Extremes for stationary regularly varying random fields over arbitrary index sets

We consider the clustering of extremes for stationary regularly varying random fields over arbitrary growing index sets. We study sufficient assumptions on the index set such that the limit of the point random fields of the exceedances above a high threshold exists. Under the so-called anti-clustering condition, the extremal dependence is only local. Thus the index set can have a general form compared to previous literature [3, 21]. However, we cannot describe the clustering of extreme values in terms of the usual spectral tail measure [23] except for hyperrectangles or index sets in the lattice case. Using the recent extension of the spectral measure for star-shaped equipped space [18], the $\upsilon$-spectral tail measure provides a natural extension that describes the clustering effect in full generality.


Introduction
Asymptotic results for extreme values of random fields have attracted much attention recently, see Samorodnistky and Wu [23], Basrak and Planinic [3], and Jakubowski and Soja-Kukieła [14], to name a few.Extending the basic results of Basrak and Segers [2] in the context of time series, the newly developed approaches focus on stationary regularly varying R d -valued random fields X = (X t ) t∈Z k : The random vectors (X t1 , ...., X tn ) are regularly varying in R nd for each t 1 , ..., t n ∈ Z k .The existence of the spectral tail random field Θ := (Θ t ) t∈Z k characterizes the limit behavior of the extremes around the origin {0} under the condition that X 0 is extreme and normalized by |X 0 |.The random field Θ characterizes the extrema of X and hence any asymptotic extreme value set.One phenomenon is the clustering of extrema, i.e., the tendency for extrema to occur locally.To formalize this phenomenon, the approach is to extend the basic result of Davis and Hsing [5] via the convergence of the point process of exceedances.More precisely, we define N n as a simple point random field of exceedances on a hyperrectangle The level of excesses is set to a n , which satisfies lim n→∞ nP(|X 0 | > a n ) = 1, as if the observations were independent.Samorodnistky and Wu [23] show that N n converges to a cluster point random field N on R d \ {0} and an explicit representation of the latter is given in Basrak and Planinic [3].More precisely, there is a spectral cluster field Q := (Q t ) t∈Z k whose distribution is derived from that of (Θ t ) and for which holds where ( t∈Z k ε Qi,t) i≥1 ) are independent and identically distributed (iid) copies of t∈Z k ε Qt , independent of the points (Γ i ) of a standard Poisson process.The random field Q is crucial since it accurately describes the asymptotic clustering phenomenon.The paper aims to introduce and analyze a new setting adapted to index sets other than the hyperrectangle C n .
Let us consider Λ n as an arbitrary index set of Z k \ {0}.Such an extension of the rectangular index set is not straightforward, as Stehr and Rónn-Nielsen [21] showed in the asymptotically independent case (Θ t = 0 for all t = 0).For index sets (Λ n ), a geometric condition must hold.To motivate the study of index sets that are not rectangular, let us describe the most common index sets in the literature.The spatio-temporal sets Λ n are typically of the form C × {T, . . ., mT }, where C is a fixed lattice of Z 2 and T ≥ 1 is the observation period through time expressed in space-time units, and m is the number of observation periods.As usual, we consider a stationary, regularly varying random field (X t ).Remark that the assumption of stationarity in time and space on X is standard in environmental statistics, even when the spatial grid C is large but finite, see the review paper by Davison, Padoan and Ribatet [10] and references therein.We first obtain the existence of a limiting point random field N Λ as follows where a Λ n comes from lim In contrast to the hyperrectangle case, the distribution of the edge point random field N Λ depends on the asymptotic lattice properties of the general lattice Λ n .Not surprisingly, the asymptotic form of the index set Λ n constrains the clustering effect.We derive the limiting distribution of the point random field under a sufficient condition that ensures that Λ n consists asymptotically of translated versions of countably many fixed sets D j .It appears that the limiting cluster point random field distribution is a mixture of expressions of the spectral tail random field over the different D j .However, a representation of the clusters using the original spectral tail random field (Θ t ) is limited to specific Λ n .We derive the representation of the limiting points similar as in Basrak and Planinic [3] only when all the D j 's are lattice.This condition is satisfied for C n , and we recover the characterization of the clusters first provided in Basrak and Planinic [3].For irregular index set Λ n , the representation of the (asymptotic) clusters does not naturally use the original spectral tail random field (Θ t ).Instead, one has to introduce the concept of the Υ-tail field that characterizes the limiting behaviour of the extremes around the region Υ given that X t is extreme over Υ and normalized by a modulus of (X t ) t∈Υ .This framework has already been developed for iid sequences by Ferreira and de Haan [8] and for time series cases by Segers et al. [18] but not for random fields.Our main contribution is to introduce a very general setting for index sets, namely Condition (D Λ ), which does not involve any topological properties.This condition allows for countably many different shapes, and it is only an asymptotic condition.In particular, it implies that some shapes repeat approximately an infinite number of times proportional to |Λ n |. Surprisingly, the shape of the asymptotic local region Υ is arbitrary.In contrast with existing results such as Stehr and Rónn-Nielsen [21] we show that convexity is not required when dealing with extremes.Under the anti-clustering condition, we deal with index sets that are local regions such as Υ reproduced over a lattice.The rest of the paper is organized as follows.In Section 2 are collected preliminaries, notation and main assumptions.That the crucial Condition (D Λ ) implies asymptotic local region reproduced over lattices is the main theoretical challenge of the paper, alleviated in Section 3. The asymptotic clusters are studied in Section 4 for any index set Λ n satisfying Condition (D Λ ).Their characterization is provided using the Υ−spectral tail field in Section 5. Two applications of this new approach are developed in Section 6, determining the extremal index and providing sufficient conditions for max-stable random fields.Section 7 contains the proofs of the results of Section 3 and the rest of the proofs are collected in Section 8 and 9.
2 Preliminaries, notation and main assumptions Let (X t ) t∈Z k be an R d -valued regularly varying stationary random field.

Spectral tail fields
Let us recall two fundamental results of Samorodnitsky and Wu [23]: the existence of the tail field and the time change formula for the tail and spectral tail fields.Theorem 1 (Theorem 2.1 in [23]).An R d -valued stationary random field (X t ) t∈Z k is jointly regularly varying with index α if and only if there exists a random field (Y t ) t∈Z k such that as x → ∞, and P(|Y 0 | > y) = y −α for y ≥ 1.We call (Y) t∈Z k the tail field of (X t ) t∈Z k .
Theorem 2 (Theorem 3.2 in [23]).Let (Y) t∈Z k be the tail field corresponding to an R d -valued stationary random field (X t ) t∈Z k that is jointly regularly varying with index α and define Θ t = Y t /|Y 0 |, t ∈ Z k .Let g : (R d ) Z k → R be a bounded measurable function.Take any s ∈ Z k .Then the following identities hold: We call (Θ t ) t∈Z k the spectral field of (X t ) t∈Z k .
We consider a complete order ≺ on Z k that is invariant: if s ≺ t for s, t ∈ Z k implies that s+i ≺ t+i for any i ∈ Z k .An example of an invariant order is the lexicographic (or dictionary) order: for s, t ∈ Z k , we say that s ≺ t if either (1) s 1 < t 1 , or (2) there exists 2 ≤ j ≤ k such that s i = t i for all i = 1, ..., j − 1, and s j < t j .
with λ i > 0 and q i=1 λ i = 1, where q ∈ N ∪ {∞} is the number of these Ds.
Condition (D Λ ) says the following.Consider a point t in Λ n .Translate the set Λ n by −t so that t is now at 0. Take an hypercube around 0 of side 2p, for p large enough, and intersect it with the positive points according to and with the translated set.Thus, we have obtained Λ (t,p) n . Now, it might happen that the same set Λ (t,p) n is exactly the same for other points in Λ n , and also for other points in Λ m with m > n.Condition (D Λ ) imposes that there are different sets (denoted D i ∩ K p , i ∈ {1, ..., q}) such that the number of points t ∈ Λ n for which Λ (t,p) n is equal to one of these sets, divided by |Λ n |, has a limit and these limits form a weighted sum.This condition provides a minimum requirement to have (at least asymptotically) a structure for studying the long-time clustering behaviour of extremes.Example 1. Imagine observing precipitations over a specific geographical area C.There are stations spread throughout the geographical region that measure the precipitation.Let Λ n lying in Z d where d is the sum of the time dimension and the space dimension (thus d = 3 or d = 4 depending on whether we consider the geographical region C to lie in Z 2 or Z 3 , respectively).In the time direction, each point is the number of rain rained in a certain amount of time, while the space direction indicates the location where this is measured.Thus, X t where t ∈ Λ n corresponds to the amount of rain measured in a certain period in a specific location.Assume that the measurements over C repeat in a constant frequency (e.g. at every week).This assumption corresponds to condition (D Λ ), where we take the order to be increasing with successive observations.In particular, imagine measuring over C infinitely many times.Denote this set by C ∞ .Then, each D is C ∞ centered at 0 (that is translated version of C ∞ by minus one of its points) and consider the points successive to 0. Notice that only q = |C| distinct D. [4] already considered similar index sets.
Example 2. The framework of [21,22] is a particular specification of our framework.Indeed, consider Assumption 1 in [21] (which is Assumption 3 in [22]): The sequence (C n ) n∈N consists of p-convex bodies (i.e.connected sets which are also unions of p convex sets), where is bounded for each j = 1, ..., d − 1, where V j (C n,i ) indicates the intrinsic volumes of the convex body C n,i .Consider the two dimensional case, so d = 2 -similar arguments apply to other dimensions.For any convex body C, we have that V 0 (C) = 1 and V 1 (C) is equal to the perimeter of C divided by π.Then, Assumption 1 in [21] states that the sum of the perimeters of the C n,i s must not grow faster than the square root of the volume of C n .There are cases where this is not true, like when one of the C n,i s is a rectangle with edges increasing with different speed.In general, this assumption ensures that the C n,i s must grow in all directions, implying that the number of points in C n away from the boundary divided by the number of points in C n tends to 1 as n → ∞.Formally it implies that, for any r ∈ N, where D is simply given by {t ∈ Z k : t 0}.Therefore, Assumption 1 in [21] is strictly stronger than condition (D Λ ).It is important to explicitly look at the differences of our framework with the one of [21,22].First, Condition (D Λ ) is only an asymptotic condition, thus the set Λ n (or C n ) does not need to satisfy any constraint for finite n.The lack of a non-asymptotic structure for Λ n is a challenge, and in particular for the proof of Theorem 17.We overcome this by imposing structures that will be satisfied asymptotically.Another feature of our setting also exacerbates this issue: the possibility of having countably many different asymptotic sets (denoted by Ds).Indeed, having countably many sets does not allow distinguishing the points in Λ n that will eventually form an asymptotic set from other points in Λ n , because this distinction happens only asymptotically.We overcome this by using that only finitely many of these sets have weights (denoted by λs) greater than ε, for any ε > 0. The third difference is the structure of the asymptotic sets.While in [21,22] the only allowed asymptotic set is {t ∈ Z k : t 0} as just shown, in our framework any possible subset of {t ∈ Z k : t 0} is allowed.For example, we might have that Λ n is a rectangle where only one side increases.
We conclude this section by pointing out that condition (D Λ ) comes from the proof of the main asymptotic results of the paper, and it is the most refined (i.e.weakest) condition we could attain.This condition is satisfied in all the previous settings (see [5,23,21,22,4]).

Mixing and anti-clustering conditions
Following the seminal work of Davis and Hsing [5] on stationary time series, we assume two complementary conditions.The anti-clustering condition avoids too strong clustering effects.The mixing condition approximates the Laplace functional of the point random field N Λ over Λ n in terms of products of Laplace functionals of copies of the point random field over a smaller index set.Such conditions were extended to random fields by Samorodnistky and Wu [23] for the specific index set C n = [1, n] k .Some care is required when considering the general index set Λ n .Take a sequence of positive integers (r n ) such that lim n→∞ := max i∈R l,Λn |X i | and consider the following anti-clustering Condition (AC Λ ): The R d -valued stationary regularly varying random field (X t : t ∈ Z k ) satisfies the (AC Λ ) condition if there exists an integer sequence r n → ∞ and Let d n := max x,y∈Λr n max j=1,..,k |x (j) − y (j) |, namely d n be the maximum distance between the points of Λ rn .Observe that . This sufficient condition is often easier to check in practice and is implied by the anti-clustering condition considered in Samorodnistky and Wu [23] that required a stronger condition on the maxima over indices a ≤ |i| ≤ b in any directions.For the mixing condition, we require extra classical notation, namely and for any E ⊂ R d , C + K (E) the class of continuous non-negative functions g on E. Further, let the Laplace functional of a point random field ξ with points (Y i ) in the space E ⊂ R d be denoted by We adopt the notation

Lattice properties
Before giving the main results, we need to investigate further the lattice properties of the index sets D j appearing in Condition (D Λ ) on Λ n .We will distinguish two settings, lattice properties on the upper orthant and the whole index set.Condition (D Λ ) implicitly involves the upper orthant and it would have been possible to focus on the whole index set by adapting Condition (D Λ ) accordingly.This approach would have been entirely equivalent to ours.But notice that the two settings are crucial for our main results, and one cannot make the economy of one of them.

Lattice properties on the upper orthant
Recall that D 1 , D 2 , ... are the subsets of the upper-orthant Z k + that appear in Condition (D Λ ).
(I) For every D j and D i with j = i there exists a p large enough s.t.D j ∩ K p = D i ∩ K p .Further, for every D j and every p ∈ N we have the identity where The empty set is a possible D. (III) For every D j , there exist b j many different Ds, where b j ∈ N s.t.b j ≤ 1/λ j − 1, which we denote by D l1 , ..., D l b j such that, for every z ∈ D j , D li = ((D j ) −z ) + for some i = 1, ..., b j and then λ i ≥ λ j .
Point (III) of Proposition 3 suggests that D j contains shifted versions of potentially different D .In order to exhibit the lattice property of D j , we define G j as the set of the shifts that yields the same D j , namely For every 1 ≤ j ≤ q, one can partition the set D j using the lattice sets L li , i = 1, ..., b j : then the set L j is a lattice on Z k .For i = 1, ..., b j , denoting z li any point in D j such that D li = ((D j ) −z l i ) + we have the partition Further, for every D j , we have that L li ⊇ L j , and L li and L j have the same rank for i = 1, ..., b j .
In particular, D j is bounded if and only if L j = {0} and in this case D j = bj i=1 {z li }.Building on partition (4) we want to exhibit some translation invariant properties of D j .Fix any j = 1, . . ., q and denote l 0 = j for convenience, then any i ∈ {0, 1, ..., b j } satisfies the Translation Invariance Property (TIP j ) if it has the following property: Further, we let W j denote the subset of {0, ..., b j } satisfying (TIP j ) and let Dj := h∈Wj D l h .Proposition 5. Let Λ n satisfy |Λ n | → ∞ as n → ∞ together with Condition (D Λ ).Fix any j = 1, . . ., q.If i ∈ {0, 1, ..., b j } satisfies (TIP j ) then L li = L j and λ li = λ j .In particular, when L j is a full rank lattice the (TIP j ) condition is satisfied for all i = 0, ..., b j and when D j is bounded the (TIP j ) condition is never satisfied.
Further, for every i ∈ W j we have Dli = Dj and Dj ∪ {0} ∪ − Dj is translation invariant for every point in L j .Remark 1.The case of (TIP j ) not holding for some l = l 1 , ..., l bj is equivalent to the case of G j lying on the hyperplane determined by the order .For example, this is the case when we are in R 2 , the order goes along the horizontal lines (informally (0, 0) ≺ (0, 1) ≺ (0, 2) ≺ ... ≺ (0, ∞) ≺ (1, −∞) ≺ ... ≺ (1, 0) ≺ ...), and Λ n draws two lines which are parallel to the horizontal axis, see Figure 1 for an illustration.It is possible to see that in this case one D (say D1 ) is is simply given by x-axis, while for the other ( D2 ) we have the set provided in Figure 2.These sets are translation invariant with respect to the points in the respective L i and in this example L 1 and L 2 are both equal to x-axis.

Lattice properties on the whole index set
In this subsection we consider subsets Ξ j of the whole index set Z k that are the equivalent of the subsets D j of the upper-orthant.As Condition (D Λ ) defined only the D j , the existence of the Ξ j , shown in the next result, is deduced from it.Proposition 6.Let Λ n satisfy |Λ n | → ∞ as n → ∞ together with Condition (D Λ ).For any p ∈ N and any Ξ subset of Z k with 0 ∈ Ξ we have that the limits We consider an order that increases along the horizontal axis and then upward.Red chopped half-line starting at (1, 0) is D 1 , D 1 plus the red line above is D 2 also in red.Both Ds have the same G = D 1 ∪ {(0, 0)} thus the same L which coincides with the x-axis.We check that D 2 is partitioned into L + and L z where z is any point with 1 as the second coordinate.However, none of the points of L z precedes any point in G, and the TIP property fails.Notice E 1 is any couple of points {(0, 0), z} in blue.exist.Moreover, any such Ξ such that satisfies Ξ + = D j for some j = 1, ..., q.
It is important to notice that the translations of bj i=0 (L li ) z l i can coincide.A careful analysis is done in order to describe the distinct translations.Recall that l 0 = j and that L j = L l0 ⊂ L li for every i = 1, ..., b j , are same rank lattices by Proposition 4. Thus there exists a finite number of translations of L j covering any L li , i = 1, ..., b j .Denote n j ≥ 1 this number.Moreover, let x nj } where, for the sake of clarity, we include by convention {0} in E j so that 0 is always the lowest (according to ) point in E j .Notice that a certain arbitrary choice is still possible when choosing E j , see Figure 1 for an example.Any Ξ m contains {0} by definition.Thus, the different Ξ m s correspond to the different translated versions of bj i=0 (L li ) z l i containing {0}.Having in mind the identity s∈Ej (L j ) s = bj i=0 (L li ) z l i , the number of different translations is thus n j and the shifts are the elements of E j (and thus n j = |E j |).Denote I * the set of the indices j = 1, . . ., q satisfying For every j ∈ I * , let Ξ * j := s∈Ej (L j ) s and γ * j be the positive limit in (5) associated to Ξ * j .For every j ∈ I * we have (Ξ * j ) + = D j , that there exists an m = 1, . . ., q such that Ξ * j = Ξ m and that any Ξ m , m = 1 . . ., q , are translated versions of Ξ * j , j ∈ I * .It essentially means that the Ξ * j for j ∈ I * are the only relevant structures in the asymptotic of (Λ n ) as the other ones are translated versions of them:

By definition we have
for any j ∈ I * .
In the following statement, we link the asymptotic behaviour of (Λ n ) with specific non-asymptotic properties of some of its subsets.In particular, we extract from (Λ n ) specific disjoint subsets, which have helpful non-asymptotic properties(for the proof of Theorem 17) and show that these subsets asymptotically describe the whole (Λ n ) satisfying Condition (D Λ ).
We introduce the following notation.Let l ∈ N. Consider the maximum of the m ∈ N ∪ {0} such that D i ∩ K l = D j ∩ K l for every i, j ∈ I * with i, j < m.Denote this maximum by ml,1 .
Consider the maximum of the m ∈ N ∪ {0} such that D w ∩ K l/2 = D l h ∩ K l/2 for every w, v ∈ I * with w, v < m and where D w is bounded, D v is unbounded and l h is any index l 1 , ..., l bv .Denote this maximum by ml,2 .Consider the maximum of the m ∈ N ∪ {0} such that E s ⊂ K l/4 for every s ∈ I * with s < m.Denote this maximum by ml,3 .Then, we define m l as follows m l := min( ml,1 , ml,2 , ml,3 ).Notice that such m l exists because l is finite and b j is finite for every unbounded D j .Further, let S i,l := {t ∈ Λ n : (Λ n ) −t ∩ K l = Ξ * i ∩ K l } for every i ∈ I * and l, n ∈ N. Notice that S i,l depends on n, but we omit the dependency to lighten the notation.Proposition 9. Let Λ n satisfy |Λ n | → ∞ as n → ∞ together with Condition (D Λ ).Then for every n ∈ N, j, i ∈ I * with j, i < m 4l and i = j, t ∈ S j,4l and s ∈ S i,4l , we have that Further, there exists a set S j,4l , with S j,4l ⊂ S j,4l , such that for every t ∈ S i,l and that lim n→∞ |S j,4l |/|Λ n | = γ * j for every j ∈ I * with j < m 4l .Finally, we obtain that We remark that even if the Condition (D Λ ) is asymptotic, the sets S j,4l and S j,4l have both asymptotic and non-asymptotic properties.
Example 3 (Continuing Example 1).Recall that in this example, the observations formed a pattern C that repeated itself with a certain frequency in order to constitute C ∞ .Using the notation of this section, we see that such frequency is represented by G 0 .The number of D's is b 0 + 1 = |D|.The Translation Invariance Property (TIP j ) is satisfied for every i = 0, ..., b j and j = 1, ..., |D|.The infinite union of translated patterns is what we denote by Ξ in this section.Notice that any different centering of C ∞ at 0 corresponds to a different Ξ m , namely one of the Ξ's which we denoted Ξ * h for some h = 1, ..., |C|.Further, notice that I * = {h}, that C = E h and that γ * h = 1/|C|.A non-trivial result, even in this simple example, is the last statement in Proposition 5. Suppose we take the union of all the D's and {0} and their negative counterpart, then this union is translation invariant along with L 0 , namely along that certain frequencies with which the observations repeat their pattern.Moreover, Proposition 9 states that it is important that the observations (Λ n ) repeat the pattern C for a sufficiently long time.For instance, a finite number of observations (for instance, cases where the station is working intermittently -this is typical of non-automatic weather stations) does not matter.
Finally, we refer to the lattice case when every Ξ's are lattices, meaning that E j = {0} and Ξ * j = L j for every j = 1, . . ., q.
4 Main results expressed using the spectral tail field 4.1 Laplace functional of the limiting point random field The first result states the convergence of the Laplace functionals to some Laplace functional without an explicit description of the point random field.The proof of the result is based on a telescoping sum argument developed initially in the time series setting by Jakubowski and co-authors [13,1] together with lattice property (I) from Proposition 3.
→ N Λ on the state space R d \ {0} and the limit random measure has Laplace functional for g ∈ C + K , given by Remark 2. Notice that by Tonelli's theorem and by the monotone convergence theorem, the Laplace transform is the one of a mixture distribution Remark 3. In the asymptotically independent case, we have that |Θ t | = 0 for every t = 0 and so the limit random measure has Laplace functional Thus, it coincides with the case of one D and in particular D = ∅.

The spectral cluster random field in the lattice case
Define for any set A ⊂ Z k , any sequence x = (x t ) t∈Z k and any α > 0, For the spectral tail random field (Θ t ) t∈Z k of a regularly varying stationary random field we use as the normalisation constant.When x A,α < ∞ a.s., We define the spectral cluster random field by Using the lattice properties investigated in Proposition 3, we show the existence of the spectral tail random field over some lattice index sets.

Cluster point random field expressed using the spectral cluster field in the lattice case
Now, we present an explicit formulation of the asymptotic Laplace functional as a mixture of cluster random fields when the Ds are lattices (on the positive points).Theorem 12. Consider an R d -valued stationary regularly varying random fields (X t ) t∈Z k with index α > 0. We assume conditions (D Λ ), (AC Λ ) and A Λ (a Λ n ).Assume also that we are in the lattice case.Then, N Λ n d → N Λ on R d 0 and the limit admits the cluster point random field representation where , is an iid sequence of point random fields with state space R d , and where (Γ j,i ) i∈N are the points of a unit rate homogeneous Poisson process on (0, ∞) independent of is a sequence of independent point random fields with state space R d .We extend the characterization of the clusters first provided in Basrak and Planinic [3] on the whole index set Ξ * 1 = Z k , q = 1 to potential mixtures of lattices with q > 1.For instance, when the observations grow frequently along the axis.In this case, we have q = k, Ξ * j = {0} j−1 ×Z×{0} k−j , and γ * j = λ j , for every j = 1, ..., k.The value of the weights depends on fast the observations grow along one axis relative to the others, e.g. if on axis j there are twice the observations on axis i (as 5 Point random field convergence using Υ−spectral tail field For general index set Λ n satisfying Condition (D Λ ) that are non necessarily lattice, we introduce new spectral tail fields.

The Υ−spectral tail field
Let ρ be a modulus of continuity on (R d ) Z k and for any finite Υ ⊂ Z k let ρ Υ the truncation of ρ to R dΥ .In the following, we extend some of the results of Basrak and Segers [2] to the case of the random fields.In the time series case, the following result is contained in Theorem 5.1 of Segers et al. [18].Proposition 13.Let (X t ) t∈Z k be a regularly varying of index α random field in R d , with α ∈ (0, ∞).Let Υ be a finite subset of Z k .Then there exists a random field Moreover, there exists a random field It is possible to see that Θ Υ in distribution is given by Y Υ /ρ Υ (Y).For stationary regularly varying random fields it is possible to extend the time change formula of Theorem 3.2 in Samorodnitsky and Wu [23] to Υ-spectral tail field: Proposition 14.Let (Y Υ,t ) t∈Z k be the tail random field in Proposition 13 and consider Remark 4. It is possible to see that by definition ρ Υ (Θ Υ ) = 1 a.s..

Asymptotic Laplace functional expressed using the Υ−spectral tail field
We start with a simple result on the relation between the uniform norm and the other modulus of continuity.Lemma 15.Let Υ be a finite subset of Z k .There exists two positive constant C and D with C ≤ D such that, for every > 0, max Proof.From Lemma 15 we have that ρ Υ (x) < 1 + δ implies that max t∈Υ |x t | < (1 + δ)D for every δ > 0.
We let C m and D m denote the constants of Lemma 15 for E m , for every m ∈ I * .Notice that C m and D m depends on the chosen ρ, but we do not write the dependency explicitly in the notation because it does not create confusion and it lightens the notation.Remark 5. Our setting, and in particular the following Theorem 17, is general enough to allow for countable infinitely different moduli of continuity to be used at the same time, one different ρ j for each E j as in Lemma 28.With some abuse of notation we denote ρ j,Ej by ρ Ej .
Now, consider the following assumption on the modulus of continuity.
This condition is satisfied in many cases.For example, if the modulus of continuity is unique and coincides with the uniform norm then D j = 1 and c j ≤ |E j |, and so We remark that such condition is needed to implement a dominated convergence theorem in the proof of Theorem 17 and so, as it happens in most of the cases where a dominated convergence theorem is used, it might be possible to obtain the result for a specific ρ even if condition A Λ ρ is not satisfied.We are now ready to state an anti-clustering condition tailored for conditioning on the modulii of X being large over a local subset and not necessarily X 0 .For every j ∈ I * , let R Remark 6.We remark that condition (AC Λ ,I * ) is weaker than assuming that for every j ∈ I * lim l→∞ lim sup ) is m-dependent then the anti-clustering conditions considered in this paper, namely (AC Λ ) and (AC Λ ,I * ), are satisfied.Moreover, it is possible to see that in some cases condition (AC Λ ,I * ) is strictly weaker than condition (AC Λ ).As we see in the following example.

Example 4 (Continuing Example 1).
Recall that in setting of Example 1 I * = {h} and that we denote E h by C. Thus, the condition (AC Λ ,I * ) in this setting is: In case we know that the pattern of observations will be the same, namely which in practice means that the weather stations perform regularly, then R On the other hand we have that R l,Λn is given by t∈Λn l,Λn .Then, it is possible to see that (AC Λ ,I * ) is strictly weaker than condition (AC Λ ).
Let Dj = s∈Gj \{0} (E j ) s .We are now ready to present of the main results of this paper.
Theorem 17.Consider an R d -valued stationary regularly varying random field (X t ) t∈Z k with index → N Λ on the state space R d \ {0} and the limit random measure has Laplace functional for g ∈ C + K , given by where c j = lim n→∞ P(ρ E j (X)>an) P(|X0|>an) .

The Υ−spectral cluster field
Recall that G j is the lattice intersected with the non negative points associated to L j and that the extension G j to the whole Z k is just given by L j = G j ∪ −G j .For every E j , denote by Notice that H j coincides with Ξ * j for j ∈ I * .Proposition 18.Consider an R d -valued stationary regularly varying random fields (X t ) t∈Z k with index α > 0. We assume conditions (AC Λ ,I * ) and (A Λ ρ ).Then, |Θ Ej ,t | → 0 a.s.for any |t| → ∞ and t ∈ D j , and so t∈Lj ρ (Ej )t (Θ) α < ∞ a.s. and t∈Hj |Θ Ej ,t | α < ∞ a.s.for every j = 1, ..., q.
Let Υ be a finite subset of Z k and A a subset Z k and ρ a modulus of continuity.Define as the normalisation constant.We define the spectral cluster random field by A,α , where the dependence on ρ is implicit.Notice that when the modulus of continuity is ρ α , Υ is E j , and A is L j then Observe that for bounded D j we have that E j = {0} ∪ D j = H j and that G j = L j = {0}.We remark that when Υ = {0} we have that Θ {0} , Θ {0} ρα,A,α , and Q {0},A are simply given by Θ, Θ A,α , and Q A (see Section 4.2).

5.4
Cluster point random field expressed using the Υ−spectral cluster field Theorem 19.Consider an R d -valued stationary regularly varying random fields (X t ) t∈Z k with index α > 0. We assume conditions (D Λ ), (AC and the limit has Laplace functional for g ∈ C + K , with the following expression: Proof.It follows from Theorem 17 using the same arguments as in the proof of Theorem 12, the time change formula (10) and the fact that from Proposition 18 we have that t∈Lj ρ (Ej )t (Θ) α < ∞ a.s., for every j ∈ I * .
In the following corollary we consider Theorems 17 and 19 when the modulus of continuity is ρ α .Corollary 20.Let the modulus of continuity be ρ α .Consider an R d -valued stationary regularly varying random fields (X t ) t∈Z k with index α > 0. We assume conditions (D Λ ), (AC Λ ,I * ) and 0 admitting, for g ∈ C + K , the Laplace functional: where Proof.The result follows from Theorems 17 and 19 and from the fact that when the modulus of continuity is ρ α we have that: Moreover, we have the following result on the representation of N Λ .Proposition 21.Consider N Λ given in Theorems 17 and 19, then where , is an iid sequence of point random fields with state space R d , and where (Γ j,i ) i∈N are the points of a unit rate homogeneous Poisson process on (0, ∞) independent of (Q Ej ,Lj ,i,t ) t∈Ξ * j , for every j ∈ I * .Moreover, is a sequence of independent point random fields with state space R d .Finally, in the setting of Corollary 20 we have where l∈N ε Qi,l i∈N , is an iid sequence of point random fields with state space R d with mixing distribution ) for every i ∈ N, and where (Γ i ) i∈N are the points of a unit rate homogeneous Poisson process on (0, ∞) independent of ( Ql ) l∈N .
Proof.The result follows from Theorems 17 and 19 identifying the limiting Laplace functionals such as the ones of cluster Poisson random fields.
Remark 8. Notice that the chosen order does not affect the spectral tail random field.This fact is a clear advantage of the Υ−spectral cluster field approach compared to the approach of Section 4 because it allows the asymptotic representation (12) in full generality, not just in the lattice case.
Example 5 (Continuing Example 1).Armed with the results of this and the previous section we can present the extreme asymptotic behaviour of N Λ n , where Λ n is described in Example 1.Then, where we consider ρ α as the modulus of continuity.We remark that such a clean result is not achievable when the anti-clustering condition (AC Λ ,I * ) does not hold.Since in this case (AC Λ ,I * ) is strictly weaker than (AC Λ ), this representation is equivalent to Moreover, we can see that it is not important which reference point h ∈ C we consider, since our results enjoy certain translation properties.In particular, this representation is equivalent to where s is any element of C and Ξ * s is any different centering of C ∞ at 0 (in our example L h = L s for every h, s = 1, ..., |C|).

Applications 6.1 The extremal index
In this section we investigate properties of the extremal index for random fields; see the work of Hashorva [6] for max-stable random fields.First, let us define it.Definition 1 (Λ-extremal index).Consider an R d -valued stationary random field (X t ) t∈Z k .Assume that for each positive τ there exists a sequence (u n (τ )) such that lim X is the Λ-extremal index of (X t ).As shown by Samorodnistky and Wu [23] when Λ n = [1, n] k , the extremal index is connected with the so called block extremal index.In particular, let For the sake of simplicity, we provide our results for random fields with respect to the modulus ρ α in this section.Thus, D j = 1 and i∈I * γ * i |E j | = 1.For every j ∈ I * , generalizing the approach of Janssen [9] for processes to random fields, let T * j be defined as follows: for t ∈ L j define ) is satisfied then T * j is well defined thanks to the summability proved in Proposition 18 (see also the end of the proof of Lemma 29 for the connection between the summability of ρ and the one of the max norm).If (AC Λ ) is satisfied and all H j 's are lattices (namely all D ∪ {0} ∪ −D's and all the Ξ's are lattices) then T * j is also well defined thanks to the summability proved in Proposition 11.Observe that when H j is a lattice then E j = {0} and H j = L j .
Theorem 22. Consider an R d -valued stationary random field (X t ) t∈Z k with index α and a sequence (Λ n ) satisfying the condition (D Λ ).
(1) If the anti-clustering condition (AC Λ ) holds, then the limit θ Λ b := lim n→∞ θ Λ n exists, is positive and has the representations (2) If also all the Ξ * j 's are lattices then θ Λ b admits the representations (3) In any case (1) or (2), if also the mixing condition where (k n and (r n ) are as in the anti-clustering condition (AC Λ ), is satisfied, then θ Λ X exists and coincides with θ Λ b .
It is possible to see that Theorem 22 (2) only applies to D's that are lattices.It is natural to ask whether or not a similar result holds for any D. The answer is positive if a different anti-clustering condition is assumed.In particular, we have the equivalent result when the anti-clustering condition (AC Λ ,I * ) holds.Theorem 23.Consider an R d -valued stationary random field (X t ) t∈Z k with index α and a sequence (Λ n ) satisfying the condition (D Λ ).
(1) If the anti-clustering condition (AC Λ ,I * ) holds, then the limit θ Λ b := lim n→∞ θ Λ n exists, is positive and has the representations (2) If also the mixing condition (15) where (k n ) and (r n ) are as in the anti-clustering condition (AC Λ ,I * ), is satisfied, then θ Λ X exists and coincides with θ Λ b .From the two previous results we obtain the following immedaite corollary.Corollary 24.Assume that (X t ) t∈Z k is an R d -valued stationary random field with index α, a sequence (Λ n ) satisfying the condition (D Λ ) and either (AC Λ ) or (AC Λ ,I * ) and (15).Then the extremal index θ X exists, is positive, and where Φ α (x) = e −x −α , x > 0, is the standard Fréchet distribution function and θ X is given in either (13) or (16) depending on which anti-clustering and mixing conditions are satisfied, (14) being available only when the Ξ * j are lattices.

Max-stable random fields
Consider a non negative stationary random field X = (X t ) t∈Z k (with state space E = R + and d = 1).
A fundamental representation theorem by de Haan [11] states that any stochastically continuous max-stable (real valued) random field X can be represented (in finite dimensional distributions) as where (U i ) i∈N is a decreasing enumeration of the points of a Poisson point process on (0, +∞) with intensity measure u −2 du, (V i ) i∈N are i.i.d.copies of a non-negative random field < +∞ for all t ∈ Z k , the sequences (U i ) i∈N and (V i ) i∈N are independent.Observe that the above definition implies that the marginal distributions of X are 1-Fréchet, that is The aim of this Section is to find a necessary an sufficient condition for the anti-clustering condition (AC Λ ,I * ) to hold for stationary max-stable random fields.We recall some notation: H j = s∈Lj (E j ) s where, for every j ≥ 1, E j are finite subsets of Z k including 0 and L j are any lattice of Z k (possibly degenerate).The following result is an extension of results in Samarodnistky and Wu [23].Notice that the limit (12) motivates the introduction of a mixing distribution on V t as in the second assertion below.Proposition 25.Let (X t ) t∈Z k be a stationary max-stable random field with non-negative values.Consider a sequence Λ n of subsets of translated j≥1 H j satisfying the condition (D Λ ) then (X t ) t∈Z k satisfies the (AC Λ ,I * ) condition for any r n → ∞ s.t.n/r n → ∞ if for any i ≥ 1 and j ∈ I * lim |t|→∞,t∈(Hi) is supported by a subset of a translation of a unique H j , j ≥ 1 then the condition (18) simplifies to lim |t|→∞,t∈(Hj for any j ∈ I * . Notice that these specific max-stable random fields could be used to model any asymptotic clustering due to our result (12).
Remark 9.Under Condition (D Λ ), the index set Λ n fills up the translated asymptotic index set j≥1 H j .Checking the conditions (18) and (19) requires the knowledge of the E j , j ≥ 1, beforehand.Thus it requires some prior knowledge on the grid of the observations.Example 6 (Continuing Example 1).Max-stable random fields have been widely used for modeling extremal phenomena such as storm, starting with the pioneer work of [20].The spatial model [19], called spectrally stationary, is widely used to model space dependence because of its simplicity.It is defined as follows: consider an iid sequence of stationary random fields V i,s , s ∈ Z 2 which are not null.Then X space s = max i≥1 U i V i,s is a stationary max-stable random field.However it does not satisfy conditions (18) nor (19) in any direction of Z 2 because it would contradict the stationary assumption on V i,s , s ∈ Z 2 .The M3 representation of [12,15] was introduced to bypass this issue.Consider now a state-space model with space defined over H 0 = t∈L0 (C) t where C is a finite subset of Z 2 .It is sufficient to check (18) and ( 19) where the limit is taken along the time direction only.Thus a stationary space-time process X t such that its space distribution is the one of X space t can satisfy conditions (18) and (19) when its extremes are sufficiently independent over time.A basic example is an iid process in time for which the condition max t∈E0 V t = 0 = max t∈C×{0} V t = 0 forces that V t = 0 for any other component t = (s, k), k = 0.Such spectrally stationary models in space were not attainable in previous studies, see Remark 5 (i) of [4], because they are not ergodic as shown in [7].
7 Proofs in Section 3

Proof of Proposition 3
Assume that the first statement of point (I) is false for at least some D j and D i with j = i.Then, for every p ∈ N, D j ∩ K p = D i ∩ K p , which implies that D j = D i contradicting Condition (D Λ ).By Condition (D Λ ) we infer on one hand that λ j,p ≥ i∈I (j) Fix ε > 0. As q i=1 λ i < ∞, there exists some m sufficiently large such that i≥m λ i < ε.Moreover, there exists p sufficiently large so that D j ∩ K p = D i ∩ K p for any i, j ≤ m from the reasoning above.Thus and dividing both sides by |Λ n | and letting n → ∞ we obtain As it holds for every ε > 0 it implies the desired relation λ j,p ≥ i∈I (j) p λ i .On the other hand λ i,p cannot be strictly greater than i∈I (j) p λ i .Indeed, defining the equivalence relation i ∼ j ⇔ i ∈ (I (j) p ) 1≤j≤q , one considers the partition of {1, . . ., q} generated by the equivalence classes for some j ∈ {1, . . ., q} belonging to the class ∈ P p then one gets which yields a contradiction.
For point (II), consider the case of (Λ rn ) n∈N whose points have a distance between each other which increases as n increases.The increase of the distance within the points of Λ rn as n increases allows the following fact: when we consider K p around one of the points, for every fixed p, all the other points are outside K p for every n large enough.Then, in this case there is only one D for (Λ rn ) n∈N and it is given by the empty set.
For point (III) we need the following Lemma.Lemma 26.For any 1 ≤ j ≤ q and z ∈ D j , there exists 1 ≤ i ≤ q such that D i = ((D j ) −z ) + and λ i ≥ λ j .
Proof.Consider z ∈ D j , 1 ≤ j ≤ q and let D := ((D j ) −z ) + .Notice that for every q ∈ N there exists a p ∈ N such that K p ⊃ ((K q ) −z ) + .Thus D j ∩ ((K q ) −z ) + ⊂ D j ∩ K p and thus z + s belongs to D j ∩ K p for any s ∈ D ∩ K q .Thus for every q ∈ N we have Assume that D does not coincide with any D i , 1 ≤ i ≤ q.Fix ε > 0 so small that it satisfies λ j > ε.Let m satisfies i>m λ i < ε as above in the proof of point (I) (thus j ≤ m) and p sufficiently large such that Using the notation introduced in the proof of point (I), we have which is in contradiction with (20).Therefore, D = D i for some 1 ≤ i ≤ q and we have and the relation λ i ≥ λ j follows from (20).
To prove Point (III) observe first that for any D j there exists b j ∈ N ∪ {∞}, with b j ≤ |D j | + 1, distinct sets (D j ) + −z , z ∈ D j .The sum of the corresponding weights λ i being smaller than 1 and larger than b j λ j , by Lemma 26 we get the constraint b j λ j ≤ 1 and Point (III) follows.

Proof of Proposition 4
First, for any 1 ≤ j ≤ q, let us show that G j is invariant by addition in the sense that if z ∈ G j and z ∈ G j we infer that z + z ∈ G j \ {0}.Indeed, z ∈ D j = ((D j ) −z ) + so that necessarily z + z ∈ D j .Moreover we have It shows that G j is invariant by addition on (Z k ) + .Thus G j is given by a lattice, namely given k (not necessarily linearly independent) distinct vectors v 1 , ..., v k ∈ Z k (i.e. a basis of Z l for l ∈ {1, ..., k} called the rank) we have the identity We will refer to the degenerate case G j = {0} as the case of null rank k = 0. Thus, L j is a lattice on Z k .Let us now we prove the existence of the partition (4).We have to show that for any z ∈ D j there exists a unique 1 ≤ i ≤ b j so that z − z li ∈ L li .We know that we have a unique D li such that D li = ((D j ) −z ) + = ((D j ) −z l i ) + for some 1 ≤ i ≤ b j .Assume without loss of generality that z z li .Thus, either z = z li and then z − z li = 0 ∈ G li .Or z − z li 0 and for any s ∈ D li we have s + z ∈ D j and thus s + z − z li ∈ D li .That z − z li ∈ G li follows by definition of G li .Since z is an arbitrary point in D j and since l i is unique as D li , we obtain the desired partition.Consider any D l so that there exists z ∈ D j satisfying D l = ((D j ) −z ) + .Then for any s ∈ G j \ {0} we have Since z ∈ D j = ((D j ) −s ) + then z + s ∈ D j and s ∈ D l .Thus we proved that s ∈ {z ∈ D l ∪ {0} : ((D l ) −z ) + = D l } =: G l and that G j ⊆ G l .Further, we show now that G j and G l , l = l 1 , ..., l bj , have the same rank.Assume the contrary.Thus, let G j = { m l=1 a l v l : a l ∈ Z} + where v 1 , ..., v m ∈ Z k are linearly independent and G l = { p l=1 a l v l : a l ∈ Z} + where v 1 , ..., v p ∈ Z k are linearly independent, with k ≥ p > m.Since G j ⊆ G l we know that v i = c i v i for some c i ∈ Z, for every i = 1, ..., m.Since a h v h ∈ G l \ G j for any a h ∈ Z such that a h v h 0 and since G j ⊆ G l (and G l is a lattice), we have that (G j ) a h v h ⊂ G l , and again by the lattice structure of G l we have Now, consider (D j ∩ K q ) \ i∈Gj \{0} (D j ∩ K q ) i , namely the points in D j ∩ K q without K + q (i) for every i ∈ G j \ {0}.By (21) we have am+1∈Z am+2∈Z where z l is defined in the statement of Point (V).Thus, |(D j ∩ K q ) \ i∈Gj \{0} (D j ∩ K q ) i | → ∞ as q → ∞ monotonically.In particular, there is a q * large enough such that there are λ j,q * |Λ n ||(D j ∩ K q ) \ i∈Gj \{0} (D j ∩ K q * ) i | asymptotically many points in Λ n , but since λ j,q * ≥ λ j we have that λ j,q * |Λ n ||(D j ∩ K q ) \ i∈Gj \{0} (D j ∩ K q * ) i | > |Λ n |, which leads to a contradiction.Thus, G j and G l , l = l 1 , ..., l bj , have the same rank.Finally, if D j is bounded then by definition of G j we have that G j = {0}.If G j = {0} then G li = {0} because G j and G li have the same rank, for every i = 1, ..., b j .Since b j is finite, we conclude that D j is finite.

Proof of Proposition 5
Since G j ⊆ G li , it remains to show that G j ⊇ G li considering that i satisfies (TIP j ).We notice that for x ∈ (L li ) + So that ((D j ) −x ) + is the unique D l , l = 1, . . ., b j associated to the lattice G l = G li and it coincides with D li .Then y − x ∈ D li , with y ∈ G j , is such that ((D li ) −(y−x) ) + = D j .We then obtain that G j ⊇ G li by exchanging the role of D li with the one of D j in the proof of G j ⊆ G li in Point (V).Further, by applying Lemma 26 to D li we conclude that λ j = λ li .If G j is a full rank lattices then it is spanned by k linearly independent vectors and there always exists a point s ∈ G j such that s z l for every l = l 1 , ..., l bj .This implies that the (LC l ) condition must be satisfied and G j = G l for every l = l 1 , ..., l bj .This concludes the proof of the first statement.
Let us now prove the second statement.By (TIP j ), for every i = 0, ..., b j , and points (V) and (VI) we have that there exists z ∈ D li such that ((D li ) −z ) + = D j .Hence, z ∈ D li contains a translated copy of D j hence a translated copy of any D l h , h = 1, ..., b j , already contained in D j .Thus (l h ; h = 0, ..., b j ) = (l h ; h = 0, ..., b li ) so that (l h ; h ∈ W j ) = (l h ; h ∈ W li ) and then Dli is the union of the same sets than Dj Dli = Concerning the translation invariance property, we need to check that Dj ∪ {0} ∪ − Dj is invariant to the translation by every point in the lattice L j , that is Dj ∪{0}∪− Dj = ( Dj ∪{0}∪− Dj ) s for every s ∈ L j .With no loss of generality consider s ∈ G + j so that for any h ∈ W j we have

Proof of Propositon 6
Let Ξ be a subset of Z k such that 0 ∈ Ξ and let p ∈ N. Denote by −v be the lowest point (according to ) of Ξ ∩ K p and denote the points of K p \ {t ∈ Z k : t −v} by −w 1 , ..., −w u , for some u ∈ N which depends on p.Let Φ 1 , ..., Φ v , for some v ∈ N, be the subsets of K + 2p such that for each h = 1, ..., v we have Φ h ∩ (K p ) v = ((Ξ ∩ K p ) v ) + .Similarly, for every i = 1, ..., u, let Ψ i,1 , ..., Ψ i,vi , for some v i ∈ N, be the subsets of K + 2p such that for each h = 1, ..., v i we have Ψ i,h ∩ (K p ) wi = (Ξ ∩ K p ) wi .Moreover, for every i = 1, ..., u, denote by Π i,1 , ..., Π i,si the nonempty subsets of K p \ {t ∈ Z k : t v} with highest point (according to ) given by −w i .Observe that ((Ξ ∩ K p ) v ) + = (Ξ ∩ K p ) v \ {0} and ((Ξ ∩ K p ) wi ) + = (Ξ ∩ K p ) wi for every i = 1, ..., u.First, for every n ∈ N we have that Second, we have the following identities and since for every i = 1, ..., u where the last equality follows by the definition of the Φ's and the Ψ's and the fact that Φ l = Φ m for every l, m = 1, ..., v with l = m, and that Ψ i,h = Ψ i,k for every i = 1, ..., u and h, k = 1, ..., v i with h = k.Now, thanks to Point (I) in Proposition 3 we have that for l = 1, ..., v, and similarly for Ψ i,h for i = 1, ..., u and h = 1, ..., v i .Then, we obtain that the following limit exists Further, observe that for p > p we have the inclusion thus, the following limit exists This concludes the first part of the statement.Now, let Ξ + = D j for every 1 ≤ j ≤ q and fix ε > 0. As q i=1 λ i = 1, there exists some m sufficiently large such that i≥m λ i < ε and there exists p sufficiently large so that Ξ∩K + p = D i ∩K p for any i ≤ m.Then, lim which concludes the proof.

Proof of Proposition 7
Consider first the case of bounded Ξ b .In this case, we have that |Ξ b | < 1/γ b + 1; otherwise we will have a contradiction because we will asymptotically end up with more points than the ones in Λ n .Further, denote by −z its lowest point according to , then we have which implies that (Ξ b ) z = D j for some j = 1, ..., q.Further, since lim and since by (4) L j ∪ bj i=1 (L li ) z l i = {0} ∪ D j we obtain that Ξ b = ({0} ∪ D j ) −z and the first statement follows.Now, let Ξ b be unbounded.We show that Ξ b is a finite union of translated lattices.Consider any point s in Ξ b .Let g p ∈ N be such that and since this holds for every p large enough, we get Then, we have that ((Ξ b ) −s ) + = D k for some k = 1, ..., q.By Proposition 4, we deduce that Ξ b is a union of translated lattices.Further, this union is finite because γ b is strictly positive.Now, consider a point r on the most preceding lattice of Ξ b .Then, ((Ξ b ) −r ) + = D j , for some j = 1, ..., q, and so ((Ξ b ) −r ) = bj i=0 (L li ) z l i , which concludes the proof of the first statement.Let us now prove the second statement.Let p ∈ N. Define the equivalence relation i ∼ j ⇔ i ∈ (F (j) p ) 1≤j≤q , one considers the partition of {1, . . ., q } generated by the equivalence classes P p = {1, . . ., q }\ ∼.Recall the definition of P p from the proof of point (I) in Proposition 3.For every l ∈ P p , let P p,l ⊂ P p such that i ∈ P p,l if where Π 1 , ..., Π u , u ∈ N, are the subsets of K − p \ {0}, we obtain that λ p,j = i∈P p,j γ p,i .Thus, we have that 1 = j∈Pp λ p,j = j∈Pp i∈P p,j γ p,i = i∈P p γ p,i .By applying Fatou's lemma, we get that Hence, 1 ≥ q j=1 γ j .By applying the same arguments as the ones used in the proof of point (I) in Proposition 3 we have that i∈F (j) p γ i ≥ γ j,p for every j ∈ P p , which implies that Therefore, combining the two results we have that q i=1 γ i = 1 and i∈F (j) p γ i = γ j,p for every j ∈ P p .

Proof of Proposition 8
Let j ∈ I * .Consider (Ξ * j ) −x for every x ∈ E j and observe that these are the only possible Ξs that can be formed by translations of Ξ * j .The weights of the sets (Ξ * j ) −x , x ∈ E j , are all equal to γ * j .This is because of the following arguments.Let γ b be the weight of (Ξ * j ) −x .Consider a point x ∈ E j \ {0}.For any p ∈ N, let g p ∈ N be such that Ξ Thus, for each j ∈ I * we have that the sum of the weights of the Ξs composed by the translations of , where the last equality comes from Proposition 7.

Proof of Proposition 9
First, we have that m l → ∞ as l → ∞ and, since by Lemma 8 we know that For every n ∈ N and j ∈ I * with j < m 4l , consider the set S j,4l .We let the dependency on n be implicit.Let j, i ∈ I * with j, i < m 4l and i = j.In the following we show that for every t ∈ S j,4l and s ∈ S i,4l we have that The idea behind the following proof is that by taking points in Λ n with certain structure on K 4l around them (i.e.Ξ ∩ K 4l for i ∈ I * with i < m 4l ) where l is large enough (see above), we ensure that the sets K + 2l around them do not intersect for different structures (i.e.(D j ∩ K 2l ) t ∩ (D i ∩ K 2l ) s = ∅, for every t ∈ S j,4l and s ∈ S i,4l and every i, j ∈ I * with i, j < m 4l and i = j).First, consider the case of D i and D j bounded.Notice that t = s because D j ∩ K 4l = D i ∩ K 4l and so Ξ * j ∩ K 4l = Ξ * i ∩ K 4l .Thus, if (D j ∩ K 2l ) t and (D i ∩ K 2l ) s have an intersection then one of the two Ξ * ∩ K 4l 's will have at least one point in K − 4l \ {0} (in particular at s − t if t s or at t − s if s t) which is impossible by definition of bounded Ξ * 's because its lowest point (according to ) is {0}.Second, consider the case of D i bounded and D j unbounded.Then, as before t = s.Moreover, if (D j ∩ K 2l ) t and (D i ∩ K 2l ) s have an intersection and s t then Ξ * i ∩ K 4l will have at least one point in K − 4l \ {0} which is impossible.If they have an intersection and t s, then we have for some l h = l 1 , ..., l bj .However, the equality is impossible by construction.Third, consider the case of D i and D j unbounded.Then, as before t = s.Further, if t s, then we have for some l h = l 1 , ..., l bj as in the previous paragraph, which is impossible by construction.We conclude by observing that the case s t is specular to the case t s.Now, we bound the number of points in S i,4l for every i < m 4l and i ∈ I * .For every D i bounded with i < m 4l and i ∈ I * , let For the unbounded case we have the following.Consider any D i unbounded with i < m 4l and i ∈ I * .Notice that (E i ) s ∩ (E i ) t = ∅ for every s, t ∈ S i,4l because E i ⊂ K + l ∪ {0} and because we are considering Ξ i ∩ K 4l and thus an intersection would violate the structure of Ξ i ∩ K 4l .For any t ∈ S i,4l , consider the set and denote it by Si,4l .We remark that because by construction for every t ∈ S i,4l and s ∈ S j,4l , where j ∈ I * with j = i and j < m 4l , we have that and we arbitrarily take out a point x ∈ S i,4l then we might end up taking out more than one point in Si,4l because the points in Si,4l need the existence of certain points in S i,4l around them.Thus, we need to show that it is possible to find a procedure in which by taking out a certain point in S i,4l we only take out one (and only one) point in Si,4l .
Consider the lowest point in Si,4l according to and denote it by w.Then, by taking out s w,1 from S i,4l we only take out w from Si,4l (but not from S i,4l ).This is because s w,1 is the lowest among s w,1 , ..., s w,v and since s w,1 is the lowest point in Si,4l , this implies that s w,1 = s t,h for every t ∈ Si,4l \ {w} and every h = 1, ..., v. Thus, by taking out s w,1 from S i,4l we are not taking out any other point in Si,4l apart from w. .
Concerning the asymptotic behaviour of S i,4l , in the bounded case, since lim , by continuity of the minimum function we obtain that In the unbounded case, notice that S i,4l ⊃ Si,4l ⊃ S i,p for every p ≥ 8l and every n ∈ N. Since lim 8 Proofs in Sections 4 and 5

Proof of Theorem 10
First, by A Λ (a Λ n ) it suffices to show that for any g ∈ C + K , (Ψ Ñ Λ rn (g)) kn converges to (8) as n → ∞.Then, by regular variation of |X| and the definition of (a n ) as n → ∞.So by Taylor expansion it suffices to prove that k n (1 − Ψ Ñ Λ rn (g)) converges to the logarithm of (8) as n → ∞.Denote t |Λr n | the highest element of Λ rn according to ≺, by t |Λr n |−1 the second highest one,..., by t 1 the lowest one.Let Recall that K l = {x ∈ Z k : x ∈ {−l, ..., l} k }.Using the stationarity of X and the fact that ≺ is shift invariant, we have where J Now, for the every point in Λ rn we have that To lighten the notation assume that q in Condition (D Λ ) is ∞, so that there are infinitely many Ds.By point (ii) in the construction of Λ rn only the points Observe that there are finitely many different subsets of K l .We denote their total number by τ l and denote them by . Thus, we have where µ (j) l }|.Recall from the proof of Point (I) in Proposition 3, that by defining the equivalence relation i ∼ j ⇔ i ∈ (I (j) p ) 1≤j≤q , one considers the partition of {1, . . ., q} generated by the equivalence classes P p = {1, . . ., q}\ ∼.Then, by (i) and (ii) and in particular by point (I) in Proposition 3 we have Therefore, we have that Notice that the above arguments hold for every l large enough.By monotone convergence theorem we have that Finally, the existence of the limiting random measure N Λ is ensured by Corollary 4.14 in [16].

Proof of Proposition 11
In order to prove Proposition 11 we need the following Lemma.
Lemma 27.Let (Y t : t ∈ Z k ) be an R d -valued random field such that the time change formula (1) is satisfied.
Let Υ be a subset of {t ∈ Z k : t 0} containing {0} and assume that Υ ∪ −Υ is translation invariant along the points of a (not necessarily full rank) lattice.
Proof.The proof is divided in two parts.In the first part we show that Denote by L the lattice and let G := L ∩ {t ∈ Z k : t 0}.We stress that {0} ∈ G. Let > 0.
Suppose that , and observe that the sets for every C > 0.Then, we have that for every 0 < D ≤ 1 and for every D > 1 we have that By the time change formula (1) we get where b is the number of points of Υ inside the fundamental parallelotope of L. Notice that we used that for every t ∈ G and h ∈ −Υ (i.e.−h ∈ Υ) we have that t x where x is one of the b different points in the fundamental parallelotope, which we denote by B to be consistent with the notation of the proof of Proposition 3. Thus, we have a contradiction and so being the smallest of these finitely many points according to .That is for every t ∈ Υ ∪ −Υ we have By construction |Θ T * | is a measurable function.Since the difference of two measurable functions is measurable and the intersection of two measurable sets is also measurable we have that {ω : T * (ω) = t} is a measurable set.Further, for any subset A of Z k , since (T * ) −1 (A) = ∪ t∈A {ω : T * (ω) = t} and since the union of measurable sets is measurable we have that (T * ) −1 (A) is measurable.Thus, T * is a well defined random variable.Using the same arguments we can construct T * L , where the supremum is taken over L instead of Υ ∪ −Υ.In the same way we can construct T * (L)x where the supremum is taken over (L) x where x ∈ B.

Consider any
Now, we generalise the arguments adopted in the proof of Lemma 3.3 in [23].For each i ∈ L define a function Observe that g i is a bounded measurable function and observe that, for any t ∈ Z k , g i (• − t) = 1 when and zero otherwise.Then, by time change formula we have which is a contradiction.Notice that we used the fact that by construction, for every t ∈ (L) x , we have (L x where r is the highest point in the closure of the fundamental parallelotope (as defined in the proof of Proposition 3), and that t + i ∈ (L) r−x for any i ∈ L. Thus, we have t∈(L)x |Θ t | α < ∞ almost surely.The same arguments can be repeated for every x ∈ B and use the fact proven in the proof of Proposition 3 that for every x ∈ B we know that r−x ∈ B. Therefore, since B is finite we conclude that t∈Υ∪ Therefore, condition (AC Λ ) implies the following anti-clustering condition: Now, for any z > 0, by the regular variation of |X 0 | and by (22) we have that In other words, for any > 0 and z > 0, there exists l > 0 such that for all w > l This, implies that P( lim |t|→∞ |Y t | = 0) = 1 and so P( lim Then, from Lemmas 3 and 27 we obtain the statement.

Proof of Theorem 12
By changes of variables we have From the time-change formula, we obtain that for any h ∈ D j ∪ −D j , Since h is a lattice point then One deduces the following expression of the Laplace transform of N λ

Poof of Proposition 13
The first statement follows from similar arguments as the ones used in the proof of Theorem 2.1 in [2] and in Lemma 3.1 in [18].In particular, it is easy to see that for all s, t ∈ Z k with s ≤ t .
Let μ be the tail measure of X with auxiliary regularly varying function P(|X 0 | > x).By the definition of regular variation of X, by homogeneity of the tail measure, and assuming w.l.o.g. that {0} ∈ Υ we have (see also Lemma 3.1 in [18]) ∈ (0, ∞).
It is possible to see that (ν s,t ) s,t∈Z k is a family of consistent probability measures and by Kolmogorov extension theorem we obtain the first statement.The second statement follows from the first and the continuous mapping theorem.

Proof of Proposition 14
This follows from similar arguments used in the proofs of Theorem 3.1 in [2] and of Theorem 3.2 in [23].Consider any s ∈ Z k and any i, j ∈ Z k such that Υ ⊂ {t ∈ Z k : i ≤ t ≤ j}.Then, following the proof of Theorem 3.1 in [2] we define the spaces: Define the bijection T : E i,j → (0, ∞) × S i,j by T (y i , ..., y j ) = ρ Υ (y), y i ρ Υ (y) , ..., y j ρ Υ (y) .
Concerning point (ii), consider any i, j ∈ Z k and let g : (R d ) Z k → R be a bounded and continuous function.By stationarity where The last equality follows from the consistency of the measures µ l,k , l, k ∈ Z k and the fact that µ l,k restricted on the set {(y l , ..., y k ) : ρ (Υ)s (y) > 1} is a probability measure, call it νs,t , and this holds for any s ∈ Z k ; indeed by stationarity As a side note observe that Y Υ is not necessarily stationary because different restrictions (i.e.different s) of µ correspond to potentially different probability measures.That is ν s,t , which is the probability measure given by µ l,k restricted on {(y l , ..., y k ) : ρ (Υ)s (y) > 1} (as introduced in the proof of Proposition 13) is potentially a different from νs,t .Now, by (24) applied to µ l,k we obtain that (25) is equal to Following the monotone convergence arguments of the proof of Theorem 3.2 in [23] we send ε → 0 and drop the continuity assumption of g.Since i and j were arbitrary we obtain the result.Finally point (iii) follows from (9) applied to the function g(y) = g(y/ρ (Υ)s (y))1(ρ (Υ)s (y) = 0).

Proof of Theorem 17
Before proving Theorem 17 we present the following result on the connection between tail random fields for different sets Υ 1 and Υ 2 and different moduli of continuity ρ 1 and ρ 2 .Lemma 28.Let Υ 1 and Υ 2 be two finite subset of Z k and consider ρ 1 and ρ 2 be two moduli of continuity on R dZ k .Let C 1 and C 2 be the constants such that, for every > 0, ρ 1,Υ1 (x) > implies ρ 2,Υ2 (x) > C2 , and ρ 2,Υ2 (x) > implies ρ 1,Υ1 (x) > C 1 .Then, Proof.Let Ξ be a finite subset of Z k and let g : R d|Ξ| → R be a bounded and continuous function.
Then, by homogeneity we have where Thus, we have Since g and Ξ were arbitrary we obtain the first stated result.The same arguments apply to the second one.
For notational purpose we consider the general case of countably many Ds and so of Es.For the first part of this proof we follow similar arguments as the ones used in the proof of Theorem 10.By A Λ (a n ) it suffices to show that for any g ∈ C + K , (Ψ Ñ Λ rn (g)) kn converges to (11) as n → ∞: It suffices to prove that k n (1 − Ψ Ñ Λ rn (g)) converges to the logarithm of (11) as n → ∞.
Recall the sets S i,4l , S i,4l , and Si,4l from Proposition 9 and its proof.Further, let Si,4l := S i,4l \ S i,4l .For notational consistency let Si,4l := ∅ for i corresponding to D i bounded.Now, we apply a telescoping sum argument which generalises the one used in the proof of Theorem 10.
Let u := | i∈I * ,i<m 4l S i,4l | (we omit the dependency on l and on n in u) and let s 1 ≺ s 2 ≺ ... ≺ s u denote the points in i∈I * ,i<m 4l S i,4l .Denote by E j1 , ..., E ju the Es associated to s 1 , ..., s u .Let ū := | i∈I * ,i<m 4l Si,4l | and let s1 ≺ s2 ≺ ... ≺ sū denote the points in i∈I * ,i<m 4l Si,4l .Denote by Ej 1 , ..., Ej ū the Es associated to s1 , ..., sū .Let s1 , ..., sũ , for some ũ ∈ N ∪ {0}, be the ordered points in In this case we associate the set {0} to any point sh , h = 1, ..., ũ.Denote by û := u + ū + ũ and by ŝ1 , ..., ŝû the points s 1 , ...s u , s1 , ...s u , s1 , ..., sũ indexed such that ŝ1 ≺ ŝ2 ≺ ... ≺ ŝû , and denote by Ê1 , ..., Êû , the corresponding sets E j1 , ..., E ju , Ej 1 , ..., Ej ū , {0}, ..., {0} so that for every l ∈ N we obtain 1 − Ψ Ñrn (g) = û m=1 Ψl,m+1 (g) − Ψl,m (g).By the stationarity of X, we have Consider now only the J l,m where m is such that ŝm = s i for some i = 1, ..., u.We have that and since by dominated convergence theorem we have that (28) is equal to where the last equality follows by the anti-clustering condition (AC Λ ,I * ).Now, let us focus on (26) where m is such that ŝm = si for some i = 1, ..., ū or ŝm = sl for some l = 1, ..., ũ.Since by Proposition 9 By combining this with the fact that (26) is bounded by Now, by construction (recall (7)) when ŝm = s k for some k = 1, ..., u we get where we used that by definition (D * i ∩K 2l )\ s∈{0}∪−Gi (E i ) s = Di ∩K 2l .Therefore, by combining (29) and (31) we have that Thus, for the remaining part of the proof we focus on From Lemma 15 we have that for every i and thus we have By assumption A Λ ρ we can apply the dominated convergence theorem (twice) as follows.First since the integrand is bounded by the constant j∈I * γ * j c j D α j for every l ∈ N and since this constant is bounded by condition A Λ ρ , which makes it an integrable function for the integral ∞ δ d(−y −α ) for any δ > 0, we can put the limit of l going to infinity inside the integral.Second, consider the finite counting measure j∈I * γ * j c j D α j ε j (•), where ε is the Dirac delta measure.Since the integrand is bounded by 1 and 1 is an integrable function with respect to this finite counting measure, then we can apply again the dominated convergence theorem.Hence, we obtain Since ρ Ej (Θ) = 1 a.s., by Corollary 16 we have that max s∈Ej |Θ s | ≤ D j a.s.. Further, recall that g has compact support, in particular for any x ∈ R d with |x| < δ we have that g(x) = 0.Then, we obtain that if y < δ then y Dj Θ Ej ,t < δ a.s., and so g( y Dj Θ Ej ,t ) = 0 a.s.. Thus, (34) is equal to Finally, Corollary 4.14 in [16] ensures the existence of the limiting random measure N Λ and so N Λ has the stated Laplace formulation.

P(
which is a contradiction.Notice that we used the fact that by construction, for every i, t ∈ L, we have i − t ∈ L. Given that lim l→∞ θ (l) = θ Λ b exists, Theorem 30 point (a) ensures that for (u n ) = (u n (τ )) and some τ > 0 we have that θ Λ b = lim n→∞ θ Λ n exists and is positive.Thus, from Theorem 30 point (b) for (u n ) = (u n (τ )) and arbitrary τ > 0 we obtain that θ Λ X exists, is positive and it is equal to θ Λ b .From these arguments we obtain the first equality in (16) V ti y i , V 0 which converges to (44) as x → ∞.

x y 0 D2Figure 2 :
Figure 2: Representation of D2 .It is possible to see that D2 is translation invariant for the points in L 2 which in this case is given by the x-axis.
Thus, we have t∈L ρ (Υ)t (Θ) α < ∞ a.s., and by homogeneity and continuity we have that t∈L max s∈Υ |Θ Υ,t+s | α < ∞ a.s., and since Υ is finite we obtain that t∈ s∈L (Υ)s |Θ Υ,t | α < ∞ a.s.Proof of Theorem 23.Recall (42) and letθ (l) := lim n→∞ h∈I * ,h<m 4l P( max t∈ Dh ∩K 2l X t ≤ u n | max t∈E h X t > u n ) |S h,4l | |Λ rn | P(max t∈E h X t > u n ) P(X > u n ), for l ∈ N. Using the arguments in the proof of Theorem 17, we have thatP(max t∈ Dh ∩K 2l X t ≤ u n , max t∈E h X t > u n , ρ Ej i (X) > un Y E h ,t > D h , max t∈ Dh ∩K 2l Y E h ,t ≤ D h Θ E h ,t > D h , y max t∈ Dh ∩K 2l Θ E h ,t ≤ D h d(−y −α ), Θ E h ,t > D h , y max t∈ Dh ∩K 2l Θ E h ,t ≤ D h d(−y −α ) Θ E h ,t > D h , y sup t∈ Dh Θ E h ,t ≤ D h d(−y −α ) h Θ E h ,t > D h , y sup t∈ Dh Θ E h ,t ≤ D h d(−y −α ).
[23]e for the others we have that we have that Finally, by the time-change formula applied tof t ((Θ Ej ,s ) s∈Hj ) = max z∈(Ej )t Θ αand shifting t to 0, for every t ∈ L j , we have For every j ≥ 1 we also denoteR Λn)−s⊃Ej } ((Λ n ) −t ∩ {s ∈ Z k : s 0}) \K l and M Λ,X,(j) i so that Condition (AC Λ ,I * ) is satisfied if for every j ∈ I *We start by showing the following Lemma Lemma 32.Let (X t ) t∈Z k be a stationary max-stable random field.Then (X t ) t∈Z k is jointly regularly varying and the finite-dimensional distributions of its tail field (Y t ) t∈Z k is given byP(Y t1 < y 1 , ..., Y tn < y n )for t 1 , ..., t n ∈ Z k and y 1 , ..., y n ∈ (0, ∞).Proof.It follows from similar computations as the ones in the proof of Proposition 6.1 in[23].For any x 1 , ..., x n ∈ (0, ∞) we have (seeExamples 1.5.4 and 4.4.3 in [17])P(X t1 ≤ x 1 , ..., X tn ≤ x n ) = exp −E max P(x −1 X t1 ≤ y 1 , ..., x −1 X tn ≤ y n | X 0 > x)= P(X t1 ≤ xy 1 , ..., X tn ≤ xy n ) − P(X t1 ≤ xy 1 , ..., X tn ≤ xy n , X 0 ≤ x) P(X 0 > x) t∈Ei Θ Ei,t > D i , Y sup t∈Ei Θ Ei,t > D i − P Y max t∈Ei Θ Ei,t > D i , Y sup t∈Ei Θ Ei,t > D i − P y max t∈Ei Θ Ei,t > D i , y sup t∈ Di Θ Ei,t > D i d(−y −α )where we used the fact that since ρ Ei (Θ) = 1 a.s., then max s∈Ei Θ(s ≤ D i a.s.byCorollary 16.Moreover, applying a change of variable and the time change formula we get the representationj∈I * γ * j c j E sup t∈Hj Θ α Ej ,t s∈Hj |Θ Ej ,s | α = j∈I * γ * j c j E sup t∈Hj Q α Ej ,Lj ,t .X t∈Ej X t > a Λ n x = 0.