2. Preliminaries

An element $\textbf{n}\in{\mathbb Z}^d$ is often denoted by $(n_1,n_2,\ldots,n_d)$ and $\|\textbf{n}\|$ is its sup norm. We write $\textbf{i}\leq \textbf{j}$ and $\textbf{n}\to\pmb{\infty}$ whenever $i_l\leq j_l$ and $n_l\to\infty$ , respectively, for all $l\in\{1,2,\ldots, d\}$ . We put $\mathbf{0}:= (0,0,\ldots,0)$ , $\mathbf{1}:= (1,1,\ldots,1)$ , and $\pmb{\infty}:= (\infty,\infty,\ldots,\infty)$ .

In our considerations $\{X_{\textbf{n}} \colon \textbf{n}\in{\mathbb Z}^d\}$ is a d-dimensional stationary random field. We ask for the asymptotics of ${\mathbb{P}}(M_{\textbf{N}(n)}\leq v_n)$ as $n\to\infty$ , for $\textbf{N} = \{\textbf{N}(n) \colon n\in{\mathbb N}\}\subset{\mathbb N}^d$ , such that

(2.1) \begin{equation}\textbf{N}(n)\to\pmb{\infty}\quad \text{and} \quad N^*(n):= N_1(n)N_2(n)\cdots N_d(n)\sim n^d\end{equation}

and $\{v_n \colon n\in{\mathbb N}\} \subset {\mathbb R}$ .

We are interested in weakly dependent fields. We assume that

(2.2) \begin{equation}{\mathbb{P}}(M_{\textbf{N}(n)}\leq v_n) = {\mathbb{P}}(M_{\textbf{p}(n)}\leq v_n)^{k_n^d}+ {\text{o}}(1)\end{equation}

is satisfied for some $r_n\to\infty$ and all $k_n\to\infty$ such that $k_n= {\text{o}}(r_n)$ , with

(2.3) \begin{equation}\textbf{p}(n):= (\lfloor N_1(n)/k_n\rfloor,\lfloor N_2(n)/k_n\rfloor,\ldots,\lfloor N_d(n)/k_n\rfloor ).\end{equation}

Applying the classical fact (see e.g. O’Brien [Reference O’Brien17])

(2.4) \begin{equation}(a_n)^n-\exp(\!-n(1-a_n))\to 0 \quad\text{as}\ n\to\infty \quad \text{for} \ a_n\in[0,1],\end{equation}

we obtain that (2.2) implies

(2.5) \begin{equation}{\mathbb{P}}(M_{\textbf{N}(n)}\leq v_n) = \exp(\!-k_n^d {\mathbb{P}}(M_{\textbf{p}(n)}> v_n))+ {\text{o}}(1).\end{equation}

Above, $p_l(n)= {\text{o}}(N_l(n))$ for $l\in\{1,2,\ldots,d\}$ , which we denote by $\textbf{p}(n)= {\text{o}}(\textbf{N}(n))$ .

Let $\preccurlyeq$ be an arbitrary total order on ${\mathbb Z}^d$ which is translation-invariant, that is, $\textbf{i} \preccurlyeq \textbf{j}$ implies $\textbf{i} + \textbf{k} \preccurlyeq \textbf{j} + \textbf{k}$ . An example of such an order is the lexicographic order:

\begin{equation*} \textbf{i} \preccurlyeq \textbf{j} \quad {\text{if and only if}} \ (\textbf{i} = \textbf{j}\ \text{or}\ i_l < j_l\ \text{for the first}\ \textit{l}\ \text{where}\ i_l\ \text{and}\ j_l\ \text{differ}).\end{equation*}

We will write $\textbf{i} \prec \textbf{j}$ whenever $\textbf{i} \preccurlyeq \textbf{j}$ and $\textbf{i}\neq\textbf{j}$ . For technical requirements in further sections, we define the set $A(\textbf{p})\subset {\mathbb Z}^d$ for each $\textbf{p}\in{\mathbb N}^d$ as follows:

(2.6) \begin{equation}A(\textbf{p}) := \{\textbf{j}\in{\mathbb Z}^d \colon -\textbf{p} \leq \textbf{j} \leq \textbf{p} \ \text{and}\ \mathbf{0} \prec \textbf{j} \}.\end{equation}

3. Main theorem

In the following the main result of the paper is presented. The asymptotic behaviour of ${\mathbb{P}}(M_{\textbf{N}(n)}\leq v_n)$ as $n\to\infty$ , for weakly dependent $\{X_{\textbf{n}}\}$ and for $\{\textbf{N}(n)\}$ and $\{v_n\}$ as in Section 2, is described.

Theorem 3.1. Let $\{X_\textbf{n}\}$ satisfy (2.2) for some $r_n\to\infty$ and all $k_n\to\infty$ such that $k_n={\text{o}}(r_n)$ . If

(3.1) \begin{equation}\liminf_{n\to\infty} {\mathbb{P}} (M_{\textbf{N}(n)} \leq v_n) > 0,\end{equation}

then, for every $\{k_n\}$ as above, we obtain

(3.2) \begin{equation}{\mathbb{P}} (M_{\textbf{N}(n)} \leq v_n) = \exp(\!- n^d {\mathbb{P}}(X_{\mathbf{0}} > v_n, M_{A(\textbf{p}(n))}\leq v_n)) + {\text{o}}(1){,}\end{equation}

with $\textbf{p}(n)$ and $A(\textbf{p}(n))$ given by (2.3) and (2.6), respectively.

Remark 3.1. If (2.2) holds for some $k_n\to\infty$ , then (3.1) is implied by the condition

(3.3) \begin{equation}\limsup_{n\to\infty} n^d{\mathbb{P}}(X_{\mathbf{0}} > v_n) < \infty.\end{equation}

This follows from (2.5) and the inequality

\begin{equation*}k_n^d{\mathbb{P}}(M_{\textbf{p}(n)}> v_n) \leq N^*(n) {\mathbb{P}}(X_{\mathbf{0}} >v_n) \sim n^d {\mathbb{P}}(X_{\mathbf{0}} >v_n).\end{equation*}

The proof of the theorem generalizes the reasoning proposed by O’Brien [Reference O’Brien17, Theorem 2.1] for sequences. A way of dividing the event $\{M_{\textbf{p}(n)} > v_n\}$ into $p^*(n):= p_1(n)p_2(n)\cdots p_d(n)$ mutually exclusive events determined by $\preccurlyeq$ (which are similar in some sense) plays a key role in the proof. An analogous technique was used by French and Davis [Reference French and Davis9, Lemma 4] in the two-dimensional Gaussian case. Recently, Ling [Reference Ling15, Lemma 3.1] expanded their result to some non-Gaussian fields. In both papers the authors restrict themselves to the lexicographic order on ${\mathbb Z}^2$ .

Proof of Theorem 3.1 . Let the assumptions of the theorem be satisfied. Then (2.5) holds. Dividing the set $\{M_{\textbf{p}(n)} > v_n\}$ into $p^*(n)=p_1(n)p_2(n)\cdots p_d(n)$ disjoint sets and applying monotonicity and stationarity, we obtain

\begin{align*} {\mathbb{P}}(M_{\textbf{p}(n)} > v_n)& = \sum_{\mathbf{1}\leq \textbf{j} \leq \textbf{p}(n)} {\mathbb{P}}(X_\textbf{j} > v_n, X_{\textbf{i}} \leq v_n\ \text{for all}\ \textbf{i} \succ \textbf{j}\ \text{such that}\ \mathbf{1} \leq \textbf{i} \leq \textbf{p}(n) )\\& \geq \sum_{\mathbf{1}\leq \textbf{j} \leq \textbf{p}(n)} {\mathbb{P}} (X_\textbf{j} > v_n, X_{\textbf{i}} \leq v_n\ \text{ for all }\ \textbf{i} \in A(\textbf{p}(n))+\textbf{j} ) \\& = p^*(n) {\mathbb{P}} (X_{\mathbf{0}} > v_n, M_{A(\textbf{p}(n))} \leq v_n ),\end{align*}

which, combined with (2.5) and the fact that $k_n^dp^*(n) \sim n^d$ , gives

(3.4) \begin{equation}{\mathbb{P}} (M_{\textbf{N}(n)}\leq v_n) \leq \exp(\!-n^d {\mathbb{P}}(X_{\mathbf{0}} > v_n, M_{A(\textbf{p}(n))} \leq v_n)) + {\text{o}}(1).\end{equation}

In the second step of the proof we will show that the reverse inequality to (3.4) also holds. It is sufficient to consider the case ${\mathbb{P}}(M_{\textbf{N}(n)} \leq v_n) \to \gamma$ for $\gamma \in [0,1]$ , and we do so. Since $\gamma=0$ is excluded by assumption (3.1) and for $\gamma =1$ the proven inequality is obvious, we focus on $\gamma\in(0,1)$ . Let us choose $\{t_n\}\subset{\mathbb N}_+$ so that $t_n\to\infty$ and $t_n={\text{o}}(k_n)$ . Put

\begin{equation*} \textbf{s}(n):= (\lfloor N_1(n)/t_n\rfloor,\lfloor N_2(n)/t_n\rfloor , \ldots, \lfloor N_d(n)/t_n\rfloor )\quad\text{and}\quad s^*(n):= s_1(n)s_2(n)\cdots s_d(n).\end{equation*}

Since $t_n= {\text{o}}(r_n)$ , (2.5) holds with $k_n$ replaced by $t_n$ and $\textbf{p}(n)$ replaced by $\textbf{s}(n)$ . Also, $\textbf{p}(n) = {\text{o}}(\textbf{s}(n))$ and $\textbf{s}(n)= {\text{o}}(\textbf{N}(n))$ . Moreover, for the sets

\begin{equation*}C(\textbf{p}(n), \textbf{s}(n)):= \{\textbf{j} \in {\mathbb Z}^d \colon \textbf{p}(n)+\mathbf{1} \leq \textbf{j} \leq \textbf{s}(n)-\textbf{p}(n)\}\end{equation*}

and

\begin{equation*}B(\textbf{p}(n), \textbf{s}(n)):= \{\textbf{j}\in{\mathbb Z}^d \colon \mathbf{1} \leq \textbf{j} \leq \textbf{s}(n)\} \backslash \, C(\textbf{p}(n), \textbf{s}(n)){,}\end{equation*}

we obtain

\begin{align*}\dfrac{{\mathbb{P}}(M_{\textbf{s}(n)} > v_n, M_{B(\textbf{p}(n), \textbf{s}(n))} \leq v_n)}{{\mathbb{P}}(M_{B(\textbf{p}(n), \textbf{s}(n))} > v_n)}& = \dfrac{{\mathbb{P}}(M_{\textbf{s}(n)} > v_n) - {\mathbb{P}}(M_{B(\textbf{p}(n), \textbf{s}(n))} > v_n)}{{\mathbb{P}}(M_{B(\textbf{p}(n), \textbf{s}(n))} > v_n)} \\[6pt]& = \dfrac{{\mathbb{P}}(M_{\textbf{s}(n)} > v_n)}{{\mathbb{P}}(M_{B(\textbf{p}(n), \textbf{s}(n))} > v_n)} - 1\\& = \dfrac{{\mathbb{P}}(M_{\textbf{s}(n)} > v_n)}{ {\text{o}}(s^*(n)/p^*(n)) \cdot {\mathbb{P}}(M_{\textbf{p}(n)} > v_n)} - 1\\& = \dfrac{1+{\text{o}}(1)}{{\text{o}}(1)} \cdot \dfrac{t_n^d {\mathbb{P}}(M_{\textbf{s}(n)} > v_n)}{k_n^d {\mathbb{P}}(M_{\textbf{p}(n)} > v_n)} - 1.\end{align*}

Applying (2.5) twice, we get

\begin{equation*}\dfrac{t_n^d {\mathbb{P}}(M_{\textbf{s}(n)} > v_n)}{k_n^d {\mathbb{P}}(M_{\textbf{p}(n)} > v_n)} \to \dfrac{-\log \gamma}{- \log \gamma} = 1 \quad \text{as} \ n\to\infty,\end{equation*}

and consequently

(3.5) \begin{equation}\dfrac{{\mathbb{P}}(M_{B(\textbf{p}(n), \textbf{s}(n))} > v_n)}{{\mathbb{P}} (M_{\textbf{s}(n)} > v_n, M_{B(\textbf{p}(n), \textbf{s}(n))} \leq v_n) } \to 0 \quad \text{as} \ n\to\infty.\end{equation}

Now, observe that

\begin{align*}{\mathbb{P}} (M_{\textbf{s}(n)} > v_n) & = {\mathbb{P}} (M_{\textbf{s}(n)} > v_n, M_{B(\textbf{p}(n), \textbf{s}(n))} \leq v_n) + {\mathbb{P}} (M_{B(\textbf{p}(n), \textbf{s}(n))} > v_n) \\& = {\mathbb{P}} (M_{\textbf{s}(n)} > v_n, M_{B(\textbf{p}(n), \textbf{s}(n))} \leq v_n) (1+ {\text{o}}(1))\\& \leq \sum_{\textbf{j} \in C(\textbf{p}(n), \textbf{s}(n))} {\mathbb{P}} (X_{\mathbf{j}} > v_n, M_{A(\textbf{p}(n)) + \textbf{j}} \leq v_n) \cdot (1+ {\text{o}}(1))\\& \leq s^*(n) {\mathbb{P}} (X_{\mathbf{0}} > v_n, M_{A(\textbf{p}(n))} \leq v_n) (1+ {\text{o}}(1)),\end{align*}

by property (3.5), subadditivity and monotonicity of probability, and by stationarity of the field $\{X_{\textbf{n}}\}$ . Applying (2.5) with $(k_n, \textbf{p}(n))$ replaced by $(t_n, \textbf{s}(n))$ and the fact that $t_n^d s^*(n) \sim n^d$ , we conclude that

(3.6) \begin{align}{\mathbb{P}} (M_{\textbf{N}(n)}\leq v_n) & \geq \exp (\!-t_n^d s^*(n) {\mathbb{P}} (X_{\mathbf{0}} > v_n, M_{A(\textbf{p}(n))} \leq v_n) (1+ {\text{o}}(1))) + {\text{o}}(1)\notag \\&= \exp (\!-n^d {\mathbb{P}}(X_{\mathbf{0}} > v_n, M_{A(\textbf{p}(n))} \leq v_n)) + {\text{o}}(1).\end{align}

Since inequalities (3.4) and (3.6) are both satisfied, the proof is complete.

Theorem 3.1 immediately yields the following generalization of the result established by Chernick, Hsing, and McCormick [Reference Chernick, Hsing and McCormick5, Proposition 1.1] for $d=1$ . Assumption (3.7) given below is a multidimensional counterpart of the local mixing condition $D^{(m+1)}(v_n)$ defined in [Reference Chernick, Hsing and McCormick5] for sequences and it is satisfied by m-dependent fields, for example (see Section 5.1).

Corollary 3.1. Let the assumptions of Theorem 3.1 be satisfied. Then

\begin{equation*}{\mathbb{P}} (M_{\textbf{N}(n)} \leq v_n) = \exp (\!- n^d {\mathbb{P}} (X_{\mathbf{0}} > v_n, M_{A((m,m,\ldots,m))}\leq v_n ) ) + {\text{o}}(1)\end{equation*}

if and only if

(3.7) \begin{equation}n^d {\mathbb{P}} (X_{\mathbf{0}} > v_{n} \geq M_{A((m,m,\ldots,m))}, M_{A(\textbf{p}(n)) \backslash A((m,m,\ldots,m))} > v_n ) \xrightarrow[n\to\infty]{} 0,\end{equation}

where $k_n\to\infty$ is such that $k_n = {\text{o}}(r_n)$ .

We point out that Corollary 3.1 reforms a faulty formula for m-dependent fields proposed by Ferreira and Pereira [Reference Ferreira and Pereira8, Proposition 2.1]; see [Reference Jakubowski and Soja-Kukieła12, Example 5.5]. We also suggest comparing the above condition (3.7) with assumption $D^{\prime\prime}(v_n, \mathcal{B}_n,\mathcal{V})$ proposed by Pereira, Martins, and Ferreira [Reference Pereira, Martins and Ferreira18, Definition 3.1].

Remark 3.2. There exists a close relationship between Theorem 3.1 and compound Poisson approximations in the spirit of Arratia, Goldstein, and Gordon [Reference Arratia, Goldstein and Gordon1, Section 4.2.1]. The random variable

\Lambda _n^{(1)}: = \sum\limits_{{\bf{1}} \le {\bf{k}} \le {\bf{N}}(n)} {{_{\{ {X_{\bf{k}}} > {v_n},{M_{{\bf{k}} + A({\bf{p}}(n))}} \le {v_n}\} }},}\

with the expectation ${\lambda^{(1)}_n}:= N^*(n) {\mathbb{P}} (X_{\mathbf{0}} > v_n, M_{A(\textbf{p}(n))}\leq v_n ) $ , estimates the number of clusters of exceedances over $v_n$ in the set $\{\textbf{k} \colon \mathbf{1} \leq \textbf{k} \leq \textbf{N}(n)\}$ , and we have

\begin{equation*}{\mathbb{P}}(M_{\textbf{N}(n)} \leq v_n) = \exp(\!-{\lambda^{(1)}_n}) + {\text{o}}(1). \end{equation*}

4. Extremal index

In this part we use the results given in Section 3 to establish formulas (4.2) and (4.3) for the extremal index $\theta$ for random fields. We refer to Choi [Reference Choi6] or Jakubowski and Soja-Kukieła [Reference Jakubowski and Soja-Kukieła12] for definitions and some considerations on the extremal index in the multidimensional setting.

Here we present a method for calculating the number $\theta\in[0,1]$ satisfying

(4.1) \begin{equation}{\mathbb{P}}(M_{\textbf{N}(n)} \leq v_n) - {\mathbb{P}}(X_{\mathbf{0}} \leq v_n) ^{\theta n^d} \to 0 \quad \text{as}\ n\to\infty,\end{equation}

whenever such a $\theta$ exists. Let us observe that according to (2.4) we have

\begin{equation*}{\mathbb{P}}(X_{\mathbf{0}} \leq v_n) ^{n^d} = \exp (\!-n^d {\mathbb{P}}(X_{\mathbf{0}} > v_n)) + {\text{o}}(1)\end{equation*}

and, moreover, Theorem 3.1 yields

\begin{equation*}{\mathbb{P}}(M_{\textbf{N}(n)} \leq v_n) = \exp (\!- n^d {\mathbb{P}} (X_{\mathbf{0}} > v_n, M_{A(\textbf{p}(n))}\leq v_n) ) + {\text{o}}(1)\end{equation*}

for $\textbf{N}(n)$ , $v_n$ and $\textbf{p}(n)$ satisfying appropriate assumptions. Hence, provided that

\begin{equation*} 0 < \liminf_{n\to\infty} n^d{\mathbb{P}}(X_{\mathbf{0}} > v_n) \leq \limsup_{n\to\infty }n^d{\mathbb{P}}(X_{\mathbf{0}} > v_n) < \infty,\end{equation*}

condition (4.1) is satisfied if and only if

(4.2) \begin{equation}\theta = \lim_{n\to\infty}{\mathbb{P}} (M_{A(\textbf{p}(n))}\leq v_n \mid X_{\mathbf{0}} > v_n) .\end{equation}

Formula (4.2), allowing computation of extremal indices $\theta$ for random fields, is a multidimensional generalization of (1.2). In the special case when assumption (3.7) is satisfied, it is easy to show that formula (4.2) simplifies to the following:

(4.3) \begin{equation}\theta = \lim_{n\to\infty} {\mathbb{P}} (M_{A((m,m,\ldots,m))}\leq v_n \mid X_{\mathbf{0}} > v_n) .\end{equation}

The above formulas are in line with the interpretation of $\theta$ as the reciprocal of the mean number of high threshold exceedances in a cluster. Indeed, they answer the question: What is the asymptotic probability that a given element of a cluster is the distinguished element of the cluster?, where the distinguished element in a cluster is the greatest one with respect to the order $\preccurlyeq$ . This identification of a unique representative for each cluster is called declustering, declumping, or anchoring, and has much in common with compound Poisson approximations (see e.g. [Reference Arratia, Goldstein and Gordon1], [Reference Barbour and Chryssaphinou2], and [Reference Basrak and Planinić4]).

Remark 4.1. Formula (4.2) justifies the following definition of the runs estimator $\hat{\theta}^R_{\textbf{N}(n)}$ for the extremal index $\theta$ :

\begin{equation*}\hat{\theta}^R_{\textbf{N}(n)} := S_n^{-1}\sum_{\mathbf{1}+\textbf{p} (n) \, \leq \, \textbf{k} \, \leq \, \textbf{N}(n)-\textbf{p}(n)} { \mathbb{I}_{\{X_{\mathbf{k}} > v_n, M_{\textbf{k} + A(\textbf{p}(n))}\leq v_n\}} },\end{equation*}

where $S_n$ is the number of exceedances over $v_n$ in the set $\{\textbf{k}\in{\mathbb Z}^d \colon \mathbf{1} \leq \textbf{k} \leq \textbf{N}(n)\}$ .

5. Maxima of m-dependent fields

In this section we focus on m-dependent fields. We recall that $\{X_\textbf{n}\}$ is m-dependent for some $m\in{\mathbb N} $ if the families $\{X_{\textbf{i}}\colon \textbf{i}\in U \}$ and $\{X_{\textbf{j}} \colon \textbf{j}\in V\}$ are independent for all pairs of finite sets $ U , V \subset{\mathbb Z}^d$ satisfying $\min \{\|\textbf{i}-\textbf{j}\| \colon \textbf{i}\in U ,\,\textbf{j}\in V \} > m$ .

Let us assume that $\{X_{\textbf{n}}\}$ is m-dependent and satisfies (3.1) for some sequence $\{v_n\}\subset{\mathbb R} $ . Then it is easy to show that condition (3.3) holds too (see [Reference Jakubowski and Soja-Kukieła12, Remark 4.2]). Below, we present two methods that can be used to calculate the limit of ${\mathbb{P}}(M_{\textbf{N}(n)} \leq v_n)$ . A direct connection between them can be given and we illustrate it in the case $d=2$ .

5.1. First method

The first of the methods is a consequence of the main result presented in the paper. Since the field $\{X_{\textbf{n}}\}$ is m-dependent, it satisfies (2.2) for each $k_n\to\infty$ such that $k_n= {\text{o}}(r_n)$ , for some $r_n\to\infty$ (see e.g. [Reference Jakubowski and Soja-Kukieła12]). Moreover, the inequality

\begin{align*}& n^d {\mathbb{P}} (X_{\mathbf{0}} > v_{n} \geq M_{A((m,m,\ldots,m))}, M_{A(\textbf{p}(n)) \backslash A((m,m,\ldots,m))} > v_n ) \\& \qquad \leq n^d \sum_{\textbf{i} \in A(\textbf{p}(n)), \|\textbf{i}\| > m} {\mathbb{P}}(X_{\mathbf{0}} > v_{n}, X_{\textbf{i}} > v_{n})\\& \qquad = n^d \sum_{\textbf{i} \in A(\textbf{p}(n)), \|\textbf{i}\| > m} {\mathbb{P}}(X_{\mathbf{0}} > v_{n}){\mathbb{P}}(X_{\textbf{i}} > v_{n})\\& \qquad \leq n^d \cdot \dfrac{n^d}{k_n^d} \cdot {\mathbb{P}}(X_{\mathbf{0}} > v_{n})^2 (1+ {\text{o}}(1))\end{align*}

holds with the right-hand side tending to zero by (3.3). From Corollary 3.1, we obtain

(5.1) \begin{equation}{\mathbb{P}} (M_{\textbf{N}(n)} \leq v_n) = \exp (- n^d {\mathbb{P}}(X_{\mathbf{0}} > v_n, M_{A((m,m,\ldots,m))}\leq v_n) + {\text{o}}(1).\end{equation}

5.2. Second method

The second formula comes from Jakubowski and Soja-Kukieła [Reference Jakubowski and Soja-Kukieła12, Theorem 2.1]. It states that we have

(5.2) \begin{equation}{\mathbb{P}} (M_{\textbf{N} (n)} \leq v_n)=\exp\biggl( -n^d \sum_{\pmb{\varepsilon}\in\{0,1\}^d} (\!-1)^{\varepsilon_1+\varepsilon_2+\ldots+\varepsilon_d}{\mathbb{P}} ( M_{\pmb{\varepsilon},(m,m,\ldots,m)} > v_n) \biggr) + {\text{o}}(1)\end{equation}

under the above assumptions on $\{X_{\textbf{n}}\}$ . This result is a consequence of the Bonferroni-type inequality from Jakubowski and Rosiński [Reference Jakubowski and Rosiński11, Theorem 2.1].

5.3. Comparison

For $d=1$ both of the formulas simplify to the well-known result of Newell [Reference Newell16]:

\begin{equation*}{\mathbb{P}}(M_n \leq v_n) = \exp(\!-n{\mathbb{P}}(X_0> v_n, M_{1,m} \leq v_n)) + {\text{o}}(1).\end{equation*}

Each of them allows us to describe the asymptotic behaviour of maxima on the base of tail properties of joint distribution of a fixed finite dimension. To apply the first method, one uses the distribution of the $(1 + ((2m+1)^d-1)/2)$ -element family

\begin{equation*}\{X_{\textbf{n}}\colon \textbf{n}\in \{\mathbf{0}\} \cup A((m,m,\ldots,m))\}.\end{equation*}

To involve the second method, we use the distribution of the $(m+1)^d$ -element family $\{X_{\textbf{n}}\colon \mathbf{0} \leq \textbf{n} \leq (m,m,\ldots,m)\}$ .

Below, a link between the two formulas is described in two ways: a more conceptual one, and one that is shorter but perhaps less intuitive. To avoid annoying technicalities, we focus on $d=2$ .

5.3.1 First approach: counting clusters

Our aim is to calculate the number of clusters of exceedances over $v_n$ in the window $W:= \{\textbf{k}\in{\mathbb Z}^2 \colon \mathbf{1} \leq \textbf{k} \leq \textbf{N}(n)\}$ in two different ways and obtain, as a consequence, the equivalence of (5.1) and (5.2).

Let the random set $J_n$ be given as $J_n:= \{\textbf{k}\in W \colon X_{\textbf{k}}>v_n\}$ and let $\leftrightarrow$ be the equivalence relation on $J_n$ , defined as follows:

\begin{gather*} \textbf{i} \leftrightarrow \textbf{j} \ {\text{whenever there exist}\ {l\in\mathbb{N}}\ \text{and}\ {\textbf{k}_0,\textbf{k}_1, \ldots, \textbf{k}_l,\textbf{k}_{l+1}\in J_n, \textbf{k}_0=\textbf{i}, \textbf{k}_{l+1}=\textbf{j}}} \\ {\text{such that} }\ \max_{0\leq h\leq l}\|\textbf{k}_{h+1}-\textbf{k}_h\| \leq m,\end{gather*}

for $\textbf{i},\textbf{j}\in J_n$ . We define a cluster as an equivalence class of $\leftrightarrow$ and obtain the partition $\mathcal{C}_n:= J_n/ _{\leftrightarrow}$ of $J_n$ into $\Lambda_n:= \# \mathcal{C}_n$ clusters. We put

\begin{gather*}\lambda_n := {\mathbb{E}} (\Lambda_n),\quad\mathcal{C}^{\prime}_n:= \Bigl\{C\in\mathcal{C}_n \colon \max_{\textbf{i}, \textbf{j} \in C}\|\textbf{i}-\textbf{j}\| \leq m\Bigr\},\\\mathcal{C}^{\prime\prime}_{n}:= \mathcal{C}_n \backslash \mathcal{C}^{\prime}_n,\quad\Lambda^{\prime}_n:= \#\mathcal{C}^{\prime}_n,\quad\lambda^{\prime}_n:= {\mathbb{E}} (\Lambda^{\prime}_n).\end{gather*}

Let $\Lambda^{(1)}_n$ and $\lambda^{(1)}_n$ , associated with the method presented in Section 5.1, be defined as in Remark 3.2 with $\textbf{p} (n):= (m,m)$ . Recall that we have

\begin{equation*}A(m,m)=\{\textbf{j}\in{\mathbb Z}^2 \colon (\!-m,-m) \leq \textbf{j} \leq (m,m) \text{ and } (0,0)\prec \textbf{j}\}.\end{equation*}

Analogously (see [Reference Jakubowski and Soja-Kukieła12, Remark 2.2]) we define $\Lambda^{(2)}_n$ and $\lambda^{(2)}_n$ related to the method from Section 5.2 as follows:

\begin{align*}\Lambda^{(2)}_n &:= \sum_{\textbf{k} \in W}\sum_{\pmb{\varepsilon}\in\{0,1\}^2} (\!-1)^{\varepsilon_1 + \varepsilon_2} \mathbb{I}_{\{M_{\textbf{k}+\pmb{\varepsilon}, \textbf{k} + (m,m)} > v_n\}},\\ \lambda^{(2)}_n &:= {\mathbb{E}} (\Lambda^{(2)}_n) = N^*(n) \sum_{\pmb{\varepsilon}\in\{0,1\}^2} (\!-1)^{\varepsilon_1 + \varepsilon_2} {\mathbb{P}}(M_{\pmb{\varepsilon}, (m,m)} > v_n).\end{align*}

Assume that $C\in\mathcal{C}^{\prime}_n$ . Let

\begin{equation*}B(C,m):={\textbf{j}\in\mathbb{Z}^2 \colon \|\textbf{j} - \textbf{k}\| \leq m \text{ for some } \textbf{k} \in C\}\end{equation*}

and suppose we have $M_{B(C,m) \backslash C} \leq v_n$ (which is obviously satisfied in the typical case $B(C,m)\subset W$ ). Observe that for such C we have

(5.3) \begin{equation}\sum_{\textbf{k} \in C} \mathbb{I}_{\{M_{A(m,m) + \textbf{k}}\leq v_n \}} = \sum_{\textbf{k} \in C} \mathbb{I}_{\{ \textbf{k} \text{ is the largest element of \textit{C} with respect to } \preccurlyeq\}} = 1\end{equation}

and

(5.4) \begin{equation}\sum_{\textbf{k} \in \bar{C}} \sum_{\pmb{\varepsilon}\in\{0,1\}^2} (\!-1)^{\varepsilon_1 + \varepsilon_2} \mathbb{I}_{\{M_{\textbf{k}+\pmb{\varepsilon},\textbf{k}+(m,m)} > v_n \}}= \mathbb{I}_{\{M_{\textbf{k}(C), \textbf{k}(C)+(m,m)} > v_n\}} = 1,\end{equation}

where

\begin{equation*}\bar{C}:= \{\textbf{k} \in {\mathbb Z}^2 \colon \textbf{k} + \textbf{i} \in C \text{ for some } \mathbf{0} \leq \textbf{i} \leq (m,m)\}\end{equation*}

and $\textbf{k}(C)$ satisfies the condition

\begin{equation*}C\subset \{ \textbf{k} \in {\mathbb Z}^2 \colon \textbf{k}(C) \leq \textbf{k} \leq \textbf{k}(C) + (m,m)\}.\end{equation*}

We will show that each of the following equalities holds:

(5.5) \begin{align}\delta_n &:= {\mathbb{E}}|\Lambda_n - \Lambda^{\prime}_n| = {\mathbb{E}}(\Lambda_n - \Lambda^{\prime}_n) = {\text{o}}(1), \end{align}

(5.6) \begin{align}\delta^{(1)}_n &:= {\mathbb{E}}|\Lambda^{(1)}_n - \Lambda^{\prime}_n| = {\text{o}}(1), \end{align}

(5.7) \begin{align}\delta^{(2)}_n &:= {\mathbb{E}}|\Lambda^{(2)}_n - \Lambda^{\prime}_n| = {\text{o}}(1). \end{align}

This will entail the condition $\lambda_n=\lambda^{\prime}_n + {\text{o}}(1) = \lambda^{(1)}_n +{\text{o}}(1) = \lambda^{(2)}_n +{\text{o}}(1)$ and complete this section.

To show (5.5), observe that the event $\{\# \mathcal{C}^{\prime\prime}_n = l\}$ , for $l\in\mathbb{N}_+$ , implies that there exist pairs $\textbf{j}(i,a), \textbf{j}(i,b) \in J_n$ , for $i\in\{1,2,\ldots,l\}$ , such that $m < \|\textbf{j}(i,a) - \textbf{j}(i,b)\|\leq 2m$ holds for each i and $\|\textbf{j}(i_1,c_1)-\textbf{j}(i_2,c_2)\| > m$ is satisfied for $i_1\neq i_2$ and $c_1, c_2\in\{a,b\}$ . Thus we have

\begin{equation*} \delta_n = \sum_{l=1}^{\infty} l{\mathbb{P}}(\# \mathcal{C}^{\prime\prime}_n = l) \leq \sum_{l=1}^{\infty} l (N^*(n) ((4m+1)^2-(2m+1)^2) {\mathbb{P}}(X_\mathbf{0} > v_n)^2) ^l.\end{equation*}

Since

\begin{equation*} q_n:= N^*(n)((4m+1)^2-(2m+1)^2) {\mathbb{P}}(X_\mathbf{0} > v_n)^2={\text{o}}(1) \end{equation*}

by (3.3), we obtain

\begin{equation*}\delta_n \leq \sum_{l=1}^{\infty} l(q_n)^{l} = q_n (1-q_n)^{-2}\quad\text{for all large {\textit{n}}}\end{equation*}

and finally $\delta_n={\text{o}}(1)$ .

Before we establish (5.6) and (5.7), we will give an upper bound for the probability that a fixed $\textbf{k}\in W$ belongs to a large cluster. Note that we have

(5.8) \begin{align}& {\mathbb{P}} (\textbf{k} \in C \text{ for some } C\in\mathcal{C}_n^{\prime\prime}) \notag\\[6pt] & \qquad = {\mathbb{P}} (\textbf{k} \in C \text{ and } \|\textbf{j}-\textbf{k}\|\leq m \text{ for all }\textbf{j}\in C, \text{ for some } C\in\mathcal{C}_n^{\prime\prime}) \notag\\[6pt] & \qquad\quad\, + {\mathbb{P}} (\textbf{k} \in C \text{ and } \|\textbf{j}-\textbf{k}\|> m \text{ for some }\textbf{j}\in C, \text{ for some } C\in\mathcal{C}_n^{\prime\prime}) \notag \\[6pt] & \qquad \leq {\mathbb{P}} (\|\textbf{i} - \textbf{k}\| \leq m \text{ and } \|\textbf{j} - \textbf{k}\|\leq m \text{ and } \|\textbf{i} - \textbf{j}\|>m, \text{ for some }\textbf{i},\textbf{j}\in J_n) \notag\\[6pt] &\qquad\quad\, + {\mathbb{P}} (\textbf{k} \in J_n \text{ and } m<\|\textbf{k}-\textbf{j}\|\leq 2m \text{ for some }\textbf{j} \in J_n) \notag\\[6pt] &\qquad \leq ((2m+1)^4 + ((4m+1)^2 - (2m+1)^2)) {\mathbb{P}}(X_{\mathbf{0}} > v_n)^2 = a(m){\mathbb{P}}(X_{\mathbf{0}} > v_n)^2\end{align}

with $a(m):= (2m+1)^4 + 4m(3m+1)$ .

Applying observation (5.3), property (5.8) and taking into account estimation errors for clusters situated near the edges of the window W, we obtain

\begin{align*}\delta_n^{(1)} & = {\mathbb{E}} \biggl|\sum_{\textbf{k} \in W} \bigl(\mathbb{I}_{\{\textbf{k} \in \bigcup \mathcal{C}^{\prime}_n, M_{\textbf{k} + A(m,m)}\leq v_n \}}+ \mathbb{I}_{\{\textbf{k} \in \bigcup \mathcal{C}^{\prime\prime}_n , M_{\textbf{k} + A(m,m)}\leq v_n\}}\bigr) - \Lambda^{\prime}_n \biggr|\\& \leq {\mathbb{E}}\biggl|\sum_{C\in\mathcal{C}^{\prime}_n} \sum_{\textbf{k}\in C} \mathbb{I}_{\{M_{\textbf{k} + A(m,m)}\leq v_n\}} - \Lambda^{\prime}_n\biggr|+ {\mathbb{E}} \biggl(\sum_{\textbf{k} \in W} \mathbb{I}_{\{\textbf{k} \in \bigcup \mathcal{C}^{\prime\prime}_n \}}\biggr)\\& \leq 2m(N_1(n)+N_2(n)) {\mathbb{P}}(X_{\mathbf{0}} > v_n)+ a(m)N^*(n){\mathbb{P}}(X_{\mathbf{0}} > v_n)^2,\end{align*}

which combined with assumption (3.3) implies (5.6). Quite similarly, using (5.4) instead of (5.3) and writing $\bar{W}:= \{\textbf{k}\in {\mathbb Z}^2 \colon \mathbf{1}-(m,m) \leq \textbf{k} \leq \textbf{N}(n)\}$ , we conclude that

\begin{align*}\delta_n^{(2)}& \leq {\mathbb{E}} \biggl| \sum_{\textbf{k}\in \bar{W} } \mathbb{I}_{\{\textbf{k}\in\bar{C} \text{ for some }C\in\mathcal{C}^{\prime}_n \}} \sum_{\pmb{\varepsilon}\in\{0,1\}^2} (\!-1)^{\varepsilon_1 + \varepsilon_2} \mathbb{I}_{\{M_{\textbf{k}+\pmb{\varepsilon}, \textbf{k} + (m,m)} > v_n\}} - \Lambda^{\prime}_n\biggr| \\& \quad\, + {\mathbb{E}} \biggl(\sum_{\textbf{k}\in\bar{W}\backslash W} \sum_{\pmb{\varepsilon}\in\{0,1\}^2} \bigl| (\!-1)^{\varepsilon_1 + \varepsilon_2} \mathbb{I}_{\{M_{\textbf{k}+\pmb{\varepsilon}, \textbf{k} + (m,m)} > v_n\}}\bigr|\biggr) \\& \quad\, + 2{\mathbb{E}} \biggl(\sum_{\textbf{k}\in W} \mathbb{I}_{\{\textbf{k}\in\bar{C} \text{ for some }C\in\mathcal{C}^{\prime\prime}_n \}} \biggr)\\& \leq {\mathbb{E}}\biggl| \sum_{C\in \mathcal{C}^{\prime}_n} \sum_{\textbf{k}\in \bar{C}} \sum_{\pmb{\varepsilon}\in\{0,1\}^2} (\!-1)^{\varepsilon_1 + \varepsilon_2} \mathbb{I}_{\{M_{\textbf{k}+\pmb{\varepsilon}, \textbf{k} + (m,m)} > v_n\}} - \Lambda^{\prime}_n \biggr|\\& \quad\, + \sum_{\textbf{k}\in\bar{W}\backslash W} (2m+1){\mathbb{P}}(X_{\mathbf{0}} > v_n) + 2N^*(n){\mathbb{P}} (\textbf{k}\in\bar{C} \text{ for some }C\in\mathcal{C}^{\prime\prime}_n)\\& \leq 2m(2m+1)(N_1(n)+N_2(n)){\mathbb{P}}(X_\mathbf{0}>v_n)\\& \quad\, + m(2m + 1)(N_1(n) + N_2(n) + m){\mathbb{P}}(X_\mathbf{0} > v_n) + 2(m + 1)^2a(m)N^*(n){\mathbb{P}}(X_\mathbf{0} > v_n)^2.\end{align*}

Since the right-hand side tends to zero by (3.3), property (5.7) follows.

5.3.2. Second approach: direct verification

In this part we assume that $\preccurlyeq$ is the lexicographic order on ${\mathbb Z}^2$ . Let us note that

\begin{align*}& {\mathbb{P}} (M_{{(0,0)},(m,m)} > v_n) - {\mathbb{P}} (M_{(1,0),(m,m)} > v_n) - {\mathbb{P}} (M_{(0,1),(m,m)} > v_n) + {\mathbb{P}} (M_{(1,1),(m,m)} > v_n) \\[6pt] & \qquad = {\mathbb{P}} (X_{(0,0)} > v_n, M_{R((m,m))} \leq v_n) \\[6pt] & \qquad\quad\, - {\mathbb{P}} (M_{(0,1), (0,m){}} > v_n, M_{(1,0),(m,0)} >v_n, M_{(1,1), (m,m)} \leq v_n)\end{align*}

is true with $R((p_1,p_2)):= A((p_1,p_2))\cap {\mathbb N}^2$ , where on the left-hand side of the equation the sum of probabilities from (5.2) for $d=2$ appears. Next, let us look at the exponent in (5.1) and observe that

\begin{align*}& {\mathbb{P}} (X_{(0,0)} > v_n,M_{A((m,m))} \leq v_n) \\& \qquad = {\mathbb{P}} (X_{(0,0)} > v_n, M_{R((m,m))} \leq v_n) \\& \qquad\quad\, - {\mathbb{P}} (X_{(0,0)} > v_n, M_{R((m,m))} \leq v_n, M_{(1,-m),(m,-1)} > v_n)\end{align*}

and, moreover, the second summand of the right-hand side satisfies

\begin{align*}& {\mathbb{P}} (X_{(0,0)} > v_n, M_{R((m,m))} \leq v_n, M_{(1,-m),(m,-1)} > v_n) \\& \qquad = \sum_{l=1}^{m} {\mathbb{P}} (X_{(0,0)} > v_n, M_{R((m,m))} \leq v_n, M_{(1,-l),(m,-l)} > v_n, M_{(1,-l+1), (m,-1)} \leq v_n) \\& \qquad = \sum_{l=1}^{m} {\mathbb{P}} (X_{(0,0)} >\! v_n, M_{R((m,m-l))} \leq v_n, M_{(1,-l),(m,-l)} > v_n, M_{(1,-l+1), (m,-1)} \leq v_n) + {\text{o}}(n^{-2}) \\& \qquad = \sum_{l=1}^{m} {\mathbb{P}} (X_{(0,l)} > v_n, M_{(0,l) + R((m,m-l))} \leq v_n, M_{(1,0),(m,0)} > v_n, M_{(1,1), (m,l-1)} \leq v_n) + {\text{o}}(n^{-2}) \\& \qquad = {\mathbb{P}} (M_{(0,1), (0,m)} > v_n, M_{(1,0),(m,0)} >v_n, M_{(1,1), (m,m)} \leq v_n ) + {\text{o}}(n^{-2}) .\end{align*}

In the above statement, probabilities of mutually exclusive events are summed up, and m-dependence, assumption (3.3), and stationarity are applied. Finally, we obtain

\begin{align*}& {\mathbb{P}} (M_{{(0,0)},(m,m)} > v_n) - {\mathbb{P}} (M_{(1,0),(m,m)} > v_n) - {\mathbb{P}} (M_{(0,1),(m,m)} > v_n) + {\mathbb{P}} (M_{(1,1),(m,m)} > v_n) \\& \qquad = {\mathbb{P}} (X_{(0,0)} > v_n,M_{A((m,m))} \leq v_n) + {\text{o}}(n^{-2}) .\end{align*}

Summarizing, we have confirmed that both presented methods lead to the same result.

Remark 5.1. The above reasoning for m-dependent fields can also be applied in the general setting. Suppose that formula (1.3), with $\preccurlyeq$ the lexicographic order on ${\mathbb Z}^2$ , describes the asymptotics of partial maxima of the stationary field $\{X_\textbf{n} \colon \textbf{n}\in{\mathbb Z}^2\}$ . Then

(5.9) \begin{equation}{\mathbb{P}} (M_{\textbf{N} (n)} \leq v_n)=\exp\biggl( -n^2 \sum_{\pmb{\varepsilon}\in\{0,1\}^2} (\!-1)^{\varepsilon_1+\varepsilon_2}{\mathbb{P}} ( M_{\pmb{\varepsilon},\textbf{p}(n)} > v_n) \biggr) + {\text{o}}(1)\end{equation}

holds if and only if $\{X_\textbf{n}\}$ satisfies the following condition:

(5.10) \begin{equation}\sum_{l=1}^{p_2(n)} {\mathbb{P}} (X_{\mathbf{0}} > v_n, M_{U_l(\textbf{p} (n))} > v_n, M_{V_l(\textbf{p} (n))} > v_n, M_{W_l(\textbf{p} (n))} \leq v_n) = {\text{o}}(n^{-2}),\end{equation}

where

\begin{align*}U_l(\textbf{p}) & := \{0,\ldots, p_1\} \times \{p_2-l+1, \ldots, p_2\},\\[3pt] V_l(\textbf{p}) & := \{1,\ldots, p_1\} \times \{-l\},\\[3pt] W_l(\textbf{p}) & := A(\textbf{p}) \cap ({\mathbb Z}\times \{-l+1,\ldots, p_2-l\}).\end{align*}

Formula (5.9) generalizes (5.2). In the present section we have used the fact that m-dependent fields satisfy (5.10) with $\textbf{p}(n):= (m,m)$ .

6. Example: moving maxima

Below, we use the results from Sections 3 and 4 to describe the asymptotics of partial maxima for the moving maximum field. We note that approaches to the problem using different methods can be found in Basrak and Tafro [Reference Basrak and Tafro3] or Jakubowski and Soja-Kukieła [Reference Jakubowski and Soja-Kukieła12]. In the first paper compound Poisson point process approximation is applied, while in the second paper the authors combine a Bonferroni-like inequality and max-m-approximability.

In the following, $\{Z_\textbf{n}\}$ is an array of independent, identically distributed random variables satisfying

\begin{equation*} {\mathbb{P}}(|Z_\mathbf{0}| > x) = x^{-\alpha} L(x)\end{equation*}

for some index $\alpha >0$ and slowly varying function L, and

\begin{equation*} \dfrac{{\mathbb{P}}(Z_\mathbf{0} > x)}{{\mathbb{P}}(|Z_\mathbf{0}| > x)}=p\quad\text{as}\ x\to\infty \quad\text{for some}\ p\in[0,1].\end{equation*}

We define

\begin{equation*}a_n:= \inf\{y>0\colon {\mathbb{P}}(|Z_\mathbf{0}| > y) \leq n^{-d}\}\end{equation*}

and $v_n:= a_n v$ with fixed $v>0$ . Then

\begin{equation*}n^d {\mathbb{P}}(|Z_\mathbf{0}| > v_n) \to v^{-\alpha}\quad\text{as}\ n\to\infty.\end{equation*}

Let us consider the moving maximum field $\{X_\textbf{n}\}$ defined as

\begin{equation*}X_\textbf{n}=\sup_{\textbf{j}\in{\mathbb Z} ^d}c_\textbf{j} Z_{\textbf{n}+\textbf{j}},\end{equation*}

where $c_\textbf{j}\in{\mathbb R}$ , not all equal to zero, satisfy

(6.1) \begin{equation}\sum_{\textbf{j}\in{\mathbb Z} ^d} |c_\textbf{j}|^\beta <\infty \quad \text{for some} \ 0<\beta < \alpha.\end{equation}

From Cline [Reference Cline7, Lemma 2.2] it follows that the field $\{X_\textbf{n}\}$ is well-defined and

(6.2) \begin{align}\lim_{x\to\infty}\dfrac{{\mathbb{P}}(X_\mathbf{0}>x)}{{\mathbb{P}}(|Z_\mathbf{0}|>x)}& = \lim_{x\to\infty}\dfrac{{\mathbb{P}} (\sup_{\textbf{j}\in{\mathbb Z} ^d}c_\textbf{j} Z_{\textbf{j}}>x) }{{\mathbb{P}}(|Z_\mathbf{0}|>x)} \notag \\& = \lim_{x\to\infty}\dfrac{\sum_{\textbf{j}\in{\mathbb Z}^d} {\mathbb{P}}(c_\textbf{j} Z_{\textbf{j}}>x)}{{\mathbb{P}}(|Z_\mathbf{0}|>x)} \notag \\& = p \sum_{c_\textbf{j} > 0} c_\textbf{j} ^\alpha+ q \sum_{c_\textbf{j} < 0} |c_\textbf{j} |^\alpha{,}\end{align}

with $q:= 1-p$ .

Since the moving maximum field is $\max$ -m-approximable, there exists a phantom distribution function for $\{X_\textbf{n}\}$ (see Jakubowski and Soja-Kukieła [Reference Jakubowski and Soja-Kukieła12]) and hence the field is weakly dependent in the sense of (2.2). We will apply Theorem 3.1, with $\preccurlyeq$ being the lexicographic order on ${\mathbb Z}^d$ , to describe the asymptotics of partial maxima. Let us observe that the exponent in (3.2) satisfies

\begin{align*}& n^d {\mathbb{P}} (X_{\mathbf{0}} > v_n, M_{A(\textbf{p}(n))}\leq v_n) \\& \qquad = n^d{\mathbb{P}}\biggl(\bigcup_{\textbf{j}\in{\mathbb Z}^d} \{c_\textbf{j} Z_\textbf{j} > v_n \}, \bigcap_{\textbf{k}\in{\mathbb Z}^d} \Bigl\{\max_{\textbf{i}\in A(\textbf{p}(n))} (c_{\textbf{k} - \textbf{i}} Z_\textbf{k}) \leq v_n\Bigr\} \biggr)\\& \qquad = n^d \sum_{\textbf{j} \in {\mathbb Z}^d} {\mathbb{P}}\biggl( c_\textbf{j} Z_\textbf{j} > v_n , \bigcap_{\textbf{k}\in{\mathbb Z}^d} \Bigl\{\max_{\textbf{i}\in A(\textbf{p}(n))} (c_{\textbf{k} - \textbf{i}} Z_\textbf{k}) \leq v_n\Bigr\}\biggr) + {\text{o}}(1)\\& \qquad = n^d \sum_{\textbf{j} \in {\mathbb Z}^d} {\mathbb{P}}\biggl( c_\textbf{j} Z_\textbf{j} > v_n \geq \max_{\textbf{i}\in A(\textbf{p}(n))} (c_{\textbf{j} - \textbf{i}} Z_\textbf{j}), \bigcap_{\textbf{k} \neq \textbf{j}} \Bigl\{\max_{\textbf{i}\in A(\textbf{p}(n))} (c_{\textbf{k} - \textbf{i}} Z_\textbf{k}) \leq v_n\Bigr\}\biggr) + {\text{o}}(1)\\& \qquad = n^d \sum_{\textbf{j} \in {\mathbb Z}^d} {\mathbb{P}} \Bigl( c_\textbf{j} Z_\textbf{j} > v_n \geq \max_{\textbf{i}\in A(\textbf{p}(n))} (c_{\textbf{j} - \textbf{i}} Z_\textbf{j})\Bigr) {\mathbb{P}} \biggl(\bigcap_{\textbf{k} \neq \textbf{j}} \Bigl\{\max_{\textbf{i}\in A(\textbf{p}(n))} (c_{\textbf{k} - \textbf{i}} Z_\textbf{k}) \leq v_n\Bigr\}\biggr) + {\text{o}}(1),\end{align*}

as $n\to\infty$ , where the second equality follows from (6.2) combined with the choice of $\{v_n\}$ and the last equality is a consequence of the independence of $Z_\textbf{j}$ for $\textbf{j}\in{\mathbb Z}^d$ . Note that we have

\begin{align*}{\mathbb{P}}\biggl(\bigcap_{\textbf{k} \neq \textbf{j}} \Bigl\{\max_{\textbf{i}\in A(\textbf{p}(n))} (c_{\textbf{k} - \textbf{i}} Z_\textbf{k}) \leq v_n\Bigr\}\biggr) & \geq {\mathbb{P}}\biggl(\bigcap_{\textbf{k} \in {\mathbb Z}^d} \Bigl\{\max_{\textbf{i}\in A(\textbf{p}(n))} (c_{\textbf{k} - \textbf{i}} Z_\textbf{k}) \leq v_n\Bigr\}\biggr) \\& \geq {\mathbb{P}} (M_{A(\textbf{p}(n))}\leq v_n) \\& \geq 1- {\text{o}}(n^d){\mathbb{P}}(X_{\mathbf{0}} >v_n) \\& = 1 + {\text{o}}(1).\end{align*}

Moreover, for $p_{\min}(n):= \min \{p_l(n) \colon 1\leq l\leq d\}$ and for $q(n) \in {\mathbb N}$ chosen so that

\begin{equation*}q(n)\to\infty,\quad q(n) \leq p_{\min}(n)/2\quad\text{and}\quad q(n)^d n^d {\mathbb{P}}(\max\{ c_{\textbf{i}}Z_{\mathbf{0}} \colon \|\textbf{i}\| >p_{\min}(n)/2 \} > v_n) \to 0,\end{equation*}

it follows that

\begin{align*}& \biggl| n^d \sum_{\textbf{j} \in {\mathbb Z}^d} {\mathbb{P}}\Bigl( c_\textbf{j} Z_\textbf{j} > v_n \geq \max_{\textbf{i}\in A(\textbf{p}(n))} (c_{\textbf{j} - \textbf{i}} Z_\textbf{j})\Bigr)- n^d \sum_{\textbf{j}\in{\mathbb Z}^d} {\mathbb{P}}\Bigl( c_\textbf{j} Z_\textbf{j} > v_n \geq \sup_{\mathbf{0} \prec \textbf{i}} (c_{\textbf{j} - \textbf{i}} Z_\textbf{j})\Bigr) \biggr| \\& \qquad \leq n^d \sum_{\textbf{j} \in {\mathbb Z}^d} {\mathbb{P}}\Bigl( c_\textbf{j} Z_\textbf{j} > v_n, \sup_{\mathbf{0} \prec \textbf{i} \notin A(\textbf{p}(n))} (c_{\textbf{j} - \textbf{i}} Z_\textbf{j}) > v_n \Bigr)\\& \qquad \leq n^d \sum_{\textbf{j} \in {\mathbb Z}^d} {\mathbb{P}}\Bigl( c_\textbf{j} Z_\textbf{j} > v_n, \sup_{\|\textbf{i} \| > p_{\min}(n)} (c_{\textbf{j} - \textbf{i}} Z_\textbf{j}) > v_n \Bigr)\\& \qquad \leq n^d\sum_{\|\textbf{j}\| \leq q(n)} {\mathbb{P}}\Bigl(\sup_{\|\textbf{i} \| > p_{\min}(n)} (c_{\textbf{j} - \textbf{i}} Z_\textbf{j}) > v_n \Bigr) + n^d \sum_{\|\textbf{j}\| > q(n)} {\mathbb{P}}( c_\textbf{j} Z_\textbf{j} > v_n)\\& \qquad \leq {n^d}(2 q(n)+1)^d {\mathbb{P}}\Bigl(\sup_{\|\textbf{i} \| > p_{\min}(n)/2} (c_{\textbf{i}} Z_{\mathbf{0}}) > v_n \Bigr) + n^d \sum_{\|\textbf{j}\| > q(n)} {\mathbb{P}}( c_\textbf{j} Z_\textbf{j} > v_n){.}\end{align*}

The first summand on the right-hand side tends to zero due to the choice of q(n) and the second summand tends to zero by properties (6.1), (6.2), and the definition of $v_n$ . We conclude that

\begin{align*}& n^d {\mathbb{P}} (X_{\mathbf{0}} > v_n, M_{A(\textbf{p}(n))}\leq v_n) \\&\qquad = \biggl(n^d \sum_{\textbf{j} \in {\mathbb Z}^d} {\mathbb{P}}\Bigl( c_\textbf{j} Z_\textbf{j} > v_n \geq \sup_{\mathbf{0} \prec \textbf{i}} (c_{\textbf{j} - \textbf{i}} Z_\textbf{j})\Bigr) + {\text{o}}(1)\biggr) (1+ {\text{o}}(1)) + {\text{o}}(1).\end{align*}

To complete the above calculation, it is sufficient to observe that

\begin{align*}n^d \sum_{\textbf{j} \in {\mathbb Z}^d} {\mathbb{P}}\Bigl( c_\textbf{j} Z_\textbf{j} > v_n \geq \sup_{\mathbf{0} \prec \textbf{i}} (c_{\textbf{j} - \textbf{i}} Z_\textbf{j})\Bigr)& = n^d \sum_{\textbf{j} \in {\mathbb Z}^d} {\mathbb{P}}\Bigl( c_\textbf{j} Z_{\mathbf{0}} > v_n \geq \sup_{\textbf{i} \prec \textbf{j}} (c_{\textbf{i}} Z_{\mathbf{0}})\Bigr)\\& = n^d {\mathbb{P}}\Bigl(\sup_{\textbf{j} \in {\mathbb Z}^d} (c_\textbf{j} Z_{\mathbf{0}}) > v_n\Bigr) \\& \to (p(c^+)^{\alpha} + q(c^-)^{\alpha})v^{-\alpha},\end{align*}

with $c^+:= \max_{\textbf{i} \in{\mathbb Z} ^d}\max\{c_\textbf{i},0\}$ and $c^-:= \max_{\textbf{i} \in{\mathbb Z} ^d} \max\{-c_\textbf{i},0\}$ . By (3.2) we obtain

\begin{equation*} {\mathbb{P}} (M_{\textbf{N}(n)} \leq v_n) \to \exp(\!-(p(c^+)^{\alpha} + q(c^-)^{\alpha})v^{-\alpha}) \quad \text{as} \ n\to\infty. \end{equation*}

Applying formula (4.2) and property (6.2), we calculate the extremal index of $\{X_\textbf{n}\}$ as follows:

\begin{equation*}\theta=\dfrac{p(c^+)^\alpha + q(c^-)^\alpha}{ p\sum_{c_\textbf{j} > 0}c_\textbf{j} ^\alpha+ q \sum_{c_\textbf{j} < 0}|c_\textbf{j} |^\alpha},\end{equation*}

whenever the denominator is positive, which is the only interesting case.