A dependent lindeberg central limit theorem for cluster functionals on stationary random fields

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A dependent lindeberg central limit theorem for cluster functionals on stationary random fields

José Gómez-García

To cite this version:

José Gómez-García. A dependent lindeberg central limit theorem for cluster functionals on stationary random fields. 2020. �hal-02501175�

A DEPENDENT LINDEBERG CENTRAL LIMIT THEOREM FOR CLUSTER FUNCTIONALS ON STATIONARY RANDOM FIELDS

JOS ´E G. G ´OMEZ-GARC´IA^∗

Abstract. In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes (Z_n(f))_f∈F whose index set F is a family of cluster functionals valued on blocks of values of a stationary random field. The practicality and applicability of the result depends mainly on the usual Lindeberg condition and a sequence T_n which summarizes the dependence between the blocks of the random field values. Finally, as application, we use the previous result in order to show the Gaussian asymptotic behavior of the iso-extremogram estimator introduced in this paper.

1. Introduction

Recent developments in massive data processing lead us to think in a different way about certain problems in Statistics. In particular, it is of interest to develop the construction of statistics as functions of data blocs and to study their inference. On the other hand, very often, in some applications (e.g., in extremes [3] and in astronomy [8]) only very little data is relevant for the estimates, without forgetting that this is also hidden among a large mass of

“raw data”. This brings us to the idea of thinking about clusters of data deemed“relevant”

(or type extremal, in the context of extreme value theory), where we say that two relevant values belong to two different clusters if they belong to two different blocks. Moreover, these relevant values are in the cores of the blocks, where the core of a block B is defined as the smaller sub-block C(B) ofB that contains all the relevant values of B, if they exist.

In the context of this work, we consider functionals which act on these clusters of relevant values and we develop useful lemmas in order to simplify the essential step to establish a Lindeberg central limit theorem for these “cluster functionals” on stationary random fields, inspired by the works of [1], [6] and [7].

Precisely, letd∈Nand let us denoten:= (n₁, . . . , n_d),1:= (1, . . . ,1)∈N^d and [j] := [1 : j], where [i:j] :={i, i+ 1, . . . , j} ⊂Z. LetX =

X_t : t∈N^d be aR^k−valued stationary random field and letX={X_n,t : t∈ [n₁]× · · · ×[n_d]}_n∈

N^d be the corresponding normalized random observations from the random field X, defined by X_n,t = L_n(X_t)IA(X_t) for some measurable functionsL_n :R^k −→R^k, such that

P(X_n,1 ∈ · |X_n,1 ∈A) −→

n→∞G(·), (1)

Date: Submitted on March 6, 2020.

2010Mathematics Subject Classification. 60G60; 60F05; 60G70.

Key words and phrases. Central limit theorem; cluster functional; weak dependence; Lindeberg method;

extremogram.

The author was funded by the Normandy Region RIN program.

1

where G is a non-degenerate distribution and A⊆R^k\ {0} is the relevance set. Here, I^A(·) denotes the usual indicator function of a subset A and the tendency n → ∞ means that n_i → ∞ for all i ∈ [d]. In particular, the convergence (1) is fulfilled if the random vector X₁ is regularly varying. For details about regularly varying vectors one can refer to Resnick [9,10].

For each i∈[d], let r_i :=r_ni be a integer value such that r_i =o(n_i) and m_i :=dn_i/r_ie:=

max{k∈N:k ≤n_i/r_i}. We define the d−blocks (or simply blocks) ofX by Yn,j1...j_d := (Xn,t)_t∈^Qd

i=1[(ji−1)ri+1 :jiri], (2)

where (j₁, . . . , j_d)∈D_n,d:=Qd

i=1[m_i]. We have thus m₁· · ·m_dcomplete blocksY_n,j1...jd, and no more than m₁+m₂ +· · ·+m_d−d+ 1 incomplete ones which we will ignore. Besides, as usual,Qd

i=1A_i denotes the Cartesian productA₁× · · · ×A_d and, by stationarity, we will denote Y_n =^D Y_n,1 as a generic block of X.

We are now going to formally define the core of a block, cluster functional and the empirical process of cluster functionals, which are generalizations of the definitions of [6] to d−blocks.

Let y= (xt)_t∈^Qd

i=1[ri] be a d−block. The core of the block y (w.r.t. the relevance set A) is defined as

C(y) =

(x_t)_t∈^Qd

i=1[ri,I :ri,S], if x_t∈A for some t∈Qd i=1[r_i];

0, otherwise,

where, for each i∈[d],r_i,I and r_i,S are defined as r_i,I = min







j_i ∈[r_i] : x_{(j1,...,ji,...,jd)} ∈A,for some (j₁, . . . , j_i−1, j_i+1, . . . , j_d)∈Y

k∈[d]\{i}

[r_k]





 ,

ri,S = max







ji ∈[ri] : x_{(j1,...,ji,...,jd)} ∈A,for some (j1, . . . , ji−1, ji+1, . . . , jd)∈Y

k∈[d]\{i}

[rk]





 .

Let (E,E) be a measurable subspace of (R^k,B(R^k)) for some k ≥ 1 such that 0 ∈ E. Let Bl1,...,ld(E) be the set ofE−valued blocks (or arrays) of sizel₁×l₂×· · ·×l_d, withl₁, . . . , l_d∈N. Consider now the set

E_∪ :=

∞

[

l1,...,ld=1

B^l1,...,ld(E),

which is equipped with the σ−field E∪ induced by the Borel−σ−fields on Bl1,...,ld(E), for l1, . . . , ld ∈ N. A cluster functional is a measurable map f : (E∪,E∪) −→ (R,B(R)) such that

f(y) =f(C(y)), for all y∈E∪, and f(0) = 0. (3) Let F be a class of cluster functionals and let {Y_n,j1j2...jd : (j₁, . . . , j_d)∈D_n,d} be the family of blocks of size r1 ×r2 × · · · ×rd defined in (2). The empirical process Zn of

2

cluster functionals in F, is the process (Z_n(f))_f∈F defined by Z_n(f) := 1

√nnvn

X

(j1,...,jd)∈Dn,d

(f(Y_n,j1...jd)−Ef(Y_n,j1...jd)), (4) where n_n =n₁· · ·n_d and v_n :=P(X_n,1 ∈A) with A⊆E\ {0} denoting the relevance set.

Under the Lindeberg condition and the convergence to zero of a sequence T_n that sum- marizes the dependence between the blocks of values of the random field, we prove that the finite-dimensional marginal distributions (fidis) of the empirical process (4) converge to a Gaussian process. The proof basically consists of the “Lindeberg method” as in [1], but adapted here to stationary random fields.

Regarding the condition T_n −→ 0, as n → ∞, this can be fulfilled if the random field X has short range dependence properties, e.g., if the random field X is weakly dependent in the sense of Doukhan & Louhichi [5] under convenient conditions for the decay rates of the weak-dependence coefficients. These rates are calculated in [7] in the context of extreme clusters of time series.

The rest of the paper consists of two sections. In Section 2, we provide useful lemmas in order to establish the central limit theorem for the fidis of the cluster functionals empirical process (4). In Section 3 we introduce the iso-extremogram (a correlogram for extreme values of space-time processes) and we use the CLT of Section2in order to show that, under additional suitable conditions, the iso-extremogram estimator has asymptotically a Gaussian behavior.

2. Results

In this section we provide useful lemmas that simplify notably the essential step to establish a central limit theorem for the fidis of the empirical process defined in (4). The proof consists in the same techniques that Bardetet al. [1] used in the demonstrations of their dependent and independent Lindeberg lemmas, but generalized here to random fields.

In order to establish the CLT, firstly consider the following basic assumption:

(Bas) The vectorr = (r₁, . . . , r_d)∈N^d is such that r_i n_i for each i∈[d].

Besides, denotingr_n =r₁· · ·r_d,r_nv_n−→τ < ∞and n_nv_n −→ ∞, as n → ∞.

Secondly, consider the following essential convergence assumptions:

(Lin) (r_nv_n)⁻¹E

(f(Y_n)−Ef(Y_n))²I{|f(Yn)−Ef(Yn)|>√ nnvn}

=o(1), ∀ >0, ∀f ∈ F;

(Cov) (r_nv_n)⁻¹Cov (f(Y_n), g(Y_n))−→c(f, g), ∀f, g∈ F.

Consider now the random blocksY_n,j1...jd, with (j₁, . . . , j_d)∈D_n,d defined in (2). For each k−tuple of cluster functionals f_k = (f₁, . . . , f_k) and each (j₁, . . . , j_d) ∈ D_n,d, we define the random vector:

Wn,j1...j_d := 1

√n_nv_n(f1(Yn,j1...j_d)−Ef1(Yn,j1...j_d), . . . , fk(Yn,j1...j_d)−Efk(Yn,j1...j_d)). (5)

3

Without loss of generality and in order to simplify writing, we will consider d = 2 in the rest of this section.

Let (W_n,ij⁰ )_(i,j)∈Dn,2 be a sequence of zero mean independent R^k-valued random variables, independents of the sequence (W_n,ij)(i,j)∈D_n,2, such that W_n,ij⁰ ∼ N_k(0,Cov(W_n,ij)), for all (i, j) ∈ D_n,2. Denote by C_b³ the set of bounded functions h : R^k −→ R with bounded and continuous partial derivatives up to order 3. Forh ∈ C_b³ and n= (n₁, n₂)∈N², define

∆_n :=

E



h



 X

(i,j)∈D_n,2

W_n,ij



−h



 X

(i,j)∈D_n,2

W_n,ij⁰









. (6)

The following assumption will allow us to present, in a useful and simplified form, lemmas of Lindeberg under independence and dependence.

(Lin’) It exists δ ∈ (0,1] such that, for all (i, j) ∈ D_n,2, EkW_n,ijk^2+δ < ∞ for all n ∈ N² and allk−tuple of cluster functionals (f₁, . . . , f_k)∈ F^k. Moreover, denote

A_n := X

(i,j)∈Dn,2

EkW_n,ijk^2+δ.

Lemma 1 (Lindeberg under independence). Suppose that the blocks (Y_n,ij)(i,j)∈Dn,2 are in- dependents and that the random variables (W_n,ij)(i,j)∈D_n,2 defined in (5) satisfy Assumption (Lin’). Then, for all n∈N²:

∆_n ≤6 kh⁽²⁾k^1−δ_∞ kh⁽³⁾k^δ_∞ A_n. Proof. First, notice that

∆_n ≤ X

(i,j)∈Dn,2

∆_n,ij , (7)

where

∆_n,ij :=

E

h_ij(V_n,ij +W_n,ij)−h_ij(V_n,ij+W_n,ij⁰ )

, ∀(i, j)∈D_n,2 ;

Vn,ij := X

(u,v)∈Dn,2\(^Si−1l=0L^m_l ²∪L^j_i)

Wn,uv, ∀(i, j)∈Dn,2\ {(m1, m2)} , V_n,m1m2 = 0 ; and

h_ij(x) :=E

"

h x+

i−1

X

u=0 m2

X

v=1

W_n,uv⁰ +

j−1

X

v=0

W_n,iv⁰

!#

.

Besides, we set the convention W_n,ij = 0, if either i= 0 orj = 0.

Now, we will use some lines of the proof of Lemma 1 in [1].

Letv, w∈R^k. From Taylor’s formula, there exist vectors v_1,w, v_2,w ∈R^k such that:

h(v+w) =h(v) +h⁽¹⁾(v)(w) + 1

2h⁽²⁾(v1,w)(w, w)

=h(v) +h⁽¹⁾(v)(w) + 1

2h⁽²⁾(v)(w, w) + 1

6h⁽³⁾(v_2,w)(w, w, w),

4

where, for j = 1,2,3, h^(j)(v)(w₁, w₂, . . . , w_j) stands for the value of the symmetric j−linear form from h^(j) of (w₁, . . . , w_j) at v. Moreover, denote

kh^(j)(v)k₁ = sup

kw1k,...,kwjk≤1

|h^(j)(v)(w₁, . . . , w_j)| and kh^(j)k∞= sup

v∈R^k

kh^(j)(v)k₁.

Thus, forv, w, w⁰ ∈R^k, there exist some suitable vectors v1,w, v2,w, v1,w⁰, v2,w⁰ ∈R^k such that h(v+w)−h(v+w⁰) = h⁽¹⁾(v)(w−w⁰) + 1

2 h⁽²⁾(v)(w, w)−h⁽²⁾(v)(w⁰, w⁰) + 1

2 h⁽²⁾(v1,w)−h⁽²⁾(v)

(w, w)− h⁽²⁾(v1,w⁰)−h⁽²⁾(v)

(w⁰, w⁰) ,

by using the approximation of Taylor of order 2, and h(v+w)−h(v+w⁰) = h⁽¹⁾(v)(w−w⁰) + 1

2 h⁽²⁾(v)(w, w)−h⁽²⁾(v)(w⁰, w⁰) +1

6 h⁽³⁾(v_2,w)(w, w, w)−h⁽³⁾(v_2,w⁰)(w⁰, w⁰, w⁰) , by using the approximation of Taylor of order 3.

Thus,γ =h(v+w)−h(v+w⁰)−h⁽¹⁾(v)(w−w⁰)−¹₂ h⁽²⁾(v)(w, w)−h⁽²⁾(v)(w⁰, w⁰)

satisfies:

|γ| ≤ kwk²+kw⁰k²

kh⁽²⁾k∞

∧ 1

6 kwk³+kw⁰k³

kh⁽³⁾k∞

≤ kwk²kh⁽²⁾k∞

∧ 1

6kwk³kh⁽³⁾k∞

+ kwk²kh⁽²⁾k∞

∧ 1

6kw⁰k³kh⁽³⁾k∞

+ kw⁰k²kh⁽²⁾k∞

∧ 1

6kwk³kh⁽³⁾k∞

+ kw⁰k²kh⁽²⁾k∞

∧ 1

6kw⁰k³kh⁽³⁾k∞

≤ 1

6^δkh⁽²⁾k^1−δ_∞ kh⁽³⁾k^δ_∞ kwk^2+δ+kwk^2(1−δ)kw⁰k^3δ+kwk^3δkw⁰k^2(1−δ)+kw⁰k^2+δ

, (8) where (8) is given by using the inequality 1∧a≤a^δ, with a≥0 and δ∈[0,1].

Substituting h_ij, V_n,ij, W_n,ij and W_n,ij⁰ for h, v, w and w⁰ in the preceding inequality (8) and taking expectations, we will obtain a bound for ∆_n,ij. Indeed,

E

h_ij(V_n,ij+W_n,ij)−h_ij(V_n,ij+W_n,ij⁰ )

=E

h_ij(V_n,ij +W_n,ij)−h_ij(V_n,ij+W_n,ij⁰ ) + 0

=E

h_ij(V_n,ij+W_n,ij)−h_ij(V_n,ij+W_n,ij⁰ )

−E h

h⁽¹⁾_ij (V_n,ij)(W_n,ij−W_n,ij⁰ )i

− 1 2E

h

h⁽²⁾_ij (V_n,ij)(W_n,ij, W_n,ij)−h⁽²⁾_ij (V_n,ij)(W_n,ij⁰ , W_n,ij⁰ )i , because V_n,ij is independent of W_n,ij and W_n,ij⁰ , and because EW_n,ij = EW_n,ij⁰ = 0 and Cov(W_n,ij) = Cov(W_n,ij⁰ ) for all (i, j)∈D_n,2.

On the other hand, using Jensen’s inequality, we deriveEkW_n,ij⁰ k^2+δ ≤ EkW_n,ij⁰ k⁴¹₂+^δ₄

, and EkW_n,ij⁰ k⁴ ≤ 3·(EkW_n,ijk²)² because W_n,ij⁰ is a Gaussian random variable with the same covariance asWn,ij.

5

Therefore,

EkW_n,ij⁰ k^2+δ≤

3· EkW_n,ijk²2¹₂+^δ₄

= 3^12+δ4 EkW_n,ijk²1+^δ₂

≤3^12+δ4EkW_n,ijk^2+δ, (9)

EkW_n,ij⁰ k^2(1−δ)EkW_n,ijk^3δ ≤ EkW_n,ij⁰ k²1−δ

EkW_n,ijk^3δ

≤ EkW_n,ijk²1−δ

EkW_n,ijk^3δ≤EkW_n,ijk^2+δ . (10) Besides, for 3δ <2,

EkW_n,ijk^2(1−δ)EkW_n,ij⁰ k^3δ≤EkW_n,ijk^2(1−δ) EkW_n,ij⁰ k^23δ₂

≤EkW_n,ijk^2+δ, (11) else

EkW_n,ijk^2(1−δ)EkW_n,ij⁰ k^3δ ≤EkW_n,ijk^2(1−δ) EkW_n,ij⁰ k^43δ₄

, because 3δ≤4

≤3^3δ4 EkW_n,ijk^2(1−δ) EkW_n,ijk^23δ₂

≤3^12+δ4EkW_n,ijk^2+δ. (12) The inequalities (9)-(12) allow to simplify the terms between parentheses in the last inequal- ity in (8). Recall thatkh^(k)_ij k∞ ≤ kh^(k)k∞ for all (i, j)∈ D_n,2 and 0≤ k ≤3. Therefore, we obtain that

∆n,ij ≤ 2(1 + 3^12+δ4)

6^δ kh⁽²⁾k^1−δ_∞ kh⁽³⁾k^δ_∞EkWn,ijk^2+δ ≤6kh⁽²⁾k^1−δ_∞ kh⁽³⁾k^δ_∞EkWn,ijk^2+δ, because, for all δ∈[0,1], C(δ) = ²⁽¹⁺³

1 2+δ

4)

6^δ ≤C(0) = 2(1 +√

3)<6.

As a consequence, from Assumption (Lin’), ∆_n ≤6kh⁽²⁾k^1−δ_∞ kh⁽³⁾k^δ_∞ A_n.

Remark 2.1. Taking < 6kh⁽²⁾k∞ (kh⁽³⁾k∞)⁻¹ and using suitably the second inequality of (8) in the proof of Lemma 1, classical Lindeberg conditions may be used:

∆_n≤2 kh⁽²⁾k_∞ B_n() +kh⁽³⁾k_∞ a_n 4

3+p B_n()

, (13)

where

B_n() = X

(i,j)∈Dn,2

E

kW_n,ijk²I^{kWn,ijk>}

, >0, n ∈N² ; a_n = X

(i.j)∈Dn,2

EkW_n,ijk² <∞, n∈N².

Moreover, these classical Lindeberg conditions derive the conditions from Lemma 1. Indeed,

∆_n ≤2 kh⁽²⁾k∞ ^−δA_n+kh⁽³⁾k∞ a_n 4

3+^−δ/2p A_n

, for δ∈(0,1) and >0.

The proof of this remark for general independent random vectors is given in [1, p.165].

6

Remark 2.2. Observe that the assumptions (Lin) and (Cov) imply that B_n() −→

n→∞ 0 and that an = Pk

i=1(rnvn)⁻¹Cov (fi(Yn), fi(Yn)) −→

n→∞

Pk

i=1c(fi, fi) < ∞, respectively. There- fore, if the blocks (Y_n,ij)(i,j)∈D_n,2 are independent and if the assumptions (Lin) and (Cov) hold, then from Lemma 1 and Remark 2.1, the fidis of the empirical process (Z_n(f))f∈F of cluster functionals converge to the fidis of a Gaussian process (Z(f))f∈F with covariance function c.

For the dependent case, we need to consider more notations:

Let L^j_i := {(i, v) : v ∈[j]} ⊂ D_n,2 , for all (i, j) ∈ D_n,2. We set L⁰_i = L^j₀ = ∅ for any i∈[m1] and any j ∈[m2]. For each k ∈N, fk= (f1, . . . , fk)∈ F^k, t∈R^k and n∈N² ; we define

T_n,t(f_k) := X

(j1,j2)∈D_n,2

Cov





exp





iht, X

(u1,u2)∈D_n,2\(Sj1−1

l=0 L^m_l ²∪L^j_j²

1)

W_n,u1u2i





,exp (iht, W_n,j1j2i)





 .

Lemma 2 (Dependent Lindeberg lemma). Suppose that the r.v.’s (W_n,ij)(i,j)∈D_n,2 defined in (5) satisfy Assumption (Lin’). Consider the special case of complex exponential functions h(w) = exp (iht,wi) with t∈R^k. Then, for each k ∈N and each k−tuple f_k = (f₁, . . . , f_k) of cluster functionals, the following inequality holds:

∆_n ≤T_n,t(f_k) + 6ktk^2+δA_n , n∈N².

Proof. Consider (W_n,j^∗ _1j2)_(j1,j2)∈D_n,2 an array of independent random variables satisfying Assumption (Lin’) and such that (W_n,j^∗ _1j2)_(j1,j2)∈Dn,2 is independent of (W_n,j1j2)_(j1,j2)∈Dn,2 and (W_n,j⁰

1j2)_{(j1,j2)∈Dn,2}. Moreover, assume that W_n,j^∗

1j2 has the same distribution as W_n,j1j2 for (j₁, j₂)∈D_n,2.

Then, using the same decomposition (7) in the proof of the previous lemma, one can also write,

∆_n,j1j2 ≤ E

h_j1j2(V_n,j1j2 +W_n,j1j2)−h_j1j2(V_n,j1j2 +W_n,j^∗ _1j2) +

E

h_j1j2(V_n,j1j2+W_n,j^∗ _1j2)−h_j1j2(V_n,j1j2 +W_n,j⁰ _1j2)

. (14) Then, from the previous lemma, the second term of the RHS of the inequality (14) is bounded by

6kh⁽²⁾k^1−δ_∞ kh⁽³⁾k^δ_∞ EkW_n,j1j2k^2+δ≤6 ktk^2+δ EkW_n,j1j2k^2+δ.

For the first term of the RHS of the inequality (14), first notice that for aR^k−valued random vector X independent from (W_n,j⁰ _1j2)(j1,j2)∈Dn,2,

Ehj1j2(X) = E

"

h X+

j1−1

X

u=0 m2

X

v=1

W_n,uv⁰ +

j2−1

X

v=0

W_n,j⁰ _1v

!#

= exp −1 2t^T

j1−1

X

u=0 m2

X

v=1

Cn,uv+

j2−1

X

v=0

Cn,j1v

! t

!

·E[exp (iht, Xi)],

because W_n,j⁰ _1j2 ∼ N_k(0, C_n,j1j2), where C_n,j1j2 := Cov(W_n,j1j2) is the covariance matrix of the vector Wn,j1j2, for (j1, j2) ∈Dn,2. For j1 = 0 or j2 = 0, recall that Wn,j1j2 = 0. In this

7

case, we also set C_n,j1j2 = 0.

Thus, E

h_j1j2(V_n,j1j2 +W_n,j1j2)−h_j1j2(V_n,j1j2 +W_n,j^∗ _1j2)

=

exp −1 2t^T

j1−1

X

u=0 m2

X

v=1

Cn,uv+

j2−1

X

v=0

Cn,j1v

! t

!

×E

exp (iht, V_n,j1j2i)· exp (iht, W_n,j1j2i)−exp iht, W_n,j^∗ _1j2i

=

exp −1 2t^T

j1−1

X

u=0 m2

X

v=1

Cn,uv+

j2−1

X

v=0

Cn,j1v

! t

!

× |Cov (exp (iht, V_n,j1j2i),exp (iht, W_n,j1j2i))|

≤ |Cov (exp (iht, V_n,j1j2i),exp (iht, W_n,j1j2i))|. Therefore,

∆_n = X

(j1,j2)∈D_n,2

∆_n,j1j2

≤ X

(j1,j2)∈Dn,2

|Cov (exp (iht, Vn,j1j2i),exp (iht, Wn,j1j2i))|+ 6ktk^2+δEkWn,j1j2k^2+δ

=T_n,t(f_k) + 6ktk^2+δA_n.

The previous lemma together with Remark 2.1 derive the following theorem.

Theorem 3 (CLT for cluster functionals on random fields). Suppose that the basic as- sumption (Bas) holds and that the assumptions (Lin) and (Cov) are satisfied. Then, if for each k ∈ N, T_n,t(f_k) converges to zero as n → ∞, for all t ∈ R^k and all k−tuple fk = (f1, . . . , fk) ∈ F^k of cluster functionals, the fidis of the empirical process (Zn(f))f∈F

of cluster functionals converge to the fidis of a Gaussian process (Z(f))f∈F with covariance function c defined in (Cov).

Proof. The assumptions (Lin) and (Cov) imply that, as n → ∞, B_n() −→ 0 and a_n −→ Pk

s=1c(f_s, f_s) < ∞, respectively. Therefore, taking into account Remark 2.1, we obtain from Lemma 2that, for each k∈N,

∆_n= E



h



 X

(i,j)∈Dn,2

W_n,ij



−h



 X

(i,j)∈Dn,2

W_n,ij⁰









n→∞−→ 0, for allt∈R^k, withh(w) = exp(iht,wi), because by hypothesis,T_n,t(f_k) −→

n→∞ 0 for allt∈R^k and all fk= (f1, . . . , fk)∈ F^k.

Notice that

W_n⁰ := X

(i,j)∈Dn,2

W_n,ij⁰ ∼ Nk(0, m1m2Cov(Wn,11))

8

and that |E(h(W_n⁰)−h(W))| −→

n→∞0, where W ∼ N_k(0,Σ_k), with Σ_k = (c(f_i, f_j))_(i,j)∈[k]². Using triangular inequality, we deduce that

E



h



 X

(i,j)∈D_n,2

W_n,ij



−h(W)





n→∞−→ 0,

and therefore (Z_n(f₁), . . . , Z_n(f_k)) = P

(i,j)∈D_n,2W_n,ij −→^D

n→∞ W.

Remark 2.3. The previous theorem can be formulated for d = 3 as follows. Define S_i = {(u, v, w) : u ∈[i], v ∈[m₂], w ∈ [m₃]} ⊆D_n,3, for i∈ [m₁], with the convention S₀ =∅.

Moreover, L^k_ij = {(i, j, w) : w ∈ [k]}, for (i, j, k) ∈ D_n,3, and L^k_ij = ∅ if i, j or k is zero.

Then, if (Bas), (Lin), (Cov) are satisfied (for d= 3), and if for each k ∈N, T_n,t^∗ (fk) = X

(j1,j2,j3)∈Dn,3

|Cov (exp(iht, Vn,j1j2j3i),exp (iht, Wn,j1j2j3i))| (15)

converges to zero as n→ ∞ for all t∈R^k and all k−tuple f_k = (f₁, . . . , f_k)∈ F^k of cluster functionals, with

Vn,j1j2j3 := X

(u1,u2,u3)∈Dn,3\(^Sj1−1 ∪ Sj2−1 l=0 L^m_j³

1l∪L^j_j³

1j2)

Wn,u1u2u3 ,

the fidis of the empirical process (Z_n(f))f∈F of cluster functionals converge to the fidis of a Gaussian process (Z(f))_f∈F with covariance function c.

Remark 2.4. We have mentioned previously thatn = (n₁, . . . , n_d)→ ∞meansn_i → ∞for each i∈[d]. However, if the reader would like it, the limits of the sequences indexed with n, as n → ∞, could be reformulated in terms of the limits of such sequences as “n → ∞ along a monotone path on the lattice N^d”, i.e. along n = (dϑ₁(n)e, . . . ,dϑ_d(n)e) for some strictly increasing continuous functions ϑ_i : [1,∞)−→ [1,∞), with i ∈ [d], such that ϑ_i(n) −→ ∞ as n → ∞, for i∈[d].

Suppose that from each blockY_n we extract a sub-blockY_n⁰ and that the remaining parts R_n = Y_n −Y_n⁰ of the blocks Y_n do not influence the process Z_n(f). In particular, this last statement is fulfilled if, (r_nv_n)⁻¹E|∆_n(f)−E∆_n(f)|²I^{|∆n(f)−E∆n(f)|≤√

nnvn} = o(1) and P |∆_n(f)−E∆_n(f)|>√

n_nv_n

=o(r_n/n_n), where ∆_n(f) :=f(Y_n)−f(Y_n⁰). This assump- tion would allow us to considerT_n,t(f_k) (or T_n,t^∗ (f_k)) as a function of the blocksY_n⁰ (separated byl_n) instead of the blocks Y_n, in order to provide them bounds based on either the strong mixing coefficient of [11] or the weak-dependence coefficients of [5] for stationary random fields. These bounds are developed in [7] for the case of weakly-dependent time series, how- ever, we will not develop them in the random field context as this is not the aim of this work. This topic will be addressed in a forthcoming applied statistics paper with numerical simulations.

9

3. Asymptotic behavior of the extremogram for space-time processes In this section we propose a measure (in two versions) of serial dependence on space and time of extreme values of space-time processes. We provide an estimator for this measure and we use Theorem3 in order to establish an asymptotic result. This section is inspired of the extremogram for times series defined in [4].

LetX ={X_t(s) :s∈Z^d, t≥0}be a R^k−valued space-time process, which is stationary in both space and time. We define the extremogram of X for two sets A and B both bounded away from zero by

ρA,B(s, ht) := lim

x→∞P x⁻¹Xht(s)∈B

x⁻¹X0(0)∈A

, (s, ht)∈Z^d×[0,∞) ; (16) provided that the limit exists.

In estimating the extremogram, the limit on xin (16) is replaced by a high quantileu_n of the process. Definingu_n as the (1−1/k_n)−quantile of the stationary distribution of kX_t(s)k or related quantity, with k_n=o(n)−→ ∞, as n → ∞, one can redefine (16) by

ρ_A,B(s, h_t) = lim

n→∞P u⁻¹_n X_ht(s)∈B

u⁻¹_n X₀(0)∈A

, (s, h_t)∈Z^d×[0,∞). (17) The choice of such a sequence of quantiles (u_n)n∈N is not arbitrary. The main condition to guarantee the existence of the limit (17) for any two setsA andB bounded away from zero, is that it must satisfy the following convergence

k_nP u⁻¹_n X_t1(s₁), . . . , X_tp(s_p)

∈ · vague

n→∞−→ m_{(s1,t1),...,(sp,tp)}( · ), (18) for all (s_i, t_i)∈Z^d×[0,∞),i∈[p],p∈N, where

m_{(s1,t1),...,(sp,tp)}

(si,ti)∈Z^d×[0,∞), i∈[p], p∈N

is a collection of Radon measures on the Borel σ−field B(R^kp\ {0}), not all of them being the null measure, with m_{(s1,t1),...,(sp,tp)}(R^kp\R^kp) = 0. In this case,

P u⁻¹_n X_ht(s)∈B

u⁻¹_n X₀(0)∈A

=k_nP(u⁻¹_n (X₀(0), X_ht(s))∈A×B) knP(u⁻¹_n X0(0)∈A)

−→m_(0,0),(s,ht)(A×B)

m_(0,0)(A) =ρ_A,B(s, h_t), provided that m_(0,0)(A)>0.

Remark 3.1. The condition (18) is particularly satisfied if the space-time process X is regularly varying. For details and examples of regularly varying space-time processes and time series, see [3] and [2], respectively.

Note that the extremogram (17) is a function of two lags: a spatial-lag s ∈ Z^d and a non-negative time-lag h_t. Due to all the spatial values that the spatial-lag s takes, in practice, it is very complicated to analyze the results of the estimation of such extremogram.

Moreover, the calculation would be computationally very slow. In order to obtain a simpler interpretation and simplify the calculations, we will assume that the space-time process X satisfies the following “isotropy” condition:

10

(I) For each pair of non-negative integersh_t and h_s,

P(X₀(0)∈A, X_ht(s)∈B) =P(X₀(0)∈A, X_ht(s⁰)∈B), ∀s,s⁰ ∈S^d−1hs , where S^d−1h :=

s∈Z^d:ksk∞=h with h ≥0 andk(s1, . . . , sd)k∞ = maxi=1,...,d|si|.

Under this condition, the extremogram (17) can be redefined using only two non-negative integer lags: a spatial-laghs and a time-laght. Indeed, under the assumption (I), we define the iso-extremogram of X for two sets A and B both bounded away from zero by

ρ^∗_A,B(h_s, h_t) = ρ_A,B(h_s~e₁, h_t), h_s, h_t= 0,1,2, . . . (19) where~e₁ = (1,0,0,0, . . . ,0)∈R^d is the first element of the canonical basis ofR^d.

We will propose now a estimator for the iso-extremogram. For this, consider w.l.o.g. d= 2, because the cased >2 is similar.

Let X_n := {X_t(i, j) : (i, j, t)∈[n₁]×[n₂]×[n₃]} be the observations from a R^k−valued space-time process X, stationary in both space and time, and which satisfies the condi- tion (I). Let us set n=n₁n₂n₃. The sample iso-extremogram based on the observations X_n is given by

ρb^∗_A,B(h_s, h_t) :=

X

(j1,j2)∈[m1]×[m2] n3−ht

X

t=1

X

(i1,i2)∈Shs(cj1j2)

IXt+ht(i1,i2)

un ∈B, ^Xt(_un^cj1j2)∈A

#Shs(c_j1j2) X

(j1,j2)∈[m1]×[m2] n3

X

t=1

IXt(cj1j2)

un ∈A

, (20)

for h_s = 0,1,2, . . . ,d2⁻¹min{r₁, r₂}e −1, and h_t= 0, . . . , n−1, where c_ij :=

(2i−1)r₁+ 1 2

,

(2j−1)r₂+ 1 2

denotes the “center” of the block B_ij = [(i −1)r₁ + 1 : ir₁] ×[(j −1)r₂ + 1 : jr₂], for (i, j) ∈ [m₁] ×[m₂]. Besides, Sh(u, v) := {(i, j) ∈ [n₁]× [n₂] : k(u, v)− (i, j)k∞ = h}

with h ≥ 0 and #E denotes the cardinality of the set E. Remember that r_i = r_ni and m_i =dn_i/r_ie, for i= 1,2,3.

Defining the cluster functional f_A,B,h1,h2 : S∞

l1,l2,l3=1Bl1l2l3(R^k),R∪

−→ (R,B(R)), for h₁, h₂ = 0,1,2, . . ., such that

f_A,B,h1,h2 (x_(i1,i2,i3))_(i1,i2,i3)∈[l1]×[l2]×[l3]

= X

(i1,i2)∈Sh1(c) l3−h2

X

i3=1

I^A×B(x_(c,i3), x_{(i1,i2,i3+h2)})

#Sh1(c) , (21) with c= (d(l₁+ 1)/2e,d(l₂+ 1)/2e)∈[l₁]×[l₂] (the “center” of the blockB = [l₁]×[l₂]), we can rewrite the estimator (20) as:

ρb^∗_A,B(h_s, h_t) = P

(j1,j2,j3)∈Dn,3f_A,B,hs,ht(Y_n,j1j2j3) +δ_n+R_A,B,hs,ht P

(j1,j2,j3)∈Dn,3f_A,A,0,0(Y_n,j1j2j3) +R_A,A,0,0 , (22)

11