ x(s); ∀s ∈ D : ∃s n ∈ D, s n → s : lim inf n→∞ x n (s n ) ≥ x(s). (2.1)
Let (Ω, A, Pr) be a complete probability space and let B(F ) be the Borel σ-field on F = F (D × R) generated by the Fell topology. A random closed set of D × R is a Borel measurable map from Ω into F . Similarly, a usc process ξ on D is a Borel measurable map from Ω into USC(D). Equivalently, hypo ξ is a random closed subset of D × R taking values in the collection of all closed hypographs. Some authors use the term 'normal integrand' to refer to such processes, as a special case of stochastic processes with usc realizations (Salinetti and Wets (1986), p. 12). In this work, a usc process is always a function ξ :

D × Ω → [-∞, ∞] such that the map Ω → F : ω → hypo ξ( • , ω) is Borel measurable.
The capacity functional of hypo ξ is the map T : K → [0, 1] defined by

T (K) = Pr{(hypo ξ) ∩ K = ∅}, K ∈ K.
We also call T the capacity functional of ξ rather than of hypo ξ. Since D × R is LCHS, the Borel σ-field B(F ) is generated by the sets

F K = {F ∈ F : F ∩ K = ∅}, K ∈ K.
( [START_REF] Molchanov | Theory of Random Sets[END_REF], p. 2). The collection of complements of the sets F K being a π-system, the capacity functional determines the distribution of a random closed set or a usc process. We emphasize that the Borel σ-field on USC(D) is strictly larger than the one generated by the finite-dimensional cylinders. Without additional hypotheses, the finite-dimensional distributions of a usc process do not determine its distribution as a usc process. The evaluation mappings USC(D) → [-∞, ∞] : x → x(s) being hypo-measurable, a usc process ξ is also a stochastic process (i.e., a collection of random variables) with usc trajectories. The converse is not true, however: for such stochastic processes, the map Ω → USC(D) : ω → ξ( • , ω) is not necessarily hypo-measurable; see [START_REF] Rockafellar | Variational analysis[END_REF], Proposition 14.28, for a counterexample. Measurability of maps to or from USC(D) with respect to the Borel σ-field induced by the hypo-topology will be referred to as hypo-measurability.

Transformations of usc functions

Obviously, a usc function remains usc if multiplied by a positive, continuous function. More generally, we will need to prove that mappings derived from the probability integral transform or the quantile transform are hypo-measurable mappings from USC(D) to itself. To do so, we will rely on the function class U(D) defined and studied next.

Definition 3.1. A function U : D × [-∞, ∞] → [-∞, ∞]
(i) If z n hypo-converges to z in USC(D) and if s n → s in D as n → ∞, then lim sup n→∞ U(s n , z n (s n )) ≤ U(s, z(s)). (ii) For every z ∈ USC(D), the function U * (z) : D → [-∞, ∞] defined by (U * (z))(s) = U(s, z(s)) belongs to USC(D) too. (iii) For every compact K ⊂ D × R, the set {z ∈ USC(D) : hypo U * (z) ∩ K = ∅} is hypo-closed. (iv) The map U * : USC(D) → USC(D) is hypo-measurable. Proof. (i) Consider two cases: z(s) = ∞ and z(s) < ∞.
Suppose first that z(s) = ∞. By the assumptions on U, we have

lim sup n→∞ U(s n , z n (s n )) ≤ lim sup n→∞ U(s n , ∞) ≤ U(s, ∞) = U(s, z(s)).
Suppose next that z(s) < ∞. Let y > z(s). By hypo-convergence, we have lim sup n→∞ z n (s n ) ≤ z(s); see (2.1). As a consequence, there exists a positive integer n(y) such that z n (s n ) ≤ y for all n ≥ n(y). By the properties of U, we have lim sup

n→∞ U(s n , z n (s n )) ≤ lim sup n→∞ U(s n , y) ≤ U(s, y).
Since y > z(s) was arbitrary and since x → U(s, x) is non-decreasing and rightcontinuous, we find lim

sup n→∞ U(s n , z n (s n )) ≤ inf y>z(s) U(s, y) = U(s, z(s)). (ii) In statement (i), set z n = z to see that lim sup n→∞ U(s n , z(s n )) ≤ U(s, z(s)) whenever s n → s in D as n → ∞.
(iii) Let K ⊂ D × R be compact. Suppose that z n hypo-converges to z in USC(D) as n → ∞ and that hypo U * (z n ) ∩ K = ∅ for all positive integers n. We need to prove that hypo U * (z) ∩ K = ∅ too.

For each positive integer n, there exists (

s n , x n ) ∈ K such that U(s n , z n (s n )) ≥ x n . Since K is compact, we can find a subsequence n(k) such that (s n(k) , x n(k) ) → (s, x) ∈ K as k → ∞. By (i), we have U(s, z(s)) ≥ lim sup k→∞ U(s n(k) , z n(k) (s n(k) )) ≥ lim sup k→∞ x n(k) = x.
As a consequence, (s, x) ∈ hypo U * (z) ∩ K.

(iv) The collection of sets of the form {z ∈ USC(D) : hypo(z) ∩ K = ∅}, where K ranges over the compact subsets of D × R, generates the Borel σ-field on USC(D) induced by the hypo-topology. By (iii), the inverse image under U * of each such set is hypo-closed and thus hypo-measurable. We conclude that U * is hypo-measurable.

Remark 3.1. In part (b) of the definition of U(D), we did not include x = -∞. The reason is that this case is automatically included: by Proposition 3.1(ii) applied to the function z(s) ≡ -∞, the map s → U(s, -∞) is necessarily usc too.

Condition (b) is also necessary for the conclusion of Proposition 3.1(ii) to hold: given x, define z(s) ≡ x.

A convenient property of U(D) is that it is closed under an appropriate kind of composition. This allows the deconstruction of complicated transformations into more elementary ones.

Lemma 3.1. If U and V belong to U(D), the function W defined by (s, x) → V (s, U(s, x)) belongs to U(D) too, and

W * = V * • U * . Proof. First, fix s ∈ D. The map x → V (s, U(s, x)) is non-decreasing: for -∞ ≤ x ≤ y ≤ ∞, we have U(s, x) ≤ U(s, y) and thus V (s, U(s, x)) ≤ V (s, U(s, y)).
The map x → V (s, U(s, x)) is also right-continuous: if x n converges from the right to x as n → ∞, then so does u n = U(s, x n ) to u = U(s, x) and thus V (s, u n ) to V (s, u).

Next, fix x ∈ R ∪ {∞}. We need to show that the function s → V (s, U(s, x)) is usc. But the function z defined by s → U(s, x) is usc, and, by Proposition 3.1, so is the function s → V (s, z(s)) = V (s, U(s, x)).

Example 3.1. Here are some simple examples of functions U in the class U(D) and the associated mappings

U * : USC(D) → USC(D). (i) If f : [-∞, ∞] → [-∞, ∞] is non-decreasing and right-continuous, the func- tion (s, x) → f (x) belongs to U(D). The associated map USC(D) → USC(D) is z → f • z. (ii) If y ∈ USC(D), the functions (s, x) → x ∨ y(s) and (s, x) → x ∧ y(s) both belong to U(D). The associated maps USC(D) → USC(D) are z → z ∨ y and z → z ∧ y, respectively. (iii) If a : D → (0, ∞) is continuous, then the function (s, x) → a(s) x belongs to U(D). If b : D → R is usc, then the map (s, x) → x + b(s) belongs to U(D).
The associated maps USC(D) → USC(D) are z → az and z → z + b, respectively.

A more elaborate example of a function in U(D) is induced by the collection of right-continuous quantile functions of the marginal distributions of a stochastic process with usc trajectories. See Appendix A for some background on rightcontinuous quantile functions.

Lemma 3.2. Let ξ = (ξ(s) : s ∈ D) be a stochastic process indexed by D and with values in [-∞, ∞]. Let F s : x ∈ [-∞, ∞] → F s (x) = Pr[ξ(s) ≤ x] denote the right-continuous marginal distribution function of ξ(s). Define Q s (p) = sup{y ∈ R : F s (y) ≤ p}, (s, p) ∈ D × [0, 1]. If ξ has usc trajectories, the function (s, x) → Q s ((x ∨ 0) ∧ 1) belongs to U(D).
Proof. First, by Lemma A.1, the map [0, 1] ∋ p → Q s (p) is non-decreasing and right-continuous for every s ∈ D, and hence the same is true for the map

[-∞, ∞] ∋ x → Q s ((x ∨ 0) ∧ 1). Second, fix x ∈ [-∞, ∞] and write p = (x ∨ 0) ∧ 1 ∈ [0, 1]. We need to show that the map s → Q s (p) is usc. Let s n → s in D as n → ∞; we need to show that Q s (p) ≥ lim sup n→∞ Q sn (p). Let Q ′ be
the right-continuous quantile function of the random variable lim sup n→∞ ξ(s n ). The trajectories of ξ are usc, and thus

ξ(s) ≥ lim sup n→∞ ξ(s n ), which implies Q s (p) ≥ Q ′ (p). By Lemma A.2, we then find Q s (p) ≥ Q ′ (p) ≥ lim sup n→∞ Q sn (p), as required.

Sklar's theorem for usc processes

A d-variate copula is the cumulative distribution function of a d-dimensional random vector with standard uniform margins. [START_REF] Sklar | Fonctions de répartition à n dimensions et leurs marges[END_REF] (I) For every random vector (U 1 , . . . , U d ) with uniform components and for every vector F 1 , . . . , F d of univariate distribution functions, the random vector

(Q 1 (U 1 ), . . . , Q d (U d )) has marginal distributions F 1 , . . . , F d
, where Q j is the (right-or left-continuous) quantile function corresponding to F j . (II) Every random vector (X 1 , . . . , X d ) can be represented in this way. We investigate up to what extent these statements hold for usc processes too. According to Proposition 4.1, the first statement remains true provided the sections s → Q s (p) of the right-continuous quantile functions are usc. According to Proposition 4.2, the Sklar representation is valid for usc processes whose marginal distribution functions have usc sections, and even then, the equality in distribution is not guaranteed. In Section 5, we will specialize the two propositions to max-stable processes.

Proposition 4.1 (à la Sklar I for usc processes). Let Z be a usc process having standard uniform margins. Let (F s : s ∈ D) be a family of (right-continuous) distribution functions and let

Q s (p) = sup{x ∈ R : F s (x) ≤ p} for all (s, p) ∈ D × [0, 1]. Define a stochastic process ξ by ξ(s) = Q s ((Z(s) ∨ 0) ∧ 1) for s ∈ D.
Then the following two statements are equivalent:

(i) ξ is a usc process with marginal distributions given by F s .

(ii) For every p ∈ [0, 1], the function s → Q s (p) is usc.

Proof. For every s ∈ D, the random variable (Z(s) ∨ 0) ∧ 1 is equal almost surely to Z(s), so that its distribution is uniform on [0, 1] too. By Lemma A.1(iv), the distribution function of ξ(s) is given by F s . As a consequence, Q s is the rightcontinuous quantile function of the random variable ξ(s).

If (i) holds, then the trajectories s → ξ(s) are usc. By Lemma 3.2, the function

s → Q s (p) is usc for every p ∈ [0, 1].
Conversely, if (ii) holds, then the function U defined by (s, x) → Q s ((x ∨ 0) ∧ 1) belongs to U(D) defined in Definition 3.1. Let U * be the associated map USC(D) → USC(D); see Proposition 3.1. Then ξ = U * (Z) is a usc process since Z is a usc process and U * is hypo-measurable by Proposition 3.1(iv).

Proposition 4.2 (à la Sklar II for usc processes). Let ξ be a usc process. Let

F s (x) = Pr[ξ(s) ≤ x] for x ∈ [-∞, ∞] and let Q s (p) = sup{x ∈ R : F s (x) ≤ p} for p ∈ [0, 1]. Suppose the following two conditions hold: (a) For every s ∈ D, the distribution of ξ(s) has no atoms in [-∞, ∞]. (b) For every x ∈ R ∪ {+∞}, the function s → F s (x) is usc.
Then the following statements hold:

(i) The process Z defined by Z(s) = F s (ξ(s)) is a usc process with standard uniform margins. (ii) The process ξ defined by ξ(s Proof. By condition (a), we have Pr[ξ(s) < x] = F s (x) for each fixed (s, x), so that by Lemma A.1(iii), we have

) = Q s (Z(s)) = Q s (F s (ξ(s))
{s ∈ D : x ≤ Q s (p)} = {s ∈ D : F s (x) ≤ p} for every x ∈ [-∞, ∞] and p ∈ [0, 1]. Since ξ is a usc process, the function s → Q s (p) is usc for p ∈ [0, 1] (Lemma 3.2).
The set in the display is thus closed. But then the map s → F s (x) is lower semicontinuous, and thus, by (b), continuous.

Remark 4.2. Condition (a) in Proposition 4.2, continuity of the marginal distributions, can perhaps be avoided using an additional randomization device, 'smearing out' the probability masses of any atoms, as in [START_REF] Rüschendorf | On the distributional transform, Sklar's theorem, and the empirical copula process[END_REF]. However, even if this may ensure uniform margins, it may destroy upper semicontinuity, see Example 4.1. Since our interest is in extreme-value theory, in particular in max-stable distributions (Section 5), which are continuous, we do not pursue this issue further. The examples below illustrate certain aspects of Propositions 4.1 and 4.2. In the examples, we do not go into measurability issues, that is, we do not prove that the mappings ξ from the underlying probability space into USC(D) are hypomeasurable. To do so, one can for instance rely on [START_REF] Molchanov | Theory of Random Sets[END_REF](Example 1.2 on page 3 and Theorem 2.25 on page 37).

ξ(s) =      |s|X if -1 ≤ s < 0, 0 if s = 0, sY if 0 < s ≤ 1. Condition (b) in Proposition 4.2 is fulfilled: the trajectories s → F s (x) = Pr[ξ(s) ≤
x] are continuous for every x = 0 and still usc for x = 0. The marginal probability integral transform produces Z(s) = X for s < 0 and Z(s) = Y for s > 0, while Z(0) = F 0 (0) = 1. The process Z has usc trajectories but Z(0) is not uniformly distributed. If one wanted to modify the definition of Z at 0 so that the modified version, Z, would still have usc trajectories and such that Z(0) would be uniformly distributed, then one would need to have Z(0) ≥ X ∨ Y , contradicting the uniformity assumption. We conclude that it is impossible to write ξ(s) as Q s ( Z(s)), where Z is a usc process with standard uniform margins.

ξ(s) = X ∨ (Y 1 {1} (s)) = X if s = 1, X ∨ Y if s = 1.
Then ξ is a usc process and the distribution function, F s , of ξ(s) is given by

F s (x) = Pr[ξ(s) ≤ x] = x if s = 1, x 2 if s = 1,
where x ∈ [0, 1]. As a consequence, the function s → F s (x) is lsc but not usc if 0 < x < 1, so that condition (b) in Proposition 4.2 is violated. Reducing to standard uniform margins yields

Z(s) = F s (ξ(s)) = X if s = 1, (X ∨ Y ) 2 if s = 1.
The event {0 < Y 2 < X < 1} has positive probability, and on this event we have (X ∨ Y ) 2 < X, so that the trajectory s → Z(s) is not usc. Hence, statement (i) in Proposition 4.2 fails.

Some additional examples are postponed to Appendix B, which show that Condition (b) in Proposition 4.2 is not necessary, and that the usc processes ξ and ξ in Proposition 4.2 may be different in law as random elements of USC(D).

Max-stable processes

We apply Propositions 4.1 and 4.2 to max-stable usc processes. These processes and their standardized variants are introduced in Subsection 5.2, after some preliminaries on generalized extreme-value distributions in Subsection 5.1. The main results are given in Subsection 5.3, followed by some examples in Subsection 5.4.

Generalized extreme-value distributions. The distribution function of the generalized extreme-value (GEV) distribution with parameter vector

θ = (γ, µ, σ) ∈ Θ = R × R × (0, ∞) is given by F (x; θ) = exp[-{1 + γ(x -µ)/σ} -1/γ ] if γ = 0 and σ + γ(x -µ) > 0, exp[-exp{-(x -µ)/σ}] if γ = 0 and x ∈ R. (5.1)
The corresponding quantile function is

Q(p; θ) = µ + σ[{-1/ log(p)} γ -1]/γ if γ = 0, µ + σ log{-1/ log(p)} if γ = 0, (5.2) for 0 < p < 1. The support is equal to the interval {x ∈ R : σ + γ(x -µ) > 0}.
In particular, the lower endpoint is equal to

Q(0; θ) = -∞ if γ ≤ 0 and Q(0; θ) = µ -σ/γ if γ > 0.
For every n, there exist unique scalars a n,θ ∈ (0, ∞) and b n,θ ∈ R such that the following max-stability relation holds:

F n (a n,θ x + b n,θ ; θ) = F (x; θ),
x ∈ R.

(5.3)

In fact, a non-degenerate distribution is max-stable if and only if it is GEV. This property motivates the use of such distributions for modeling maxima over many variables. The location and scale sequences are given by

a n,θ = n γ , b n,θ = (σ -γµ)(n γ -1)/γ if γ = 0, σ log n if γ = 0.
(5.4)

For quantile functions, max-stability means that

Q(p 1/n ; θ) = a n,θ Q(p; θ) + b n,θ , p ∈ [0, 1]. (5.5) Note that Q(e -1 ; θ) = µ and thus Q(e -1/n ; θ) = a n µ + b n .
In Sklar's theorem, the uniform distribution on [0, 1] plays the role of pivot. Here, it is natural to standardize to a member of the GEV family. Multiple choices are possible. We opt for the unit-Fréchet distribution, given by Φ(x) = F (x; 1, 1, 1) = 0 if x ≤ 0, e -1/x if x > 0.

(5.6)

The unit-Fréchet quantile function is given by Q(p; 1, 1, 1) = -1/ log(p) for 0 < p < 1. If the law of X is unit-Fréchet, then the law of Q(Φ(X); θ) is GEV(θ). Consider now a stochastic process {ξ(s), s ∈ D}, such that ξ(s) follows a GEV distribution with parameter θ depending on s, so that Pr[ξ(s) ≤ x] = F (x; θ(s)), with F as in (5.1). In view of the conditions in Propositions 4.1 and 4.2, the following lemma is relevant. for every x ∈ [-∞, ∞] and p ∈ [0, 1]; note that GEV distribution functions are continuous functions of x. Since the map s → Q(p; θ(s)) is usc, the above sets are closed. But then the map s → F (x; θ(s)) must be lsc, and thus continuous. Finally, the map sending a GEV distribution to its parameter is continuous with respect to the weak topology (see Lemma D.1). Hence, θ is continuous. 5.2. Max-stable usc processes. Max-stable usc processes have been defined in the literature [START_REF] Molchanov | Theory of Random Sets[END_REF]; [START_REF] Vervaat | Stationary self-similar extremal processes and random semicontinuous functions[END_REF][START_REF] Vervaat | Random upper semicontinuous functions and extremal processes[END_REF] as usc processes whose distribution is invariant under the componentwise maximum operation, up to an affine rescaling involving constants which are not allowed to depend upon the index variable s ∈ D. Likewise, the processes in de [START_REF] De Haan | A spectral representation for max-stable processes[END_REF] and [START_REF] Resnick | Random usc functions, max-stable processes and continuous choice[END_REF] and have marginal distributions which are Fréchet with shape parameter γ ≡ 1 and lower endpoint µ -γ/σ ≡ 0, so that the scaling sequences are a n ≡ n and b n ≡ 0. Moreover, max-stability is defined in de [START_REF] De Haan | A spectral representation for max-stable processes[END_REF] in terms of finite-dimensional distributions only. Definition 5.1 below extends these previous approaches by allowing for an index-dependent rescaling, through scaling functions a n and b n , and by viewing the random objects as random elements in USC(D).

Definition 5.1. A usc process ξ is max-stable if, for all integer n ≥ 1, there exist functions a n : D → (0, ∞) and b n : D → R such that, for each vector of n independent and identically distributed (iid) usc processes ξ 1 , . . . , ξ n with the same law as ξ, we have (5.7)

A max-stable usc process ξ * is said to be simple if, in addition, its marginal distributions are unit-Fréchet as in (5.6). In that case, the norming functions are given by a n (s) = n and b n (s) = 0 for all n ≥ 1 and s ∈ D, i.e., for iid usc processes ξ * 1 , . . . , ξ * n with the same law as ξ * , we have

n i=1 ξ * i d = nξ * in USC(D).
(5.8)

In (5.7) and (5.8), the meaning is that the induced probability distributions on the space USC(D) equipped with the sigma-field of hypo-measurable sets are equal. In Definition 5.1, it is implicitly understood that the functions a n and b n are such that the right-hand side of (5.7) still defines a usc process. If a n is continuous and b n is usc, then this is automatically the case; see Lemma 3.1 and Example 3.1(iii).

Equation (5.7) is not necessarily the same as saying that ( n i=1 ξ i -b n )/a n is equal in distribution to ξ. The reason is that it is not clear that ( n i=1 ξ i -b n )/a n is a usc process. Whether this is the case or not remains an open question.

The evaluation map USC(D) → [-∞, ∞] : z → z(s) is hypo-measurable for all s ∈ D. Equation (5.7) then implies the following distributional equality between random variables:

n i=1 ξ i (s) d = a n (s) ξ(s) + b n (s), s ∈ D.
As a consequence, the marginal distribution of ξ(s) is max-stable and therefore GEV with some parameter vector θ(s) ∈ Θ. The normalizing functions a n and b n must then be of the form a n (s) = a n,θ(s) and b n (s) = b n,θ(s) as in (5.4). If θ : D → Θ is continuous, then the normalizing functions are continuous too, and, by Remark 5.1, the usc process ( n i=1 ξ i -b n )/a n is well-defined and equal in distribution to ξ. 5.3. Sklar's theorem for max-stable usc processes. We investigate the relation between general and simple max-stable usc processes via the pointwise probability integral transform and its inverse. Max-stability of usc processes is defined in (5.7) via an equality of distributions on USC(D) rather than of finite-dimensional distributions. It is therefore not clear from the outset that max-stability is preserved by pointwise transformations.

Proposition 5.1 gives a necessary and sufficient condition on the GEV margins to be able to construct a general max-stable usc process starting from a simple one. Propositions 5.2 and 5.3 treat the converse question, that is, when can a max-stable usc process be first reduced to a simple one and then be reconstructed from it.

Proposition 5.1 (à la Sklar I for max-stable usc processes). Let ξ * be a simple max-stable usc process. Let θ : D → Θ. Define a stochastic process ξ by ξ(s) = Q(Φ(ξ * (s)); θ(s)) for s ∈ D. Then the following two statements are equivalent:

(i) ξ is a usc process with marginal distributions GEV(θ(s)).

(ii) For every p ∈ [0, 1], the function s → Q(p; θ(s)) is usc. If these conditions hold, then ξ is a max-stable usc process with normalizing functions a n (s) = a n,θ(s) and b n (s) = b n,θ(s) .

Proof. By Proposition 4.2(i) applied to ξ * , the stochastic process Z(s) = Φ(ξ * (s)) induces a usc process whose margins are uniform on [0, 1]. The equivalence of statements (i) and (ii) then follows from Proposition 4.1.

Assume that (i) and (ii) are fulfilled. We need to show that the right-hand side in (5.7) defines a usc process and that the stated equality in distribution holds.

For positive integer n, define

U n (s, x) = Q(Φ(nx); θ(s)) for (s, x) ∈ D×[-∞, ∞].
In view of Lemma 3.1 and Example 3.1, the map U n belongs to U(D). Moreover, max-stability (5.5) implies

U n (s, x) = Q(Φ(x) 1/n ; θ(s)) = a n,θ(s) Q(Φ(x); θ(s)) + b n,θ(s) . It follows that a n,θ(s) ξ(s) + b n,θ(s) = U n (s, ξ * (s)).
By Proposition 3.1, the function s → U n (s, z(s)) belongs to USC(D) for every z ∈ USC(D), and the map U n, * from USC(D) to itself sending z ∈ USC(D) to this function is hypo-measurable. We conclude that a n ξ + b n is a usc process. Next, we prove that ξ is max-stable. Let ξ 1 , . . . , ξ n be iid usc processes with the same law as ξ. Further, let ξ * 1 , . . . , ξ * n be iid usc processes with the same law as ξ * . For every i ∈ {1, . . . , n}, we have

ξ i d = ξ = U 1, * (ξ * ) d = U 1, * (ξ * i ).
The last equality in distribution comes from the hypo-measurability of U 1, * . By independence, it follows that

(ξ 1 , . . . , ξ n ) d = (U 1, * (ξ * 1 ), . . . , U 1, * (ξ * n )) . Write ( ξ1 , . . . , ξn ) = (U 1, * (ξ * i ), . . . , U 1, * (ξ * n ))
. By monotonicity, we have, for s ∈ D,

n i=1 ξi (s) = n i=1 Q Φ(ξ * i (s)); θ(s) = Q Φ( n i=1 ξ * i (s)); θ(s) .
Since ξ * is a simple max-stable usc process, we have

n i=1 ξ * i d = nξ * in USC(D). But then also n i=1 ξ i d = n i=1 ξi d = Q Φ(nξ * ); θ = Q Φ(ξ * ) 1/n ; θ = a n,θ Q Φ(ξ * ); θ + b n,θ = a n,θ ξ + b n,θ .
We conclude that ξ is max-stable, as required.

Proposition 5.2 (à la Sklar II for usc processes with GEV margins). Let ξ be a usc process with GEV(θ(s)) margins for s ∈ D. If θ : D → Θ is continuous, then the following statements hold: (i) The process ξ * defined by ξ * (s) = -1/ log F (ξ(s); θ(s)) is a usc process with unit-Fréchet margins. (ii) The process ξ defined by ξ(s) = Q(Φ(ξ * (s)); θ(s)) is a usc process and, with probability one,

∀s ∈ D : ξ(s) = ξ(s) if ξ * (s) < ∞, ∞ if ξ * (s) = ∞.
Proof. The marginal distributions of ξ are GEV and depend continuously on s. Conditions (a) and (b) in Proposition 4.2 are therefore satisfied. By Proposition 4.2(i), the process Z defined by Z(s) = F (ξ(s); θ(s)) is a usc process with standard uniform margins. It then follows that ξ * defined by ξ * (s) = -1/ log Z(s) is a usc process too, its margins being unit-Fréchet.

Since 0 ≤ Z(s) ≤ 1 by construction, we have 0 ≤ ξ * (s) ≤ ∞ and thus Φ(ξ * (s)) = Z(s). We find ξ(s) = Q(Z(s); θ(s)) = Q(F (ξ(s); θ(s)); θ(s)). By Proposition 4.2(ii), the process ξ is a usc process.

Recall that GEV distribution functions are continuous and strictly increasing on their domains. As a consequence, for all x such that x ≥ Q(0; θ(s)), we have

Q(F (x; θ(s)); θ(s)) = x if F (x; θ(s)) < 1, ∞ if F (x; θ(s)) = 1.
Moreover, Lemma C.1 implies that ξ ≥ Q(0; θ) almost surely. Since ξ * (s) < ∞ if and only if F (ξ(s); θ(s)) < 1, we arrive at the stated formula for ξ.

Proposition 5.3 (à la Sklar II for max-stable processes). Let ξ be a usc process with GEV(θ(s)) margins for s ∈ D. Assume that for every compact K ⊂ D, we have sup s∈K F (ξ(s); θ(s)) < 1 with probability one. As in Proposition 5.2, define two usc processes ξ * and ξ by ξ * (s) = -1/ log F (ξ(s); θ(s)) and ξ(s) = Q(Φ(ξ * (s)); θ(s)), for s ∈ D. Then, almost surely, ξ = ξ. Furthermore, the following two statements are equivalent: (i) The usc process ξ is max-stable.

(ii) The usc process ξ * is simple max-stable.

Proof. The fact that ξ * is a usc process with unit-Fréchet margins is a consequence of the continuity of θ and Proposition 5.2(i). Let (K n ) n∈N be a compact cover of D; the existence of a compact cover is guaranteed by the fact that D is locally compact and second countable, whence Lindelöf. We have Pr[∀n ∈ N : sup s∈Kn F (ξ(s); θ(s)) < 1] = 1 by assumption and thus also Pr[∀s ∈ D : F (ξ(s); θ(s)) < 1] = 1. Proposition 5.2(ii) implies that ξ = ξ almost surely.

The functions s → a n,θ(s) and s → b n,θ(s) are continuous. The map D × [-∞, ∞] → [-∞, ∞] defined by (s, x) → a n,θ(s) x + b n,θ(s) belongs to U(D). By Proposition 3.1, the map from USC(D) to itself sending z to the function s → a n,θ(s) z(s) + b n,θ(s) is well-defined and hypo-measurable. It follows that a n,θ ξ + b n,θ is a usc process.

Suppose first that (ii) holds. By Proposition 5.1, the usc process ξ is max-stable with norming functions s → a n,θ(s) and s → b n,θ(s) . Since ξ and ξ are equal almost surely in USC(D), they are also in equal in law. Statement (i) follows.

Conversely, suppose that (i) holds. Let (ξ * i ) n i=1 and (ξ i ) n i=1 be vectors of iid usc processes with common laws equal to the ones of ξ * and ξ, respectively. For z ∈ USC(D), write -1/ log F (z, θ) = (-1/ log F (z(s), θ(s)) s∈D . The mapping z → -1/ log F (z, θ) from USC(D) to itself is hypo-measurable, by an argument as in the proof of Proposition 5.2. For i ∈ {1, . . . , n}, we have, in USC(D),

ξ * i d = ξ * = -1/ log F (ξ, θ) d = -1/ log F (ξ i , θ).

By independence, we thus have (ξ

* i ) n i=1 d = (-1/ log F (ξ i , θ)) n i=1
. Property (i) and max-stability (5.3) now say that

n i=1 ξ * i d = -1/ log F ( n i=1 ξ i ; θ) d = -1/ log F (a n,θ ξ + b n,θ ; θ) = -1/ log{F (ξ; θ) 1/n } = n [-1/ log F (ξ; θ)] = nξ * .
We conclude that ξ * is simple max-stable.

Remark 5.2 (Regarding the finiteness of ξ * in Proposition 5.2(ii)). Recall that usc functions reach their suprema on compacta. Let (K n ) n∈N be a compact cover of D. For every z ∈ USC(D), we have z(s) < ∞ for all s ∈ D if and only if sup s∈Kn z(s) < ∞ for all n ∈ N. The event n∈N {sup s∈Kn ξ * (s) < ∞} is thus the same as the event {∀s ∈ D : ξ * (s) < ∞}.

Remark 5.3 (Regarding the continuity assumption on θ in Proposition 5.2). According to Lemmas 3.2 and 5.1, imposing the continuity of the GEV parameter vector θ(s) as a function of s ∈ D is equivalent to imposing the upper semicontinuity of the function s → F (x, θ(s)) for each fixed x ∈ R.

Examples.

In comparison to Proposition 4.2, we have added to Proposition 5.2 the assumption that the margins be GEV. Although their distribution functions are continuous and strictly increasing on their support, this does not resolve the issues arising when the marginal distributions are not continuous in space, as in Example 4.2. In Example 5.1, which parallels Example 4.2, the GEV parameter function s → θ(s) is not continuous and pointwise standardization to unit Fréchet margins produces a stochastic process whose trajectories are no longer usc almost surely.

Example 5.1 (What may happen without the continuity of θ). Consider two independent, unit-Fréchet distributed variables X and Y . As in Example 4.2, take D = [0, 2], and define

ξ(s) = X ∨ (Y 1 {1} (s)) = X if s = 1, X ∨ Y if s = 1.
Then again, ξ is a usc process. It is even a max-stable one with normalizing functions a n ≡ n and b n ≡ 0. The marginal distribution functions are

F (x; θ(s)) = Pr[ξ(s) ≤ x] = e -1/x if s = 1, e -2/x if s = 1,
where x ≥ 0. Then the function s → F s (x) is lower rather than upper semicontinuous, and the marginal GEV parameter vector is

θ(s) = (γ(s), µ(s), σ(s)) = (1, 1, 1) if s = 1, (1, 2, 2) if s = 1,
which is not continuous as a function of s. Standardizing to Fréchet margins yields

ξ * (s) = -1/ log F (ξ(s); θ(s)) = X if s = 1, (X ∨ Y )/2 if s = 1.
The event {0 < X < Y < 2X} has positive probability, and on this event, we have (X ∨ Y )/2 = Y /2 < X, so that the trajectory s → ξ * (s) is not usc.

To conclude, we present a construction principle for simple max-stable usc processes. In combination with Proposition 5.1, this provides a device for the construction of max-stable usc processes with arbitrary GEV margins. The method is similar to the one proposed in Theorem 2 in [START_REF] Schlather | Models for stationary max-stable random fields[END_REF]. Proving maxstability of the usc process that we construct requires special care, since maxstability in USC(D) does not follow from max-stability of the finite-dimensional distributions.

Example 5.2. Let Y 1 > Y 2 > Y 3 > . . . denote the points of a Poisson point process on (0, ∞) with intensity measure y -2 dy. Let V, V 1 , V 2 , . . . be iid usc processes, independent of the point process (Y i ) i . Assume that V satisfies the following properties:

• Pr[inf s∈D V (s) ≥ 0] = 1;

• E[sup s∈D V (s)] < ∞;

• the mean function f (s) = E[V (s)] is strictly positive and continuous on D. By Lemma C.3 below, inf D V is indeed a random variable. Note that we do not impose that inf D V > 0 almost surely.

Define a stochastic process ξ on D by

ξ(s) = sup i≥1 Y i 1 f (s) V i (s), s ∈ D.
We will show that ξ is 'almost surely' a simple max-stable usc process, in the following sense: Set E = USC(D) + \ {0 D }, where 0 D denotes for the null function on D. Since USC(D) is a compact space with a countable basis (Molchanov, 2005, Theorem B.2, p. 399), the space E is locally compact with a countable basis. Classical theory of point processes applies and it is possible to define Poisson processes on E by augmentation and/or continuous mappings.

Put W i = V i /f . Then for i ∈ N, W i is a random element of USC(D) such that W i ∈ E with probability one. The point process Γ defined by

Γ = i≥1 δ (Y i ,W i ) .
is thus a Poisson process on (0, ∞) × E with mean measure dΛ(y, w) = y -2 dy ⊗ dP W (w), where P W is the law of W 1 [START_REF] Resnick | Extreme Values, Regular Variation, and Point Processes[END_REF], Proposition 3.8).

The 'product' mapping T : (0, ∞) × E → E defined by T (y, w) = yw, for y > 0 and w ∈ E, is hypo-measurable. Provided that the image measure µ = Λ • T -1 is finite on compact sets of E, we find that the point process

Π = i≥1 δ (Y i W i )
is a Poisson process with mean measure µ on E [START_REF] Resnick | Extreme Values, Regular Variation, and Point Processes[END_REF], Proposition 3.7).

To check finiteness of µ on compact sets of E, we must check that for K ⊂ D and x > 0, writing

F K×{x} = {z ∈ USC(D) + : sup s∈K z(s) ≥ x},
we have µ(F K×{x} ) < ∞. Indeed, the set F K×{x} is hypo-closed in USC(D), and since USC(D) is compact, the compact sets in E are the hypo-closed sets F in USC(D) such that F ⊂ E. Thus F K×{x} is compact. Also, any compact set in E must be contained in such a F K×{x} . Now,

µ(F K×{x} ) = -Λ (r, w) ∈ (0, ∞) × USC(D) : r max s∈K w(s) ≥ x = E ∞ 0 1 r ≥ inf s∈K x W (s) dr r 2 = 1 x E sup s∈K j W (s) ≤ 1 x E[sup D V ] inf D f ,
which is finite by assumption on V .

To show (a), we adapt an argument of [START_REF] Giné | Max-infinitely divisible and max-stable sample continuous processes[END_REF](proof of Theorem 2.1). For fixed K ⊂ D compact and x > 0, we have µ(F K×{x} ) < ∞ and thus Π(F K×{x} ) < ∞ almost surely. Thus, there exists a set Ω 1 of probability one, such that the following two statements hold for all ω ∈ Ω 1 :

(i) W i ( • , ω) ∈ E for all i ∈ N; (ii) Π(F K×{x} )(ω) < ∞ for every rational x > 0 and every compact rectangle K ⊂ D with rational vertices.

Take ω ∈ Ω 1 . We need to show that for s ∈ D and x ∈ Q such that ξ(s, ω) < x, we have lim sup t→s ξ(t, ω) ≤ x. Fix s and x as above, which implies that x > 0. Let K n ց {s} be a collection of compact rational rectangles as above such that s is in the interior of K n . Then F {(s,x)} = n∈N F Kn×{x} (the inclusion '⊂' is immediate; the inclusion '⊃' is obtained by choosing for z ∈ n∈N F Kn×{x} and for n ∈ N a point s n ∈ K n such that z(s n ) ≥ x and then observing that s n → s and thus z(s) ≥ x by upper semicontinuity). From our choice of Ω 1 , we have Π(F K 1 ×{x} , ω) < ∞, so that the downward continuity property of the measure Π( • , ω) applies and

Π(F {(s,x)} , ω) = lim n→∞ Π(F Kn×{x} , ω).
By our choice of s, x, and ω, the left-hand side in the display is zero. Since the sequence on the right-hand side is integer valued, there exists n 0 such that for all n ≥ n 0 , we have Π(F Kn×{x} , ω) = 0. This implies that for n ≥ n 0 , we have

sup t∈Kn Y i (ω)W i (t, ω) < x, i ∈ N.
Complete the proof of (a) by noting that lim sup t→s ξ(t, ω) ≤ sup

t∈Kn 0 ξ(t, ω) = sup i∈N sup t∈Kn 0 Y i (ω)W i (t, ω) ≤ x.
To show (b), we need to show that, for any compact K ⊂ D × R, the set

A = {ω ∈ Ω 1 : hypo ξ( • , ω) ∩ K = ∅} is a measurable subset of Ω. Notice first that ξ( • , ω) ∈ USC(D) + for ω ∈ Ω 1 ; use property (i) of Ω 1 . It follows that A = Ω 1 as soon as K is not a subset of D × (0, ∞). Assume K ⊂ D × (0, ∞). On Ω 1 , we have hypo ξ ∩ K = ∅ if and only if Π({z ∈ E : hypo z ∩ K = ∅}) ≥ 1. Now F K = {z ∈ E : hypo z ∩ K = ∅} is a hypo-measurable subset of E, so that X = Π(F K ) is a random variable. It follows that A = X -1 ([1, ∞]) ∩ Ω 1 is measurable, yielding (b).
A standard argument yields that the margins of ξ are unit-Fréchet. To show that ξ is simple max-stable, we need to show that, for independent random copies ξ 1 , . . . , ξ n of ξ, the capacity functionals of n i=1 ξ i and nξ are the same. That is, we need to show that, for every compact set in D × R, we have

Pr[hypo( n i=1 ξ) ∩ K = ∅] = Pr[hypo(nξ) ∩ K = ∅].
(5.9)

The left-hand side is equal to Pr

[( n i=1 hypo ξ i ) ∩ K = ∅] = Pr[∀i = 1, . . . , n : hypo ξ i ∩ K = ∅] = (Pr[hypo ξ ∩ K = ∅]) n .
(5.10)

Without loss of generality, we may assume that K = p j=1 (K j × {x j }), where p ∈ N and where K j ⊂ D is compact and x j is real, for j ∈ {1, . . . , p}. Indeed, the capacity functional of a usc process is entirely determined by its values on such compacta [START_REF] Molchanov | Theory of Random Sets[END_REF], p. 340). We may also choose x j > 0, since otherwise both sides of (5.9) vanish. For such a set K, we have

Pr[hypo ξ ∩ K = ∅] = Pr Π z ∈ E : ∃ 1 ≤ j ≤ p, max K j z ≥ x j = 0 = exp -Λ (r, w) ∈ (0, ∞) × E : ∃j ≤ p , r max s∈K j w(s) ≥ x j = exp -E ∞ 0 1 ∃j : r ≥ min s∈K j x j W (s) dr r 2 = exp -E ∞ 0 1 r ≥ min j min s∈K j x j W (s) dr r 2 = exp -E max j max s∈K j W (s)
x j .

For ξ replaced by nξ, we obtain the same result, but with x j replaced by x j /n. In view of (5.10), the desired equality (5.9) follows.

Conclusion

The aim of the paper has been to extend Sklar's theorem from random vectors to usc processes. We have stated necessary and sufficient conditions to be able to construct a usc process with general margins by applying the pointwise quantile transformation to a usc process with standard uniform margins (Propositions 4.1 and 5.1). Furthermore, we have stated sufficient conditions for the pointwise probability integral transform to be possible for usc processes (Propositions 4.2, 5.2 and 5.3). These conditions imply in particular that the marginal distribution functions are continuous with respect to the space variable (Lemma 4.1). We have also provided several examples of things that can go wrong when these conditions are not satisfied. However, finding necessary and sufficient conditions remains an open problem.

The motivation has been to extend the margins-versus-dependence paradigm used in multivariate extreme-value theory to max-stable usc processes. The next step is to show that marginal standardization is possible in max-domains of attraction too. One question, for instance, is whether the standardized weak limit of the pointwise maxima of a sequence of usc processes is equal to the weak limit of the pointwise maxima of the sequence of standardized usc processes [START_REF] Resnick | Extreme Values, Regular Variation, and Point Processes[END_REF], Proposition 5.10). Interesting difficulties arise: weak convergence of finitedimensional distributions does not imply and is not implied by weak hypoconvergence; Khinchin's convergence-of-types lemma does not apply in its full generality to unions of random closed sets [START_REF] Molchanov | Theory of Random Sets[END_REF], p. 254, 'Affine normalization'). This topic will be the subject of further work. δ > 0. For r < p + δ, we have F (Q(p) + ε) > r too, and thus Q(r) < Q(p) + ε, as required.

(iii) First suppose x < Q(p); we show that Pr[X < x] ≤ p. The case x = ∞ is impossible, and if x = -∞, then Pr[X < x] = 0 ≤ p. So suppose that x ∈ R. Using statement (i), there exists y ∈ R with x ≤ y ≤ Q(p) such that Pr[X < y] ≤ p. But then also Pr[X < x] ≤ p.

Second suppose that x > Q(p); we show that Pr[X < x] > p. Clearly, we must have Q(p) < ∞, and so we can without loss of generality assume that x is real. But then Pr[X < x] > p by statement (i).

Finally, consider

x = Q(p); we show that Pr[X < Q(p)] ≤ p. If Q(p) = -∞, then Pr[X < Q(p)] = 0 ≤ p. If Q(p) > -∞, then Pr[X < y] ≤ p for all y < Q(p), and thus Pr[X < Q(p)] = sup y<Q(p) Pr[X < y] ≤ p too.
(iv) Without loss of generality, assume that 0

≤ V ≤ 1 (if not, then replace V by (V ∨ 0) ∧ 1, which is almost surely equal to V ). Let x ∈ [-∞, ∞]. By statement (iii), we have x ≤ Q(V ) if and only if Pr[X < x] ≤ V . As a consequence, Pr[x ≤ Q(V )] = Pr[Pr[X < x] ≤ V ] = 1 -Pr[X < x] = Pr[x ≤ X].
We conclude that Q(V ) and X are identically distributed and thus that Q(V ) has distribution function F too.

(v) By definition, x ≤ Q(F (x)) for every x ∈ R. For x = -∞ and x = +∞, the same inequality is trivially fulfilled too (recall that F (∞) = 1 and Q(1) = ∞). As a consequence, X ≤ Q(F (X)).

Conversely, let P be the collection of p ∈ [0, 1] such that the set {x ∈ R : F (x) = p} has positive Lebesgue measure. These sets being disjoint for distinct p, the set P is at most countably infinite. If x < Q(F (x)), then there exists y > x such that F (x) = F (y) and thus F (x) ∈ P. However, the law of F (X) is standard uniform, so that Pr[F (X) ∈ P] = 0. Hence X = Q(F (X)) almost surely.

Lemma A.2. Let (X n ) n∈N be a sequence of random variables defined on the same probability space and taking values in [-∞, ∞]. Let Q n and Q be the rightcontinuous quantile functions (A.1) of X n and lim sup n→∞ X n , respectively. Then

Q(p) ≥ lim sup n→∞ Q n (p), p ∈ [0, 1].
Proof. If Q(p) = ∞, there is nothing to show, so suppose Q(p) < ∞. Let y > Q(p); we will show that y ≥ lim sup n→∞ Q n (p). This being true for all y > Q(p), we will have proved the proposition. By Lemma A.1(i), we have Pr[lim sup n→∞ X n < y] > p. By Fatou's lemma, there exists a positive integer n(y) such that Pr[X n < y] > p for all integer n ≥ n(y). But for such n, we have y > Q n (p) too. Hence, y ≥ lim sup n→∞ Q n (p), as required. In particular, for every x ∈ USC(D), we have x(s) ≥ z(s) for all s ∈ D if and only if x(t) ≥ z(t) for all t ∈ Q z .

Proof. The set hypo z is a subset of the metrizable separable space D × R. Hence, it is separable too. Let Q be a countable, dense subset of hypo z. Let Q z be the set of t ∈ D such that (t, x) ∈ Q for some x ∈ R. Then Q z is a countable subset of D. Let y(s) denote the right-hand side of (C.1). Since z is usc, we have z(s) ≥ y(s) for all s ∈ D.

Conversely, let s ∈ D. If z(s) = -∞, then trivially y(s) ≥ z(s). Suppose z(s) > -∞. Let -∞ < α < z(s), so that (s, α) ∈ hypo z. Find a sequence (t n , α n ) ∈ Q such that (t n , α n ) → (s, α) as n → ∞. Then (t n , α n ) ∈ hypo z and thus z(t n ) ≥ α n for all n. Moreover, t n → s and α n → α as n → ∞. It follows that y(s) ≥ α. Since this is true for all α < z(s), we find y(s) ≥ z(s).

We prove the last statement. Let x ∈ USC(D) and suppose that x(t) ≥ z(t) for all t ∈ Q z . The function x is equal to its own usc hull, i.e., Combine this formula together with (C.1) to see that x(s) ≥ z(s) for all s ∈ D.

Lemma C.3. If ξ is a usc process, then inf s∈F ξ(s) is a random variable for any closed set F ⊂ D.

Proof. Let Q be a countable, dense subset of F . Since every ξ(s) is a random variable, it suffices to show that inf s∈F ξ(s) = inf s∈Q ξ(s). The inequality '≤' is trivial. To see the other inequality, suppose that x is such that ξ(s) ≥ x for all s ∈ Q. Then (s, x) ∈ hypo ξ for all s ∈ Q, and thus (s, x) ∈ hypo ξ for all s ∈ F , since hypo ξ is closed. It follows that ξ(s) ≥ x for all s ∈ F .

Appendix D. Continuity of the GEV parameter

Recall the GEV distributions from Subsection 5.1.

Lemma D.1. Let F n = F ( • , θ n ), n ≥ 0, be GEV distribution functions with associated GEV parameters θ n = (µ n , σ n , γ n ). If (F n ) n converges weakly to F 0 , then also lim n→∞ θ n = θ 0 in R × R × (0, ∞).

Proof. By continuity of the GEV distribution, weak convergence is the same as pointwise convergence, and thus lim n→∞ F n (x) = F 0 (x) for all x ∈ R.

Recall the expression (5.2) of the quantile function Q( • ; θ). Pointwise convergence of monotone functions implies pointwise convergence of their inverses at continuity points of the limit (Resnick, 1987, Chapter 0). Setting p = e -1 , we obtain µ n = Q(e -1 ; θ n ) → Q(e -1 ; θ 0 ) = µ 0 , n → ∞.

As a consequence, for p ∈ (0, 1),

Q(p; γ n , 0, σ n ) = Q(p; θ n ) -µ n → Q(p; θ 0 ) -µ 0 = Q(p; γ 0 , 0, σ 0 ), n → ∞. (D.1)
This implies that for x, y > 0 such that y = 1, lim n→∞

x γn -1 y γn -1 = x γ 0 -1 y γ 0 -1 .

For γ n = 0, the above expressions are to be understood as log(x)/ log(y). A subsequence argument then yields that (γ n ) n must be bounded, and a second subsequence argument confirms that γ n → γ 0 as n → ∞. Combine this convergence relation with (D.1) and use the identity Q n (p; γ n , 0, σ n ) = σ n Q(p; γ n , 0, 1) to conclude that σ n → σ 0 as n → ∞.

  belongs to the class U(D) if it has the following two properties: (a) For every s ∈ D, the map x → U(s, x) is non-decreasing and right-continuous. (b) For every x ∈ R ∪ {∞}, the map s → U(s, x) is usc. See Remark 3.1 for the reason why we did not include x = -∞ in condition (b). Proposition 3.1. Let U ∈ U(D).

  celebrated theorem states two things: (I) For every copula C and every vector F 1 , . . . , F d of univariate distribution functions, the function (x 1 , . . . , x d ) → C(F 1 (x 1 ), . . . , F d (x d )) is a d-variate distribution function with margins F 1 , . . . , F d . (II) Every d-variate distribution function F can be represented in this way. Reformulated in terms of random vectors, the two statements read as follows:

  is a usc process such that Pr[ ξ(s) = ξ(s)] = 1 for every s ∈ D. In particular, the finite-dimensional distributions of ξ and ξ are identical.Proof. (i) By condition (a), the marginal distributions of Z are standard uniform. By condition (b), the function U defined by (s, x) → F s (x) belongs to U(D) (Definition 3.1) and we have Z = U * (ξ) with U * : USC(D) → USC(D) as in Proposition 3.1. By item (iv) of that proposition, the map U * is hypo-measurable, so that Z is a usc process too.(ii) By Lemma A.1(v), we have Pr[ ξ(s) = ξ(s)] = 1 for every s ∈ D. The function Q s is the right-continuous quantile function of ξ(s) and the stochastic process (ξ(s) : s ∈ D) has usc trajectories. Then requirement (ii) of Proposition 4.1 is fulfilled by an application of Lemma 3.2. By item (i) of the same proposition, ξ is a usc process.Remark 4.1. Although Pr[ ξ(s) = ξ(s)] = 1 for all s ∈ D, it is not necessarily true that Pr[ ξ = ξ] = 1, and not even that ξ and ξ have the same distribution as usc processes: see Examples B.2 and B.3. However, if Pr[∀s ∈ D : 0 < F s (ξ(s)) < 1] = 1 and if Q s (F s (x)) = x for every s and every x such that 0 < F s (x) < 1, then clearly Pr[ ξ = ξ] = 1.The following Lemma helps clarifying the meaning of the assumptions of Proposition 4.2.

  Lemma 4.1 (Regularity of the marginal distributions w.r.t. the space variable ). If ξ is a usc process, conditions (a) and (b) in Proposition 4.2 together imply that the map s → F s (x) = Pr(ξ(s) ≤ x) is continuous, for any fixed x ∈ R.

  Remark 4.3. Without condition (b) in Proposition 4.2, the trajectories of the stochastic process (Z(s) : s ∈ D) may not be usc: see Example 4.2. However, condition (b) is not necessary either: see Example B.1.

  Example 4.1 (What may happen without (a) in Proposition 4.2). Consider two independent, uniformly distributed variables X and Y on [0, 1]. Take D = [-1, 1], and define

  Example 4.2 (What may happen without (b) in Proposition 4.2). Consider two independent, uniformly distributed variables X and Y on [0, 1]. On D = [0, 2], define

Lemma 5. 1 .

 1 Let θ : D → Θ. The functions s → F (x; θ(s)) and s → Q(p; θ(s)) are usc for every x ∈ R and p ∈ [0, 1] if and only if θ is continuous. Proof. If θ is continuous, then the maps s → F (x; θ(s)) and s → Q(p; θ(s)) are continuous and thus usc. Conversely, suppose that the functions s → F (x; θ(s)) and s → Q(p; θ(s)) are usc for every x ∈ R and p ∈ [0, 1]. The argument is similar to the proof of Lemma 4.1. By Lemma A.1(iii), we have {s ∈ D : x ≤ Q(p; θ(s))} = {s ∈ D : F (x; θ(s)) ≤ p}

  Remark 5.1. If θ : D → Θ is continuous, then the normalizing functions s → a n,θ(s) and s → b n,θ(s) are continuous as well. Consider the map U n (s, x) = {xb n,θ(s) }/a n,θ(s) , for (s, x) ∈ D × [-∞, ∞]. Clearly, U n ∈ U(D). By Proposition 3.1, the transformation USC(D) → USC(D) : z → (z -b n,θ )/a n,θ is then well-defined and hypo-measurable.

=

  a n ξ + b n in USC(D).

  (a) with probability one, the trajectories of ξ are usc; (b) for some set Ω 1 ⊂ Ω of probability one on which the trajectories ξ( • , ω) are usc, the map ξ : Ω 1 → USC(D) is hypo-measurable. Consider the space of nonnegative usc functions USC(D) + = {z ∈ USC(D) : ∀s ∈ D, z(s) ≥ 0} = s∈D {z ∈ USC(D) : z(s) ≥ 0}. For each fixed s ∈ D, the set {z ∈ USC(D) : z(s) ≥ 0} is hypo-closed, since it is the complement of the hypo-open set {z ∈ USC(D) : hypo z ∩ {(s, 0)} = ∅}. The set USC(D) + is thus hypo-closed, as an intersection of hypo-closed sets.

Lemma C. 2 .

 2 For every z ∈ USC(D), there exists a countable set Q z ⊂ D such that z

Acknowledgments Part of A. Sabourin's work has been funded by the 'AGREED' project from the PEPS JCJC program (INS2I, CNRS) and by the chair 'Machine Learning for Big Data' from Télécom ParisTech. J. Segers gratefully acknowledges funding by contract "Projet d'Actions de Recherche Concertées" No. 12/17-045 of the "Communauté française de Belgique" and by IAP research network Grant P7/06 of the Belgian government (Belgian Science Policy).

Appendix A. Right-continuous quantile functions

The right-continuous quantile function, Q, of a random variable X taking values in [-∞, ∞] and with distribution function

(A.1) By convention, sup ∅ = -∞ and sup R = ∞. The fact that Q is right-continuous is stated in part (ii) of the next lemma. The function Q is denoted by F -1 + in [START_REF] Pickands | The two-dimensional Poisson process and extremal processes[END_REF], p. 749, who mentions properties (ii) and (iv) of Lemma A.1. The corresponding statements for the right-continuous inverse, p → inf{x ∈ R : F (x) ≥ p}, are well-known, see for instance Section 0.2 in [START_REF] Resnick | Extreme Values, Regular Variation, and Point Processes[END_REF].

Conversely, if Q(p) = +∞, there is nothing to prove. Assume then that Q(p) < +∞, so that p < 1. Let y > Q(p). We need to show that y > x 0 too, that is, Pr[X < y] > p. But Pr[X < y] = sup{F (z) : z < y}, and this supremum must be larger than p, since for all z > Q(p) we have F (z) > p.

(ii) The sets {x ∈ R : F (x) ≤ p} becoming larger with p, the function

and let X and V be independent random variables, X standard normal and V uniform on [0, 1]. Define

Let Φ be the standard normal cumulative distribution function and choose x ∈ R.

, 0], while for s ∈ (0, 1], we have

The function s → F s (x) is constant on s ∈ [-1, 0] while it decreases linearly from Φ(x + 1) to Φ(x) for s from 0 to 1, the right-hand side limit at 0 being equal to Φ(x + 1), which is greater than Φ(x), the value at s = 0 itself. Hence the function s → F s (x) is lower but not upper semicontinuous, and condition (b) in Proposition 4.2 does not hold. Nevertheless, the random variables Z(s) = F s (ξ(s)), s ∈ D, are given as follows:

The trajectory of Z is continuous at s ∈ [-1, 1] \ {V } and usc at s = V , hence usc overall.

Example B.2 (ξ and ξ in Proposition 4.2 may be different in law (1)). Let X and Y be independent, uniform random variables on

Since Pr[Y = s] = 0 for every s ∈ [0, 1], the law of ξ(s) is standard uniform. Conditions (a) and (b) in Proposition 4.2 are trivially fulfilled with

and thus

The processes ξ and ξ have different capacity functionals and thus a different distribution as random elements in USC(D): for

Example B.3 (ξ and ξ in Proposition 4.2 may be different in law (2)). Let X and Y be independent random variables, with X uniformly distributed on

Then Pr[ξ(s) = X] = 1 for all s ∈ [0, 1], so that the marginal distribution and quantile functions F s and Q s do not depend on s and are equal to those of X, denoted by F X and Q X . The random variable U = F X (X) is uniformly distributed on [0, 1]. Since F X (1.5) = F X (1) = 0.5, we have

On the event {X ≤ 1}, which occurs with probability one half, the hypographs of ξ and ξ are different.

Appendix C. Lower bounds of usc processes

Lemma C.1. If ξ is a usc process, then the function ℓ(s) = sup{x ∈ R : Pr[ξ(s) < x] = 0}, for s ∈ D, is usc. Moreover, ξ ∨ ℓ is a usc process too, and we have Pr

Proof. The function ℓ is equal to the function s → Q s (0), with Q s the rightcontinuous quantile function (A.1) of ξ s . By Lemma 3.2, the function ℓ is usc, and by Example 3.1, the map USC(D) → USC(D) : z → z ∨ ℓ is hypo-measurable, so that ξ ∨ ℓ is a usc process too. By Lemma C.2, there exists a countable subset, Q ℓ , of D with the following property: for all x ∈ USC(D), we have x(s) ≥ ℓ(s) for all s ∈ D if and only if x(t) ≥ ℓ(t) for all t ∈ Q ℓ . Since ℓ(t) = Q t (0), we have Pr[ξ(t) ≥ ℓ(t)] = 1 for all t ∈ Q ℓ ; see Lemma A.1(iii). Since Q ℓ is countable, also Pr[∀t ∈ Q ℓ : ξ(t) ≥ ℓ(t)] = 1. By the property of Q ℓ mentioned earlier, the event {∀t ∈ Q ℓ : ξ(t) ≥ ℓ(t)} is equal to both {ξ ≥ ℓ} and {ξ = ξ ∨ ℓ}.


Gabrielle Le
email: gabrielle.letallec@ldi.cnrs.fr

Tallec Lloret

New research perspectives in post-Guillaumian linguistics: cognematics and interlocutivity

Introduction

One of the foundations of Gustave Guillaume"s (1883Guillaume"s ( -1960) ) Psychomechanics is the idea that language is the product of the combination of two indissociable structures, the semiological structure (the signifier 1 ) and the psychic structure (the signified), linked by a "congruency" relation (Guillaume 1971: 170):

A principle adhered to by all the languages in their construction is that of congruency or, in other words, the matching between the signifier and the signified. The signifier is an act of speech, the signified an act of thought, and the structure of a language, and its very existence, presuppose a sufficient agreementwhich will never be excessive (and which may always grow) between an act of speech and an act of thought.

The idea of congruency between the semiological structure and the psychic structure constitutes a breakthrough in its time, while perfectly reproducing the "Saussurean division of the sign". We shall focus here on the relationship between both: indeed, the problem of the transition from phonation to semantics has provoked a major theoretical evolution in Hispanic linguistics of Guillaumian inspiration, from the 1980s to the 2000s, with the emergence of the field called today the "linguistics of the signifier". This paper will mainly focus on grammatical signifiers.

From Guillaume to Mo.La.Che

We should bear in mind that Guillaume insists on the prevalence of the psychic structure over the semiological structure as emphasised by Francis Tollis in La parole et le sens (1991: 45), this leads Guillaume "to look for a rigorous system only around the signified" […]. Indeed, behind a semiology only offering attempts to systematise with incomplete coherence, hides a ʻpsychic' structure which is totally and rigorously coherent [START_REF] Guillaume | Leçons de linguistique 1948-1949 A.1[END_REF]: 133, cited by Tollis 1991: 45).

With Guillaume, the prevalence of the signified is a major breakthrough in the conception of the intra-sign relationship, but it does not really provide any response to the problem of the articulation between semantics and phonation, and again emphasises the contours of the linguistics of the sign rather than of the linguistics of the signifier (Le Tallec-Lloret 2012). If a prevalence is posed, that of the psychological structure as the motivating source of the semiological structurewhat Didier Bottineau calls "the geological conception of the signifying form (Bottineau 2010: 91) everything still remains to be done, precisely on this semiological structure, and de facto, on this transition from phonation to semantics. The works of Maurice [START_REF] Toussaint | Contre l'arbitraire du signe[END_REF], then the project of the Mo.La.Che group should be placed in this perspective, regardless of how insufficient they may appear.

After the "intense pleasure", the "revelation" brought by Guillaume that "the signifiers, like the effects of meaning and the syntax, should be interpreted as consequences generated by the semantic structure", Toussaint has relentlessly looked in semiology for "the de facto status" […], the morpho-phonological "translation" of the already established organisation, independently from it, on the semantic plane. (Tollis 1991: 52) Indeed, the principle of congruency has led a group of three linguists and Hispanist researchers, Maurice Molho, Michel Launay and Jean-Claude Chevalier, more readily identifiable under the acronym Mo.La.Che, to lay in the 1980s the principles of a "linguistics of the signifier" in a first manifesto-article entitled "The reason of the signifier" (1984), exposed more amply and illustrated later on in a series of articles, as a group then individually, in particular by Launay (2003) and [START_REF] Chevalier | De Guillaume à une linguistique du signifiant[END_REF]. If we re-examine the guiding principles of Mo.La.Che, we can already observe in the very formulations, the difficult conciliation between, on a one hand, the structuralist Guillaumian legacy and, on the other, the aspiration to go beyond it.

First of all, in Mo.La.Che, the unity of the linguistic sign is asserted with strength. We shall remember the Chevalier"s formula (1996: 81) echoing the famous congruency of Guillaume: "Signifier and signified are anchored to each other [...]", where we would be wrong not to dwell on the passage below: "[...] and the signifier, everywhere and always, tells what we have read mentally and who produces the signified. Nothing to look for in the signifier that does not mark the signified". The undeniable prevalence of psychology over semiology is clearly set here, in such a way that this blatantly Guillaumian mentalism paradoxically combines with the rehabilitation of the signifier.

Indeed, in the intra-sign relationship, with respect to Guillaume (Tollis 1991: 225-292), Mo.La.Che realises a major shift by considering the meaning as a ʻproduct' of the signifier (and not as its ʻsource'), by granting de facto the status of commander" to the signifier:

There is nothing deeper in a language than its surface: for those who know how to look at it, it is the faithful translation of what happens deep inside. (Mo.La.Che 1986: 96) [...] in our view, the language does not include any other system nor other systematics than as ordered and stated by a kind of semiology which is always and everywhere motivating […] (Mo.La.Che 1984: 40).

It is substantially the whole linguistic system which is concerned by the motivational mechanisms to which we wished to call attention (1984: 32) Mo.La.Che then advocates a theoretical rebalance between the signifier and the signified, in favour of the signifier. This has two consequences:

1) It is important to stress the difference with respect to structuralism, as observed in the conclusions of M. Launay on the apocope in Spanish (Launay 1985: 438).

For a structuralist, acopoped forms are treated in a uniform way, no matter which section of the sign is deleted (bueno/buen; primero/primer; santo/san; grande/gran; reciente/recien...), since only the opposition is relevant. For Launay, the very form of the apocope does matter, since the deleted segment contains the smallest possible information, either indeterminate gender (-te, -de) or unmarked gender (-o), depending on adjectival category and morphological structure.

For a structuralist only reasoning in terms of oppositions, with the postulate of the arbitrariness of the linguistic sign, the result is admittedly the same, regardless of the form which is apocoped, since the opposition is maintained. But the way the signifiers oppose also proves noteworthy.

2) This rebalance in favour of the signifier leads to go back decidedly to the ʻphysiological' part of the sign, to the elements of which it is constituted materially: the phonemes.

We shall sum up here the guiding idea of Launay according to which the sign "in itself", i.e. "outside system", is not motivated a priori, but is driven by the system, especially via semiotic connotation. With respect to Saussure, Launay does not only lay down the idea of a system but that of a system of signifiers. Correlatively, the emphasis is on the "phonematic structure of the signifier" (Launay 1985: 430): My thesis is […] that at the level of the phonematic structure of the signifier there is also a law and also a kind of prohibition which cannot be explained by the articulatory difficulties nor the semantic hierarchy of the concepts it is still referring. The Mo.La.Che group tends to take its distance from the structuralist conception of the sign, taken as an unbreakable linguistic entity, conceived in a Language, and forebodes the decomposition of the sign in semantically relevant isolable elements, that is, lexical and grammatical submorphemic units (see below). Thus, the definition of the formant proposed by M. Molho constitutes a considerable theoretical breakthrough even if the choice of the term is debatable as it tends to induce confusion with vocalic formants: For our own part, "formants" are not acoustic frequencies, but signifying elements or particles which, intervening in the structure of a given signifier, are repeated in several others, which result in the formation of an analogy field grouping one or several morphematic series. This amounts to say that a "constituent", if it appears in a set of morphems, informs the series and confers it a general signification of which it is the cause or the root. (Molho 1988: 291) M. Molho"s uneasiness can be felt in his hypothesis of the "constituent" when the latter is reduced to a single phoneme (such is the case of the constituent *n), which carries a semantic instruction IV , whereas a phoneme as we know does not carry any meaning. M. Molho is compelled to reproduce the old structuralist opposition between negative value and positive value. Still, the constituent is indeed situated at another level than the morpheme and can be somehow assimilated to a ʻsubmorpheme'. But, in the structuralist conception passed on to Molho, the lower level comprises the phoneme, which is deprived of a positive value.

Should the element *n be considered as a signifier in the ordinary meaning of the word? We have some scruples in designating it as such, since a signifier consists of an element which must be a linguistic integer (word, prefix, suffix, radical, desinence, etc.). However, *n shares with the signifiers the property of presenting itself in the form of a physism to which is associated a mental or signified content, in the strict sense of the word. In other words, if it is not a signifier, *n can still be a signified, and as such it integrates one or several signifiers and forms it or them into a system […] by the addition of a common signification element. (Molho 1988: 299) This very uneasiness is pointed out by Bottineau (1999: 6); the difficulty felt by Molho concerns the fact of navigating only between the grammatical morpheme (positive value, signified side) and the last stepping stone, the phoneme (negative value, signifier side), whereas the emergence of constituents would require the theoretical conception of another level of abstraction taking the phonological structure of the signifier into account:

[...] more recently, Molho (1986: 50) identifies vocalic and consonantal constituents in the grammatical words of Spanish, which can be detected in synchrony by analogies which do not respect the diachronic filiations; its role is not truly to fix a positive and intrinsic invariant, but to flatten synaptically categorical differentiations (such as that of the name and of the verb) which mask common psychic construction traits. Consequently, the constituent is not linked with the phonomimesis.

If in Hispanic linguistics the sign could be broken since Mo.La.Che, the breakdown operation remains morphematic. The radicality of Mo.La.Che has undeniably marked a major stepping stone in Hispanic linguistics, while leaving unsolved the tricky issue of the transition from phonation to semantics. Moreover, it may be perceptible that its studies of Spanish morphosyntax are still more concerned with a linguistics of the sign than truly with a linguistics of the signifier. A theory of the sign taken as a linguistic entity, in Languages, within an approach still strongly marked by Saussure, leaving little room for enunciatory considerations. The postulates of the Mo.La.Che have then fuelled the works of other researchers, in particular those gathered in the seminars of Gilles Luquet in Paris 3, within the GERLHIS, Study and Research group in Hispanic Linguistics.

It was in the 2000s that the reflection on the linguistic sign was reinstated as such and marked a new stepping stone in going beyond the structuralist legacy, under a double impulse, within and outside Hispanism: that of G. Luquetwith the publication of the collection of articles Regards sur le signifiant (2001) and that of the Anglicist, trained along the Guillauman lines, Didier Bottineau, with his theory of cognemes.

From Mo.La.Che to cognematics

In the lexical field, the thesis of the motivated sign maintains that what associates the signifier to the signified to the extent of being both indissociable of the sign, is the recourse to the sensorial [START_REF] Guiraud | Structures étymologiques du lexique français[END_REF]: the acoustic image of the sound and of the sense only exists in their relation of reference to the corporeal sensations. In the field of grammatical morphosemantics, the distance with respect to Guillaumian mentalism ought to be stressed, as was done by Bottineau:

This makes obsolete a geological conception of the sign as a physical and expressive trace of the occult internal mental and requires an interactive conception which reconciles the indissociable dynamics of the corporal and mental processes. (2010a: 92) This reconciliation of the physical and of the mental, where the phonological structures are conceived as articulatory gestures, forms constitutes the basis for the theory of cognemes developed by Bottineau, who observed that:

In a large number of natural languages, Indo-European or not, it appears that the grammemes and, as the cases may be, certain lexemes, do not constitute unbreakable units, but agglomerates of isolable submorphemes which, considered individually, refer to invariant mental processes, like fundamental pieces of software for cognition, named as cognemes. (2003: 185) What is a cogneme? A phoneme which is capable of priming abstract meaning at submorphemic level by using its sensory-motor properties to activate procedural interpretive processes. This operation is not only attached to the intrinsic properties of the phoneme as a network of correlated articulatory gestures, but also to the network of oppositions and values which underlines it, such as /i/ vs /a/. From a theoretical viewpoint, this involves an additional level of abstraction beneath the phoneme, that is, the participation of sensory-motor loops in sense-making processes. Initially, Bottineau formulated cognematics as a vector between speaker and hearer, and therefore formulated the theory in instructional terms:

The theory of cognemes proposes a generalized recognition of submorphemic indicators wherever they are to be found in grammatical morphology. The word cogneme designates a semantic process that the speaker invites the hearer to implement in order to establish a relation between two preexisting semantic entities, a cognitive hinge available in the linguistic system shared by the addresser and the addressee that the former can activate in the latter"s mind by sending the relevant acoustic trigger sound in the appropriate syntactic environment. In the case of I, the instruction consists in joining or even merging two previously seperate entities. The nature of the semantic entities involved depends on the format of the syntactic units between which the cognemic submarker is operating. (2007:54-55) More recently, Bottineau (2010c) has related cognematics with the paradigm of enaction and defined cognemes as phonemes whose sensory-motor dynamics activate sense-making processes both for the speaker in reflexive thinking (endophasia) and the hearer in addressed communication (exophasia), in keeping with the paradigm of radical embodiment. Instructionalism is retained as a convenient formulation at metalinguistic level, but is not the essence of submorphemics.

In this paradigm, the cogneme is understood as a complex bundle of articulatory gestures which activates a complex bundle of perceptual events: the cogneme is multimodal on both sides, both in terms of motricity and sensoriality. For this reason, each cogneme is apt to activate a matrix of interpretive processes that can be glossed in instructional term, but in a given context, one of those traits is made salient at the expense of the others. Some cognemes activate interpretive profiles which are directly linked to the sensorimotor properties of the articulatory gestures: in romance languages, the i/a opposition, marking a contrast between proximity and distance (or, in instructional terms, processes of conjunction and disjunction), is narrowly tied to the closing and opening gestures characterizing the production of those vowels. Other cognemes activate interpretive gestures that are more loosely linked (if at all) to the articulatory pattern: in presentday English, the wh-/ th-opposition (as in where / there, when / then) deals with the availability or absence of a given entity in working memory, but one cannot relate this semantic contrast with a clearcut opposition at the level of traits, even if there may have been a stronger connection in diachrony. Even at cognematic level, the degree of solidarity between the profile of the gesture and the nature of the interpretive process can fluctuate in diachrony and dep ends on how speakers interpret the link at a given moment of the history of the language, alternating conflicting trends towards relative demotivation and relative remotivation, especially when new forms emerge under the pressure of analogy. Language change oscillates between cognematic erosion (like the disappearance of the o/a opposition in French or the destabilization of the English vocalic system by the great vowel shift) and cognematic structuration (like the homologous analogies between en and -ant in French and in and -ing in English concerning spatial and aspectual interiority (Bottineau 2014): in microdiachrony and at synchronic level, a submorphological order, underpinned by cognematics, is forever emerging and struggling against dissolution, but cognematics is never fully re-created or remotivated by speakers and is not determined by the properties of external referents.

Unlike in the case of Molho"s "constituent", both dimensions, phonological and semantic, are taken into account here, in what is akin to a trajectory, a process: from a phonological mechanism (the cogneme gives phonological "instructions", articulatory traits) cognitive and semantic instructions are triggered. What is glossed as a semantic instruction is not in itself a representation but an operation intended for combining the phonation and the meaning by what is also called by Bottineau an, a "psychophone", a "cognophone", "at a subconscious level, the phonic stimulus inducing a sensorial/mental response". This theory offers the double advantage of isolating what is used as an instrument for switching from phonation to semantics (the cogneme) while going beyond the theoretical cul-de-sac which leads Mo.La.Che in order to pour meaning into the phoneme, thereby excluding phonomimesis:

The cogneme relationship fundamentally involves a restriction in its application, the obligatory "contrasting" in the actualising process:

We are founded to consider a phoneme as a submorpheme which is semantically relevant under the following (non-cumulative) conditions: 1-it occurs in a recurring alternation [...]. 2-The word-operator in which it occurs is itself globally formed of an agglutination of markers extracted from such alternations [...]. 3-The submorpheme, combined with a root or with other submorphemes in a constant position like the initial or the final position, classifies all the operators concerned in a given category […] (Bottineau 2004: 29) The "cogneme" is somehow the missing link between phonation and semantics, this union of sound and meaning sought after by R. Jakobson (1976: 22-23). The motivation of the "signifier" considered by Guillaume as the psychic welding of the "virtual signified" and of the "sign" shows forth in the sensori-motor grounding which sometimes shapes or at least supports the articulation of the virtual cogneme. The transition from phonation to semantics is not direct: it involves the cognition, a step which precisely marks the break with structuralism: "what sounds impart to the senses and raises in the mind". We may pontificate on the cognemes in an instructional manner (these phonatory, cognitive, semantic instructions), which excludes attributing them any "signified" content. For the time being, cognematics remains a linguistic hypothesis grounded in the modelization of submorphological coherence in grammatical system supported by analogical, remotivational and neological processes in microdiachrony. How this speculative proposal can be validated by other methodologies is currently discussed (Bottineau 2014) and experimental work has been undertaken [START_REF] Thibaut | Languaging behaviour as catalytic process: steps toward a theory of living language[END_REF] for the connection between cognematics and co-speech gestures; Olivier and Bottineau 2015 for gestuo-motor compatibilities).

Linguistics of the signifier

In order to come back to Hispanism and in order to understand how cognematics and linguistics of the signifier could make contact v , we need to re-examine the works of Gilles Luquet, who very quickly adopts, in the wake of Launay, a conception of languages not as systems of "signs" but as systems of "signifiers", delivering phonological sequences to objective observation.

In his works on Spanish verbs, starting from two structuring variations (person and exochrony/endochrony relationship), Luquet observes that in the personal forms, there is no such thing, properly speaking, as a subjunctive mode on the plane of the signifier (1998: 89-90):

The sole semiological traits shared by a "present", a "future" and a subjunctive "imperfect" (in this instance the marks of the person) are usually found in other "tenses" of the Spanish conjugation and especially in certain tenses of the so-called "indicative" mode. Such is indeed the case with the absence of specific marks concerning the 1 st and 3 rd persons of the singular; […] the terminal -s of the 2 nd person of the singular; […] of the terminal morphemes -mos,is and -n terminal morphems, in charge of representing respectively the 1 st , 2 nd and 3 rd persons of the plural. These marks are not specifically subjunctive since they can also be found in the signifying structure of any indicative imperfect or of any conditional. As regards the thematic morphemes to which they relate, nothing, from a semiological viewpoint, compels to group them in a single assembly.

In 2004, his theories on modes and tenses (founded strictly on the semiology of verbal forms) marks a break with the Guillaumian approach by substituting the traditional modal opposition between indicative and subjunctive with an actualising mode with respect to an inactualising mode.

After exposing a clear opposition between the three personal verbal forms distinguishing the 1 st and 3 rd persons of the conjugation, and the others, which neutralise this distinction, he proposes to make a distinction between two types of identifications around the speaker"s figure, the keystone of this temporal architecture:

-A really temporal identification, founded on the speaker"s experience time, enabling him to locate an operation in time; -an identification of the inactual, winning over a conception of time of another nature, a conceptual, imaginary time and not an experience time any longer.

Thus, the personal verbal forms in modern Spanish are organised into two modal sub-assemblies: on the one hand, an actualising mode offering three types of representations: canto ("present"), canté ("preterit"), cantaré ("future"); on the other, an inactualising mode offering five types of representations: cantaba ("imperfect"), cantaria ("conditional"), cante (ex-"present subjunctive", cantara (ex-"imperfect subjunctive", a -ra form), cantase (ex-"imperfect subjunctive", a -se form).

Based on these three superficial observations, and contrary to the Guillaumian tradition which saw in the "present" of the indicative mode a completion vi of the chronogenetic operation (fig. 1 below), Luquet considers the enunciation present as the "source" of the verbo-temporal system. By "enunciation present" is meant "the" unique present of the being defined as the EGO, both speaker and builder of the language vii . It is as an enunciating instance that the EGO builds its representations of time, its time-images. The personal and impersonal forms represent different The present is the completion Operative time conceptions of time associated with an operation: it is this conception of time that is called the "linguistic time". Personal verbal forms in Modern Spanish are organized in two modal groups: actualising mode consisting of three types: canto, canté, cantaré, and inactualising mode consisting of five types: cantaba, cantaría, cante, cantara, cantase. This theory gives rise to the Verbo-Temporal System in Spanish as suggested by the following recapitulative figure: Indeed, in the Guillaumian tradition, it is the indicative mode that corresponds to the maximum degree of actualisation of the process, as it situates it with respect to the speaker"s person and at the moment of the act of language. The operative time in Psychomechanics requires a certain time period to unfold; it is represented horizontally by Guillaume who therein includes vertical crosssections (fig. 1). In this Guillaumian vision of time which is building up to completion, the number of verbal forms grows up to the indicative mode which can be divided into time-scales: starting from four forms in the quasi-nominal mode, we obtain ten forms in the indicative mode. Still, in the Luquet"s theory, the verbo-temporal system of Spanish takes its "source" in the present and is not its achievement: the founding, enunciation present finds itself "upside down" (fig. 2) the other lines of time then marking, one after the other, a mental distance with respect to the experienced time of the speaker, following the parameter as it is being abandoned: from the threepart vision resulting from the spoken word founded the present and rooted in experience ("actualising mode"), there is a transition towards a mental universe marking a break from the experienced present ("inactualising mode").

Conclusion

Luquet"s theory, reveals, beyond the debate on the existence or not of a language structure, the idea of a system of signifiers (here the verbo-temporal subsystem of Spanish). It places de facto at the forefront the oppositions authorised by the language and that the speaker exploits, in his linguistic subconscious, both as a builder and a user of language.

Between the actuality and the inactuality of speaking, these theories on modes and times, turn interlocution, a parameter absent in Mo.La.Che, into the other theoretical key which combines with a true "reading of the signifier", today particularly active nowadays in Hispanic linguistics. Language is not the world; it is itself another world, a system with its own rules of organisation, observable even in the phonematic structure of its signifiers. From this perspective, the roll-over one can observe in Luquet (the present being turned "upside down") is not all the more a so-called "renunciating theory", taken in the classic sense of the term. It is much closer to the interlocutivity theory in the sense of Douay and[START_REF] Douay | L"interlocution comme clé du contrastif[END_REF]2014). Taking into account the signifier of the verbal forms is a cardinal parameter, resulting both from the Guillaumian legacy, the breakthroughs of Mo.La.Che, but also from taking into consideration the corporeal or enactive approach of language as modelled by Bottineau (Le Tallec-Lloret 2012: 29-36).

Today, post-Guillaumian linguistics may address a new step towards the "linguistics of the signifier", if it integrates the theoretical resolution supplied by cognematics (articulating sound and meaning by exploiting the notion of contrast) and if it seeks to articulate the necessary system and the just as necessary interlocution.

i It is furthermore important to specify that the ʻsignifier' should be understood, in the Saussurean sense of the word, designating the material face of language, and not as an equivalent of the ʻsign', as Guillaume could use it. Fully aware of the expeditious aspect of this presentation with respect to semiology and psychism, crucial issues for Guillaume, and for linguistics in general, see Tollis 2008: 179-206. ii The study of the semiological structure is called ʻpsychosemiology' while the study of the psychological structure is called ʻpsychosystematics'. Iii " [...] we were right, in that matter, to criticise the purely ʻarticulatory' explanation. But we were wrong, probably, to try and see the alpha and omega, as a reaction, in ʻsemantics': as if the signifier as such could not be the place of any law of its own, as if the laws governing the signifier should absolutely depend on the body producing it (the physiological aspect) or on the world which it may be instrumental in describing (the semantic aspect). My thesis is on the contrary that at the level of the phonematic structure of the signifier there is also a law and hence also a kind of prohibition which cannot be explained by the articulatory difficulties nor the semantic hierarchy of the concepts it is still referring", Ibid., p. 430. iv "[...] the mental content associated with n is that of the exclusion and inclusion in its double relation: more in less or less in more", (p. 301). vi The theoretical meeting concretely took place in the 12 th Colloquium of Ibero-Roman linguistics in Rennes in 2008 where Bottineau first presented his conference, «The grammatical submorphology in Spanish and the theory of the cognemes», followed by Luquet"s one, «On the iconicity of the grammatical morphemes in Spanish ». See Bottineau (2010a) and Luquet (2010). vii See Gérard Moignet: "As a language system, it displays its coherence in the progression that it shows of an increasingly precise and differentiated image of time as we go from the most virtual to the most actual" (1981: 65). viii Apart from the act of language, excluding when speaking, there are as many present tenses as human beings conceiving them.

Fig. 1 :

 1 Fig. 1 : Chronogenetic operation (Le Tallec-Lloret 2010: 98)

Fig. 2 :

 2 Fig.2: Verbo-Temporal System in Spanish -actualising mode vs. inactualising mode(Le Tallec-Lloret 2010: 118)