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

In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes $(Z_n(f))_{f\in\mathcal{F}}$ whose index set $\mathcal{F}$ 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.


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) of B 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, let d ∈ N and let us denote n := (n 1 , . . . , n d ), 1 := (1, . . . , 1) ∈ N d and [j] := [1 : j], where [i : j] := {i, i + 1, . . . , j} ⊂ Z. Let X = X t : t ∈ N d be a R k −valued stationary random field and let X = {X n,t : t ∈ [n 1 ] × · · · × [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 )I A (X t ) for some measurable functions L n : R k −→ R k , such that P ( X n,1 ∈ · | X n, 1 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 X 1 is regularly varying. For details about regularly varying vectors one can refer to Resnick [9,10].
For each i ∈ [d], let r i := r n i be a integer value such that r i = o(n i ) and m i := ⌈n i /r i ⌉ := max {k ∈ N : k ≤ n i /r i }. We define the d−blocks (or simply blocks) of X by We have thus m 1 · · · m d complete blocks Y n,j 1 ...j d , and no more than m 1 + m 2 + · · · + m d − d + 1 incomplete ones which we will ignore. Besides, as usual, d i=1 A i denotes the Cartesian product A 1 × · · · × 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.
The core of the block y (w.r.t. the relevance set A) is defined as where, for each i ∈ [d], r i,I and r i,S are defined as Let (E, E) be a measurable subspace of (R k , B(R k )) for some k ≥ 1 such that 0 ∈ E. Let B l 1 ,...,l d (E) be the set of E−valued blocks (or arrays) of size l 1 ×l 2 ×· · ·×l d , with l 1 , . . . , l d ∈ N. Consider now the set which is equipped with the σ−field E ∪ induced by the Borel−σ−fields on B l 1 ,...,l d (E), for l 1 , . . . , l d ∈ 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.
Let F be a class of cluster functionals and let {Y n,j 1 j 2 ...j d : (j 1 , . . . , j d ) ∈ D n,d } be the family of blocks of size r 1 × r 2 × · · · × r d defined in (2). The empirical process Z n of 2 cluster functionals in F , is the process (Z n (f )) f ∈F defined by where n n = n 1 · · · 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 summarizes 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 Section 2 in order to show that, under additional suitable conditions, the iso-extremogram estimator has asymptotically a Gaussian behavior.

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 Bardet et 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: Besides, denoting r n = r 1 · · · r d , r n v n −→ τ < ∞ and n n v n −→ ∞, as n → ∞. Secondly, consider the following essential convergence assumptions: Consider now the random blocks Y n,j 1 ...j d , with (j 1 , . . . , j d ) ∈ D n,d defined in (2). For each k−tuple of cluster functionals f k = (f 1 , . . . , f k ) and each (j 1 , . . . , j d ) ∈ D n,d , we define the random vector: Without loss of generality and in order to simplify writing, we will consider d = 2 in the rest of this section.
Let (W ′ n,ij ) (i,j)∈D n,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,ij ∼ N k (0, Cov(W n,ij )), for all (i, j) ∈ D n,2 . Denote by C 3 b the set of bounded functions h : R k −→ R with bounded and continuous partial derivatives up to order 3. For h ∈ C 3 b and n = (n 1 , n 2 ) ∈ N 2 , define The following assumption will allow us to present, in a useful and simplified form, lemmas of Lindeberg under independence and dependence.
Lemma 1 (Lindeberg under independence). Suppose that the blocks (Y n,ij ) (i,j)∈D n,2 are independents and that the random variables (W n,ij ) (i,j)∈D n,2 defined in (5) satisfy Assumption (Lin'). Then, for all n ∈ N 2 : where ∆ n,ij : Besides, we set the convention W n,ij = 0, if either i = 0 or j = 0. Now, we will use some lines of the proof of Lemma 1 in [1]. Let v, w ∈ R k . From Taylor's formula, there exist vectors v 1,w , v 2,w ∈ R k such that: where, for j = 1, 2, 3, h (j) (v)(w 1 , w 2 , . . . , w j ) stands for the value of the symmetric j−linear form from h (j) of (w 1 , . . . , w j ) at v. Moreover, denote by using the approximation of Taylor of order 2, and by using the approximation of Taylor of order 3.
Substituting h ij , V n,ij , W n,ij and W ′ n,ij for h, v, w and w ′ in the preceding inequality (8) and taking expectations, we will obtain a bound for ∆ n,ij . Indeed, because V n,ij is independent of W n,ij and W ′ n,ij , and because EW n,ij = EW ′ n,ij = 0 and Cov(W n,ij ) = Cov(W ′ n,ij ) for all (i, j) ∈ D n,2 . On the other hand, using Jensen's inequality, we derive E W ′ n,ij 2 ) 2 because W ′ n,ij is a Gaussian random variable with the same covariance as W n,ij . Therefore, Besides, for 3δ < 2, The inequalities (9)-(12) allow to simplify the terms between parentheses in the last inequality in (8).
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 (ǫ) −→ n→∞ 0 and respectively. Therefore, 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: 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 (i t, w ) with t ∈ R k . Then, for each k ∈ N and each k−tuple f k = (f 1 , . . . , f k ) of cluster functionals, the following inequality holds: Proof. Consider (W * n,j 1 j 2 ) (j 1 ,j 2 )∈D n,2 an array of independent random variables satisfying Assumption (Lin') and such that (W * n,j 1 j 2 ) (j 1 ,j 2 )∈D n,2 is independent of (W n,j 1 j 2 ) (j 1 ,j 2 )∈D n,2 and (W ′ n,j 1 j 2 ) (j 1 ,j 2 )∈D n,2 . Moreover, assume that W * n,j 1 j 2 has the same distribution as W n,j 1 j 2 for (j 1 , j 2 ) ∈ D n,2 . Then, using the same decomposition (7) in the proof of the previous lemma, one can also write, . (14) Then, from the previous lemma, the second term of the RHS of the inequality (14) is bounded by For the first term of the RHS of the inequality (14), first notice that for a R k −valued random vector X independent from (W ′ n,j 1 j 2 ) (j 1 ,j 2 )∈D n,2 , because W ′ n,j 1 j 2 ∼ N k (0, C n,j 1 j 2 ), where C n,j 1 j 2 := Cov(W n,j 1 j 2 ) is the covariance matrix of the vector W n,j 1 j 2 , for (j 1 , j 2 ) ∈ D n,2 . For j 1 = 0 or j 2 = 0, recall that W n,j 1 j 2 = 0. In this case, we also set C n,j 1 j 2 = 0. Thus, Therefore, 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 assumption (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 f k = (f 1 , . . . , f k ) ∈ F k of cluster functionals, 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 defined in (Cov).
Proof. The assumptions (Lin) and (Cov) imply that, as n → ∞, B n (ǫ) −→ 0 and a n −→ k s=1 c(f s , f s ) < ∞, respectively. Therefore, taking into account Remark 2.1, we obtain from Lemma 2 that, for each k ∈ N, for all t ∈ R k , with h(w) = exp(i t, w ), because by hypothesis, T n,t (f k ) −→ n→∞ 0 for all t ∈ R k and all f k = (f 1 , . . . , f k ) ∈ F k . Notice that W ′ n := (i,j)∈D n,2 W ′ n,ij ∼ N k (0, m 1 m 2 Cov(W n,11 )) Using triangular inequality, we deduce that and therefore (Z n (f 1 ), . . . , Z n (f k )) = (i,j)∈D n,2 W n,ij Remark 2.3. The previous theorem can be formulated for d = 3 as follows.
converges to zero as n → ∞ for all t ∈ R k and all k−tuple f k = (f 1 , . . . , f k ) ∈ F k of cluster functionals, with 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 that n = (n 1 , . . . , n d ) → ∞ means n 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 = (⌈ϑ 1 (n)⌉, . . . , ⌈ϑ d (n)⌉) for some strictly increasing continuous functions ϑ i : Suppose that from each block Y n we extract a sub-block Y ′ n and that the remaining parts R n = Y n − Y ′ n of the blocks Y n do not influence the process Z n (f ). In particular, this last statement is fulfilled if, . This assumption would allow us to consider T n,t (f k ) (or T * n,t (f k )) as a function of the blocks Y ′ n (separated by l 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, however, 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.

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 Theorem 3 in order to establish an asymptotic result. This section is inspired of the extremogram for times series defined in [4].
Let X = {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 provided that the limit exists.
In estimating the extremogram, the limit on x in (16) is replaced by a high quantile u n of the process. Defining u n as the (1 − 1/k n )−quantile of the stationary distribution of X t (s) or related quantity, with k n = o(n) −→ ∞, as n → ∞, one can redefine (16) by 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 sets A and B bounded away from zero, is that it must satisfy the following convergence  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: