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

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 Davis, R.A. and Mikosch, T. (2008) and in astronomy Long, J.P. and De Sousa, R.S. (2018)) 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 Bardet, J-M., Doukhan, P., Lang, G. and Ragache, N. (2007), Drees, H. and Rootzén, H. (2010) and Gómez-García, J.G. (2018).

Precisely, let $d \in N$ and let us denote $n := (n_{1}, \dots, n_{d})$ , $1 := (1, \dots, 1) \in N^{d}$ and $[j] := [1 : j]$ , where $[i : j] := {i, i + 1, \dots, j} \subset Z$ . Let $X = {X_{t} : t \in N^{d}}$ be a $R^{k} -$ valued stationary random field and let $X = {X_{n, t} : t \in$ ${[n_{1}] \times \dots \times [n_{d}]}}_{n \in N^{d}}$ 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

(1)

where $G$ is a non-degenerate distribution and $A \subseteq R^{k} ∖ {0}$ is the relevance set. Here, $I_{A} (\cdot)$ denotes the usual indicator function of a subset $A$ and the tendency $n \to \infty$ means that $n_{i} \to \infty$ for all $i \in [d]$ . 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 Resnick, S.I. (1986, 1987).

For each $i \in [d]$ , let $r_{i} := r_{n_{i}}$ be a integer value such that $r_{i} = o (n_{i})$ and $m_{i} := ⌈ n_{i} / r_{i} ⌉ := max {k \in N : k \leq n_{i} / r_{i}}$ . We define the $d -$ blocks (or simply blocks) of $X$ by

Y_{n, j_{1} \dots j_{d}} := {(X_{n, t})}_{t \in \prod_{i = 1}^{d} [(j_{i} - 1) r_{i} + 1 : j_{i} r_{i}]},

(2)

where $(j_{1}, \dots, j_{d}) \in D_{n, d} := \prod_{i = 1}^{d} [m_{i}]$ . We have thus $m_{1} \dots m_{d}$ complete blocks $Y_{n, j_{1} \dots j_{d}}$ , and no more than $m_{1} + m_{2} + \dots + m_{d} - d + 1$ incomplete ones which we will ignore. Besides, as usual, $\prod_{i = 1}^{d} A_{i}$ denotes the Cartesian product $A_{1} \times \dots \times A_{d}$ 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 Drees, H. and Rootzén, H. (2010) to $d -$ blocks.

Let $y = (x_{t})_{t \in \prod_{i = 1}^{d} [r_{i}]}$ be a $d -$ block. The core of the block $y$ (w.r.t. the relevance set $A$ ) is defined as

C (y) = {\begin{matrix} (x_{t})_{t \in \prod_{i = 1}^{d} [r_{i, I} : r_{i, S}]}, & if x_{t} \in A for some t \in \prod_{i = 1}^{d} [r_{i}]; 0, & otherwise, \end{matrix}

where, for each $i \in [d]$ , $r_{i, I}$ and $r_{i, S}$ are defined as

	$r_{i, I}$
	$r_{i, S}$

Let $(E, E)$ be a measurable subspace of $(R^{k}, B (R^{k}))$ for some $k \geq 1$ such that $0 \in E$ . Let $B_{l_{1}, \dots, l_{d}} (E)$ be the set of $E -$ valued blocks (or arrays) of size $l_{1} \times l_{2} \times \dots \times l_{d}$ , with $l_{1}, \dots, l_{d} \in N$ . Consider now the set

E_{\cup} := \infty ⋃ l_{1}, \dots, l_{d} = 1 B_{l_{1}, \dots, l_{d}} (E),

which is equipped with the $σ -$ field $E_{\cup}$ induced by the Borel $- σ -$ fields on $B_{l_{1}, \dots, l_{d}} (E)$ , for $l_{1}, \dots, l_{d} \in N$ . A cluster functional is a measurable map $f : (E_{\cup}, E_{\cup}) ⟶ (R, B (R))$ such that

f (y) = f (C (y)), for all y \in E_{\cup}, % and f (0) = 0.

(3)

Let $F$ be a class of cluster functionals and let ${Y_{n, j_{1} j_{2} \dots j_{d}} : (j_{1}, \dots, j_{d}) \in D_{n, d}}$ be the family of blocks of size $r_{1} \times r_{2} \times \dots \times r_{d}$ defined in (2). The empirical process $Z_{n}$ of cluster functionals in $F$ , is the process $(Z_{n} (f))_{f \in F}$ defined by

Z_{n} (f) := \frac{1}{\sqrt{n_{n} v_{n}}} \sum (j_{1}, \dots, j_{d}) \in D_{n, d} (f (Y_{n, j_{1} \dots j_{d}}) - E f (Y_{n, j_{1} \dots j_{d}})),

(4)

where $n_{n} = n_{1} \dots n_{d}$ and $v_{n} := P (X_{n, 1} \in A)$ with $A \subseteq E ∖ {0}$ 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 Bardet, J-M., Doukhan, P., Lang, G. and Ragache, N. (2007), but adapted here to stationary random fields.

Regarding the condition $T_{n} ⟶ 0$ , as $n \to \infty$ , 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 Doukhan, P. and Louhichi, S. (1999) under convenient conditions for the decay rates of the weak-dependence coefficients. These rates are calculated in Gómez-García, J.G. (2018) 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.

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 Bardet et al. Bardet, J-M., Doukhan, P., Lang, G. and Ragache, N. (2007) 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:

The vector $r = (r_{1}, \dots, r_{d}) \in N^{d}$ is such that $r_{i} ≪ n_{i}$ for each $i \in [d]$ .
Besides, denoting $r_{n} = r_{1} \dots r_{d}$ , $r_{n} v_{n} ⟶ τ < \infty$ and $n_{n} v_{n} ⟶ \infty$ , as $n \to \infty$ .

Secondly, consider the following essential convergence assumptions:

, $\forall ϵ > 0$ , $\forall f \in F$ ;
$(r_{n} v_{n})^{- 1} Cov (f (Y_{n}), g (Y_{n})) ⟶ c (f, g)$ , $\forall f, g \in F$ .

Consider now the random blocks $Y_{n, j_{1} \dots j_{d}}$ , with $(j_{1}, \dots, j_{d}) \in D_{n, d}$ defined in (2). For each $k -$ tuple of cluster functionals $f_{k} = (f_{1}, \dots, f_{k})$ and each $(j_{1}, \dots, j_{d}) \in D_{n, d}$ , we define the random vector:

W_{n, j_{1} \dots j_{d}} := \frac{1}{\sqrt{n_{n} v_{n}}} (f_{1} (Y_{n, j_{1} \dots j_{d}}) - E f_{1} (Y_{n, j_{1} \dots j_{d}}), \dots, f_{k} (Y_{n, j_{1} \dots j_{d}}) - E f_{k} (Y_{n, j_{1} \dots j_{d}})) .

(5)

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

Let $(W_{n, i j}^{'})_{(i, j) \in D_{n, 2}}$ be a sequence of zero mean independent $R^{k}$

-valued random variables, independents of the sequence

(W_{n, i j})_{(i, j) \in D_{n, 2}}

, such that

W_{n, i j}^{'} \sim N_{k} (0, C o v (W_{n, i j}))

, for all

(i, j) \in D_{n, 2}

. Denote by

C_{b}^{3}

the set of bounded functions

h : R^{k} ⟶ R

with bounded and continuous partial derivatives up to order

3

. For

h \in C_{b}^{3}

and

n = (n_{1}, n_{2}) \in N^{2}

, define

Δ_{n} := ∣ ∣ ∣ ∣ E ⎡ ⎢ ⎣ h ⎛ ⎜ ⎝ \sum (i, j) \in D_{n, 2} W_{n, i j} ⎞ ⎟ ⎠ - h ⎛ ⎜ ⎝ \sum (i, j) \in D_{n, 2} W_{n, i j}^{'} ⎞ ⎟ ⎠ ⎤ ⎥ ⎦ ∣ ∣ ∣ ∣ .

(6)

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

It exists $δ \in (0, 1]$ such that, for all $(i, j) \in D_{n, 2}$ , $E {∥ ∥ W_{n, i j} ∥ ∥}_{n, i j}^{2 + δ} < \infty$ for all $n \in N^{2}$ and all $k -$ tuple of cluster functionals $(f_{1}, \dots, f_{k}) \in F^{k}$ . Moreover, denote

$A_{n} := \sum (i, j) \in D_{n, 2} E {∥ ∥ W_{n, i j} ∥ ∥}_{n, i j}^{2 + δ} .$

Lemma 1 (Lindeberg under independence).

Suppose that the blocks $(Y_{n, i j})_{(i, j) \in D_{n, 2}}$ are independents and that the random variables $(W_{n, i j})_{(i, j) \in D_{n, 2}}$ defined in (5) satisfy Assumption (Lin’). Then, for all $n \in N^{2}$ :

Δ_{n} \leq 6 ∥ h^{(2)} ∥_{\infty}^{1 - δ} ∥ h^{(3)} ∥_{\infty}^{δ} A_{n} .

Proof. First, notice that

Δ_{n} \leq \sum (i, j) \in D_{n, 2} Δ_{n, i j},

(7)

where

	$Δ_{n, i j}$	$:= ∣ ∣ E [h_{i j} (V_{n, i j} + W_{n, i j}) - h_{i j} (V_{n, i j} + W_{n, i j}^{'})] ∣ ∣, \forall (i, j) \in D_{n, 2};$
	$V_{n, i j}$	$:= \sum (u, v) \in D_{n, 2} ∖ (⋃_{l = 0}^{i - 1} L_{l}^{m_{2}} \cup L_{i}^{j}) W_{n, u v}, \forall (i, j) \in D_{n, 2} ∖ {(m_{1}, m_{2})},$
	$V_{n, m_{1} m_{2}}$	$= 0; and$
	$h_{i j} (x)$	$:= E [h (x + i - 1 \sum u = 0 m_{2} \sum v = 1 W_{n, u v}^{'} + j - 1 \sum v = 0 W_{n, i v}^{'})] .$

Besides, we set the convention $W_{n, i j} = 0$ , if either $i = 0$ or $j = 0$ .

Now, we will use some lines of the proof of Lemma 1 in Bardet, J-M., Doukhan, P., Lang, G. and Ragache, N. (2007).
Let $v, w \in R^{k}$ . From Taylor’s formula, there exist vectors $v_{1, w}, v_{2, w} \in R^{k}$ such that:

	$h (v + w)$	$= h (v) + h^{(1)} (v) (w) + \frac{1}{2} h^{(2)} (v_{1, w}) (w, w)$
		$= h (v) + h^{(1)} (v) (w) + \frac{1}{2} h^{(2)} (v) (w, w) + \frac{1}{6} h^{(3)} (v_{2, w}) (w, w, w),$

where, for $j = 1, 2, 3$ , $h^{(j)} (v) (w_{1}, w_{2}, \dots, w_{j})$ stands for the value of the symmetric $j -$ linear form from $h^{(j)}$ of $(w_{1}, \dots, w_{j})$ at $v$ . Moreover, denote

∥ h^{(j)} (v) ∥_{1} = sup ∥ w_{1} ∥, \dots, ∥ w_{j} ∥ \leq 1 | h^{(j)} (v) (w_{1}, \dots, w_{j}) | and ∥ h^{(j)} ∥_{\infty} = sup v \in R^{k} ∥ h^{(j)} (v) ∥_{1} .

Thus, for $v, w, w^{'} \in R^{k}$ , there exist some suitable vectors $v_{1, w}, v_{2, w}, v_{1, w^{'}}, v_{2, w^{'}} \in R^{k}$ such that

by using the approximation of Taylor of order $2$ , and

by using the approximation of Taylor of order $3$ .
Thus, $γ = h (v + w) - h (v + w^{'}) - h^{(1)} (v) (w - w^{'}) - \frac{1}{2} (h^{(2)} (v) (w, w) - h^{(2)} (v) (w^{'}, w^{'}))$ satisfies:

$\| γ \|$	$\leq ((∥ w ∥^{2} + ∥ w^{'} ∥^{2}) ∥ h^{(2)} ∥_{\infty}) \land (\frac{1}{6} (∥ w ∥^{3} + ∥ w^{'} ∥^{3}) ∥ h^{(3)} ∥_{\infty})$
	$\leq (∥ w ∥^{2} ∥ h^{(2)} ∥_{\infty}) \land (\frac{1}{6} ∥ w ∥^{3} ∥ h^{(3)} ∥_{\infty}) + (∥ w ∥^{2} ∥ h^{(2)} ∥_{\infty}) \land (\frac{1}{6} ∥ w^{'} ∥^{3} ∥ h^{(3)} ∥_{\infty})$
	$+ (∥ w^{'} ∥^{2} ∥ h^{(2)} ∥_{\infty}) \land (\frac{1}{6} ∥ w ∥^{3} ∥ h^{(3)} ∥_{\infty}) + (∥ w^{'} ∥^{2} ∥ h^{(2)} ∥_{\infty}) \land (\frac{1}{6} ∥ w^{'} ∥^{3} ∥ h^{(3)} ∥_{\infty})$
	$\leq \frac{1}{6^{δ}} ∥ h^{(2)} ∥_{\infty}^{1 - δ} ∥ h^{(3)} ∥_{\infty}^{δ} (∥ w ∥^{2 + δ} + ∥ w ∥^{2 (1 - δ)} ∥ w^{'} ∥^{3 δ} + ∥ w ∥^{3 δ} ∥ w^{'} ∥^{2 (1 - δ)} + ∥ w^{'} ∥^{2 + δ}),$	(8)

where (2) is given by using the inequality $1 \land a \leq a^{δ}$ , with $a \geq 0$ and $δ \in [0, 1]$ .

Substituting $h_{i j}, V_{n, i j}, W_{n, i j}$ and $W_{n, i j}^{'}$ for $h, v, w$ and $w^{'}$ in the preceding inequality (2) and taking expectations, we will obtain a bound for $Δ_{n, i j}$ . Indeed,

because $V_{n, i j}$ is independent of $W_{n, i j}$ and $W_{n, i j}^{'}$ , and because $E W_{n, i j} = E W_{n, i j}^{'} = 0$ and $Cov (W_{n, i j}) = Cov (W_{n, i j}^{'})$ for all $(i, j) \in D_{n, 2}$ .
On the other hand, using Jensen’s inequality, we derive $E ∥ W_{n, i j}^{'} ∥^{2 + δ} \leq {(E ∥ W_{n, i j}^{'} ∥^{4})}^{\frac{1}{2} + \frac{δ}{4}}$ , and $E ∥ W_{n, i j}^{'} ∥^{4} \leq 3 \cdot {(E ∥ W_{n, i j} ∥^{2})}_{n, i j}^{2}$ because $W_{n, i j}^{'}$ is a Gaussian random variable with the same covariance as $W_{n, i j}$ .
Therefore,

E ∥ W_{n, i j}^{'} ∥^{2 + δ} \leq {(3 \cdot {(E ∥ W_{n, i j} ∥^{2})}_{n, i j}^{2})}^{\frac{1}{2} + \frac{δ}{4}} = 3^{\frac{1}{2} + \frac{δ}{4}} {(E ∥ W_{n, i j} ∥^{2})}_{n, i j}^{1 + \frac{δ}{2}} \leq 3^{\frac{1}{2} + \frac{δ}{4}} E ∥ W_{n, i j} ∥^{2 + δ},

(9)

	$E ∥ W_{n, i j}^{'} ∥^{2 (1 - δ)} E ∥ W_{n, i j} ∥^{3 δ}$	$\leq {(E ∥ W_{n, i j}^{'} ∥^{2})}^{1 - δ} E ∥ W_{n, i j} ∥^{3 δ}$
		$\leq {(E ∥ W_{n, i j} ∥^{2})}_{n, i j}^{1 - δ} E ∥ W_{n, i j} ∥^{3 δ} \leq E ∥ W_{n, i j} ∥^{2 + δ} .$		(10)

Besides, for $3 δ < 2$ ,

E ∥ W_{n, i j} ∥^{2 (1 - δ)} E ∥ W_{n, i j}^{'} ∥^{3 δ} \leq E ∥ W_{n, i j} ∥^{2 (1 - δ)} {(E ∥ W_{n, i j}^{'} ∥^{2})}^{\frac{3 δ}{2}} \leq E ∥ W_{n, i j} ∥^{2 + δ},

(11)

else

	$E ∥ W_{n, i j} ∥^{2 (1 - δ)} E ∥ W_{n, i j}^{'} ∥^{3 δ}$	$\leq E ∥ W_{n, i j} ∥^{2 (1 - δ)} {(E ∥ W_{n, i j}^{'} ∥^{4})}^{\frac{3 δ}{4}}, % because 3 δ \leq 4$
		$\leq 3^{\frac{3 δ}{4}} E ∥ W_{n, i j} ∥^{2 (1 - δ)} {(E ∥ W_{n, i j} ∥^{2})}_{n, i j}^{\frac{3 δ}{2}} \leq 3^{\frac{1}{2} + \frac{δ}{4}} E ∥ W_{n, i j} ∥^{2 + δ} .$		(12)

The inequalities (9)-(2) allow to simplify the terms between parentheses in the last inequality in (2). Recall that $∥ h_{i j}^{(k)} ∥_{\infty} \leq ∥ h^{(k)} ∥_{\infty}$ for all $(i, j) \in D_{n, 2}$ and $0 \leq k \leq 3$ . Therefore, we obtain that

Δ_{n, i j}

\leq \frac{2 (1 + 3^{\frac{1}{2} + \frac{δ}{4}})}{6^{δ}} ∥ h^{(2)} ∥_{\infty}^{1 - δ} ∥ h^{(3)} ∥_{\infty}^{δ} E ∥ W_{n, i j} ∥^{2 + δ} \leq 6 ∥ h^{(2)} ∥_{\infty}^{1 - δ} ∥ h^{(3)} ∥_{\infty}^{δ} E ∥ W_{n, i j} ∥^{2 + δ},

because, for all $δ \in [0, 1]$ , $C (δ) = \frac{2 (1 + 3^{\frac{1}{2} + \frac{δ}{4}})}{6^{δ}} \leq C (0) = 2 (1 + \sqrt{3}) < 6$ .

As a consequence, from Assumption (Lin’), $Δ_{n} \leq 6 ∥ h^{(2)} ∥_{\infty}^{1 - δ} ∥ h^{(3)} ∥_{\infty}^{δ} A_{n}$ . $□$

Remark 2.1.

Taking $ϵ < 6 ∥ h^{(2)} ∥_{\infty} (∥ h^{(3)} ∥_{\infty})^{- 1}$ and using suitably the second inequality of (2) in the proof of Lemma 1, classical Lindeberg conditions may be used:

Δ_{n} \leq 2 ∥ h^{(2)} ∥_{\infty} B_{n} (ϵ) + ∥ h^{(3)} ∥_{\infty} a_{n} (\frac{4}{3} ϵ + \sqrt{B_{n} (ϵ)}),

(13)

where

	$B_{n} (ϵ)$	$= \sum (i, j) \in D_{n, 2} E [∥ W_{n, i j} ∥^{2} I_{{∥ W_{n, i j} ∥ > ϵ}}], ϵ > 0, n \in N^{2};$
	$a_{n}$	$= \sum (i . j) \in D_{n, 2} E ∥ W_{n, i j} ∥^{2} < \infty, n \in N^{2} .$

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

Δ_{n} \leq 2 ∥ h^{(2)} ∥_{\infty} ϵ^{- δ} A_{n} + ∥ h^{(3)} ∥_{\infty} a_{n} (\frac{4}{3} ϵ + ϵ^{- δ / 2} \sqrt{A_{n}}),

for $δ \in (0, 1)$ and $ϵ > 0$ .

The proof of this remark for general independent random vectors is given in (Bardet, J-M., Doukhan, P., Lang, G. and Ragache, N., 2007, p.165).

Remark 2.2.

Observe that the assumptions (Lin) and (Cov) imply that $B_{n} (ϵ) ⟶ n \to \infty 0$ and that $a_{n} = \sum_{i = 1}^{k} (r_{n} v_{n})^{- 1} C o v (f_{i} (Y_{n}), f_{i} (Y_{n})) ⟶ n \to \infty \sum_{i = 1}^{k} c (f_{i}, f_{i}) < \infty$ , respectively. Therefore, if the blocks $(Y_{n, i j})_{(i, j) \in D_{n, 2}}$ are independent and if the assumptions (Lin) and (Cov) hold, then from Lemma 1 and Remark 2.1, the fidis of the empirical process $(Z_{n} (f))_{f \in F}$ of cluster functionals converge to the fidis of a Gaussian process $(Z (f))_{f \in F}$ with covariance function $c$ .

For the dependent case, we need to consider more notations:
Let $L_{i}^{j} := {(i, v) : v \in [j]} \subset D_{n, 2}$ , for all $(i, j) \in D_{n, 2}$ . We set $L_{i}^{0} = L_{0}^{j} = \emptyset$ for any $i \in [m_{1}]$ and any $j \in [m_{2}]$ . For each $k \in N$ , $f_{k} = (f_{1}, \dots, f_{k}) \in F^{k}$ , $t \in R^{k}$ and $n \in N^{2}$ ; we define

Lemma 2 (Dependent Lindeberg lemma).

Suppose that the r.v.’s $(W_{n, i j})_{(i, j) \in D_{n, 2}}$ defined in (5) satisfy Assumption (Lin’). Consider the special case of complex exponential functions $h (w) = exp (i ⟨ t, w ⟩)$ with $t \in R^{k}$ . Then, for each $k \in N$ and each $k -$ tuple $f_{k} = (f_{1}, \dots, f_{k})$ of cluster functionals, the following inequality holds:

Δ_{n} \leq T_{n, t} (f_{k}) + 6 ∥ t ∥^{2 + δ} A_{n}, n \in N^{2} .

Proof. Consider $(W_{n, j_{1} j_{2}}^{*})_{(j_{1}, j_{2}) \in D_{n, 2}}$ an array of independent random variables satisfying Assumption (Lin’) and such that $(W_{n, j_{1} j_{2}}^{*})_{(j_{1}, j_{2}) \in D_{n, 2}}$ is independent of $(W_{n, j_{1} j_{2}})_{(j_{1}, j_{2}) \in D_{n, 2}}$ and $(W_{n, j_{1} j_{2}}^{'})_{(j_{1}, j_{2}) \in D_{n, 2}}$ . Moreover, assume that $W_{n, j_{1} j_{2}}^{*}$ has the same distribution as $W_{n, j_{1} j_{2}}$ for $(j_{1}, j_{2}) \in D_{n, 2}$ .
Then, using the same decomposition (7) in the proof of the previous lemma, one can also write,

Δ_{n, j_{1} j_{2}} \leq ∣ ∣ E [h_{j_{1} j_{2}} (V_{n, j_{1} j_{2}} + W_{n, j_{1} j_{2}}) - h_{j_{1} j_{2}} (V_{n, j_{1} j_{2}} + W_{n, j_{1} j_{2}}^{*})] ∣ ∣ + ∣ ∣ E [h_{j_{1} j_{2}} (V_{n, j_{1} j_{2}} + W_{n, j_{1} j_{2}}^{*}) - h_{j_{1} j_{2}} (V_{n, j_{1} j_{2}} + W_{n, j_{1} j_{2}}^{'})] ∣ ∣ .

(14)

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

6 ∥ h^{(2)} ∥_{\infty}^{1 - δ} ∥ h^{(3)} ∥_{\infty}^{δ} E ∥ W_{n, j_{1} j_{2}} ∥^{2 + δ} \leq 6 ∥ t ∥^{2 + δ} E ∥ W_{n, j_{1} j_{2}} ∥^{2 + δ} .

For the first term of the RHS of the inequality (14), first notice that for a $R^{k} -$ valued random vector $X$ independent from $(W_{n, j_{1} j_{2}}^{'})_{(j_{1}, j_{2}) \in D_{n, 2}}$ ,

	$E h_{j_{1} j_{2}} (X)$	$= E [h (X + j_{1} - 1 \sum u = 0 m_{2} \sum v = 1 W_{n, u v}^{'} + j_{2} - 1 \sum v = 0 W_{n, j_{1} v}^{'})]$
		$= exp (- \frac{1}{2} t^{T} (j_{1} - 1 \sum u = 0 m_{2} \sum v = 1 C_{n, u v} + j_{2} - 1 \sum v = 0 C_{n, j_{1} v}) t) \cdot E [exp (i ⟨ t, X ⟩)],$

because $W_{n, j_{1} j_{2}}^{'} \sim N_{k} (0, C_{n, j_{1} j_{2}})$ , 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 $(j_{1}, j_{2}) \in D_{n, 2}$ . 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,

∣ ∣ E [h_{j_{1} j_{2}} (V_{n, j_{1} j_{2}} + W_{n, j_{1} j_{2}}) - h_{j_{1} j_{2}} (V_{n, j_{1} j_{2}} + W_{n, j_{1} j_{2}}^{*})] ∣ ∣ = ∣ ∣ ∣ ∣ exp (- \frac{1}{2} t^{T} (j_{1} - 1 \sum u = 0 m_{2} \sum v = 1 C_{n, u v} + j_{2} - 1 \sum v = 0 C_{n, j_{1} v}) t) \times E [exp (i ⟨ t, V_{n, j_{1} j_{2}} ⟩) \cdot (exp (i ⟨ t, W_{n, j_{1} j_{2}} ⟩) - exp (i ⟨ t, W_{n, j_{1} j_{2}}^{*} ⟩))] ∣ ∣ = ∣ ∣ ∣ ∣ exp (- \frac{1}{2} t^{T} (j_{1} - 1 \sum u = 0 m_{2} \sum v = 1 C_{n, u v} + j_{2} - 1 \sum v = 0 C_{n, j_{1} v}) t) ∣ ∣ ∣ ∣ \times ∣ ∣ Cov (exp (i ⟨ t, V_{n, j_{1} j_{2}} ⟩), exp (i ⟨ t, W_{n, j_{1} j_{2}} ⟩)) ∣ ∣ \leq ∣ ∣ Cov (exp (i ⟨ t, V_{n, j_{1} j_{2}} ⟩), exp (i ⟨ t, W_{n, j_{1} j_{2}} ⟩)) ∣ ∣ .

Therefore,

	$Δ_{n}$	$= \sum (j_{1}, j_{2}) \in D_{n, 2} Δ_{n, j_{1} j_{2}}$
		$\leq \sum (j_{1}, j_{2}) \in D_{n, 2} (∣ ∣ Cov (exp (i ⟨ t, V_{n, j_{1} j_{2}} ⟩), exp (i ⟨ t, W_{n, j_{1} j_{2}} ⟩)) ∣ ∣ + 6 ∥ t ∥^{2 + δ} E ∥ W_{n, j_{1} j_{2}} ∥^{2 + δ})$
		$= T_{n, t} (f_{k}) + 6 ∥ t ∥^{2 + δ} A_{n} .$

$□$

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 assumption (Bas) holds and that the assumptions (Lin) and (Cov) are satisfied. Then, if for each $k \in N$ , $T_{n, t} (f_{k})$ converges to zero as $n \to \infty$ , for all $t \in R^{k}$ and all $k -$ tuple $f_{k} = (f_{1}, \dots, f_{k}) \in F^{k}$ of cluster functionals, the fidis of the empirical process $(Z_{n} (f))_{f \in F}$ of cluster functionals converge to the fidis of a Gaussian process $(Z (f))_{f \in F}$ with covariance function $c$ defined in (Cov).

Proof. The assumptions (Lin) and (Cov) imply that, as $n \to \infty$ , $B_{n} (ϵ) ⟶ 0$ and $a_{n} ⟶ \sum_{s = 1}^{k} c (f$

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

Authors

Central limit theorems for stationary random fields under weak dependence with application to ambit and mixed moving average fields

A Dynamic Taylor's Law

On a linear functional for infinitely divisible moving average random fields

Simplified quasi-likelihood analysis for a locally asymptotically quadratic random field

On the excursion area of perturbed Gaussian fields

Marginalization in Composed Probabilistic Models

Upper tail dependence and smoothness of random fields

1. Introduction

2. Results

Lemma 1 (Lindeberg under independence).

Remark 2.1.

Remark 2.2.

Lemma 2 (Dependent Lindeberg lemma).

Theorem 3 (CLT for cluster functionals on random fields).

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

Authors

Central limit theorems for stationary random fields under weak dependence with application to ambit and mixed moving average fields

A Dynamic Taylor's Law

On a linear functional for infinitely divisible moving average random fields

Simplified quasi-likelihood analysis for a locally asymptotically quadratic random field

On the excursion area of perturbed Gaussian fields

Marginalization in Composed Probabilistic Models

Upper tail dependence and smoothness of random fields

This week in AI

1. Introduction

2. Results

Lemma 1 (Lindeberg under independence).

Remark 2.1.

Remark 2.2.

Lemma 2 (Dependent Lindeberg lemma).

Theorem 3 (CLT for cluster functionals on random fields).