# Extremes of Lévy-driven spatial random fields with regularly varying Lévy
measure

Anders Rønn-Nielsen<EMAIL_ADDRESS>Mads Stehr<EMAIL_ADDRESS>
###### Abstract

We consider an infinitely divisible random field indexed by $\mathbb{R}^{d}$,
$d\in\mathbb{N}$, given as an integral of a kernel function with respect to a
Lévy basis with a Lévy measure having a regularly varying right tail. First we
show that the tail of its supremum over any bounded set is asymptotically
equivalent to the right tail of the Lévy measure times the integral of the
kernel. Secondly, when observing the field over an appropriately increasing
sequence of continuous index sets, we obtain an extreme value theorem stating
that the running supremum converges in distribution to the Fréchet
distribution.

###### keywords:

Extreme value theory , Lévy-based modeling , regular variation , geometric
probability , random fields

###### MSC:

[2010] Primary 60G70 , 60G60; Secondary 60E07 , 60D05

††journal: Stochastic Processes and Their Applications

[cbs] organization=Department of Finance, Copenhagen Business
School,addressline=Solbjerg Plads 3, postcode=2000, city=Frederiksberg,
country=Denmark

Declarations of interest: None.
Funding: This research did not receive any specific grant from funding
agencies in the public, commercial, or not-for-profit sectors.
This work has been submitted to _Stochastic Processes and their Applications_.

##  1 Introduction

In this paper we consider the extremal behavior of a spatial Lévy-driven
moving average given by

$X_{v}=\int_{\mathbb{R}^{d}}f(v-u)\Lambda(\mathrm{d}u),\qquad
v\in{\mathbb{R}^{d}},$ (1)

where $\Lambda$ is an infinitely divisible, independently scattered random
measure on ${\mathbb{R}^{d}}$ (i.e. a Lévy basis) and $f$ is an appropriate
kernel function; see [17, Theorem 2.7] for necessary and sufficient conditions
guaranteeing its existence. Lévy-driven models as in (1) form a very general
modeling framework used for a wide range of purposes, including modeling of
financial assets ([4]), turbulent flows ([3]), brain imaging data ([13]) and
wind power prices ([5]). Estimators for its mean and variogram are suggested
in [20], and central limit theorems for these estimators are also presented.

In this paper we assume that the Lévy measure $\rho$ of the basis $\Lambda$
has a regularly varying right tail; see for instance [10, Appendix A.3]. The
regularly varying distributions are in particular subexponential, and the
class covers many interesting heavy-tailed distributions such as the Pareto,
Cauchy, Loggamma, stable (of index $<2$) and, in particular, Fréchet
distribution. Moreover (and essential to this paper), the set of regularly
varying distributions coincide with the maximum domain of attraction of the
Fréchet distribution; see [10, Chapter 3].

In addition to assuming integrability of $f$ and regular variation of the
right tail of $\rho$, we consider two different assumptions on the pair
$(\Lambda,f)$. In Assumption 1 we have minimal requirements on $\rho$ but we
consider a Hölder continuous kernel $f$. In Assumption 2 we have minimal
requirements on $f$, whereas we restrict consideration to Lévy bases of finite
variation, meaning in particular that the Lévy measure $\rho$ has finite first
moment in a neighborhood of $0$.

The first part of the paper concerns the asymptotic representation of the tail
of $\sup_{v\in B}X_{v}$ for some fixed and bounded index set
$B\subseteq{\mathbb{R}^{d}}$. With $\alpha>0$ denoting the regularly varying
index of $\rho$ we show that

$\mathbb{P}\bigl{(}\sup_{v\in
B}X_{v}>x\bigr{)}\sim\rho((x,\infty))\int_{\mathbb{R}^{d}}\sup_{v\in
B}f^{\alpha}(v-u)\mathrm{d}u$ (2)

as $x\to\infty$, where $\sim$ denotes asymptotic equivalence. In fact we show
the more general result that the equivalence holds true even if we replace $X$
with $X+Y^{1}$, where $Y^{1}$ is an independent field with right tail lighter
than $X$. To obtain the representation (2) we use a result from [22], where we
explicitly use that the right tail of $\rho$ is subexponential.

Results similar to (2) are found in the literature for one-dimensional
regularly varying processes, and for spatial fields with lighter tails: In
[11] a field

$X_{t}=\int_{\mathbb{R}_{+}\times\mathbb{R}}f(r,t-s)\Lambda(\mathrm{d}r,\mathrm{d}s)$
(3)

is studied under the assumption of an underlying regularly varying Lévy
measure. Replacing ${\mathbb{R}^{d}}$ with $\mathbb{R}$ and $B$ with the fixed
interval $[0,h]$, the claim (2) is shown for the one-dimensional moving
average. If, instead, the spatial Lévy-driven field $X_{v}$ has a convolution
equivalent Lévy measure $\rho$ of strictly positive index ([7, 8, 15]), it is
shown in [19] that

$\mathbb{P}\bigl{(}\sup_{v\in B}X_{v}>x\bigr{)}\sim\rho((x,\infty))K\lvert
B\rvert,$

where $\lvert B\rvert$ is Lebesgue measure of $B\subseteq\mathbb{R}^{d}$ and
$K$ is a computable constant. A slightly more general result is given in [28],
in which a space-time Lévy model $X_{v,t}$ with a convolution equivalent Lévy
measure is considered. For a large class of functionals $\Psi$ acting on the
field, it is shown that there are constants $c,C$ such that

$\mathbb{P}\bigl{(}\Psi(X_{v,t})>x\bigr{)}\sim C\rho((x/c,\infty))$

as $x\to\infty$. For Gaussian random fields the distribution of the supremum
can be approximated by the expected Euler characteristic of an excursion set
(see [1] and the references therein).

The second part of the paper concerns the asymptotic distribution of
$\sup_{v\in C_{n}}X_{v}$ as $n\to\infty$, where $(C_{n})$ is a sequence of
index sets in ${\mathbb{R}^{d}}$ increasing appropriately. From extreme value
theory for dependent stationary fields we know, assuming some mixing and anti-
clustering conditions, that the distribution of the running maximum of a
stationary field is determined by its marginal tail; see [10, 14, 18] for
detailed treatments of classical extreme value theory, and see [12, 26, 27]
for generalizations to stationary, discretely indexed $d$-dimensional spatial
fields. In particular, if the marginal tail is in the maximum domain of
attraction of the Fréchet distribution (or equivalently it is regularly
varying), then the running maximum of the field converges to the Fréchet
distribution. Thus, with (2) in mind, we expect (and show) that the
distribution of $\sup_{v\in C_{n}}X_{v}$ converges to the Fréchet distribution
if the index sets $C_{n}$ behave nicely: We require that $C_{n}$ is a union of
a fixed number of connected convex bodies (convex, compact sets with non-empty
interior) with intrinsic volumes sufficiently bounded relative to the volume
of $C_{n}$; see [25, Chapter 4] for an exposition of convex bodies and their
intrinsic volumes. This includes the useful situation where a union of fixed
convex bodies is scaled by an increasing real sequence. With $\alpha>0$
denoting the regularly varying index of $\rho$ we then show that there are
norming constants $(a_{n})$ such that

$\mathbb{P}\bigl{(}a_{n}^{-1}\sup_{v\in
C_{n}}X_{v}>x\bigr{)}\to\exp\bigl{(}-x^{-\alpha}\rho((1,\infty))\bigr{)}$ (4)

as $n\to\infty$. The result does not follow by the aforementioned papers on
discretely indexed fields, as the appropriate mixing and anti-clustering
conditions do not easily show for our continuously indexed field. Instead, the
proof is based on a conditioning argument, where we apply an independent
decomposition $X=Z+Y$ and condition on $Y=y$. Here $Z$ is a compound Poisson
sum, which determines the supremum of $X$, and $Y$ represents the light-tailed
and high-activity part of $X$. The limit (4) is then obtained by first
establishing the conditional result for $Z+y$ and then applying ergodic
properties of the light-tailed field $Y$. This proof technique almost
immediately implies the extended result that (4) is also satisfied with $X$
replaced by $X+Y^{1}+Y^{2}$, where the stationary and independent fields
$Y^{1}$ and $Y^{2}$ satisfy that $Y^{1}$ is ergodic and that $X$ has heavier
right tails than $Y^{1}$ and $\lvert Y^{2}\rvert$.

Extremal results related to the one in the present paper are found in the
literature in various forms. In [11] the running supremum
$\sup_{t\in[0,T]}X_{t}$ of the one-dimensional moving average (3) is shown to
converge to the Fréchet distribution as $T\to\infty$. A slightly less general
result is given in [21], in which the underlying Lévy measure is assumed to be
stable with index $<2$.

In [27] a $d$-dimensional Lévy-driven field with a convolution equivalent Lévy
measure is considered under an asymptotic regime that, regarding the sequence
of index sets, is identical to the present. Convolution equivalent
distributions are in particular in the maximum domain of attraction of the
Gumbel distribution, and thus, in this case, the running supremum $\sup_{v\in
C_{n}}X_{v}$ converges to a power of the Gumbel distribution function
$x\mapsto\exp(\mathrm{e}^{-x})$. The proof structure in [27] has some
similarities with the proofs in the present paper, as both rely on
conditioning on the light-tailed and heavy-activity part of the field.
However, the obvious differences in both tail behavior and limits make the
proofs substantially different.

The paper is organized as follows. In Section 2 we formally define our Lévy-
driven field and provide assumptions on the basis, integration kernel and
increasing index sets, before presenting the main results of the paper. In
Section 3 we give some useful results for regularly varying distributions,
which are then used to show the tail representation of the supremum of the
field in Section 4. Section 5 is devoted to geometric proofs related to the
expansion of the index sets $(C_{n})$, and proofs for the result on the
running supremum $\sup_{v\in C_{n}}X_{v}$ are found in Section 6.

##  2 Definitions and main results

In this section we formally define our random field and state sufficient
assumptions after which we can present the main results. Before doing so, we
briefly clarify some notation used throughout the paper. We let
$\lvert{}\cdot{}\rvert$ denote size in the following sense: $\lvert v\rvert$
is the Euclidean norm for a single (one- or multi-dimensional) point $v$,
$\lvert A\rvert$ is Lebesgue measure of a full-dimensional set
$A\subseteq{\mathbb{R}^{d}}$, and $\lvert A\rvert$ is the number of points in
a discrete set $A\subseteq{\mathbb{Z}^{d}}$. However, we will at times also
use the notation $m$ for Lebesgue measure.

We consider a stationary Lévy-driven random field
$(X_{v})_{v\in{\mathbb{R}^{d}}}$ given as a integral of a kernel function with
respect to a Lévy basis. A Lévy basis is an infinitely divisible and
independently scattered random measure. The random measure $\Lambda$ on
${\mathbb{R}^{d}}$ is independently scattered if for all disjoint Borel sets
$(A_{n})_{n\in\mathbb{N}}\subseteq{\mathbb{R}^{d}}$ the random variables
$(\Lambda(A_{n}))_{n\in\mathbb{N}}$ are independent and furthermore satisfy
$\Lambda(\cup_{n\in\mathbb{N}}A_{n})=\sum_{n\in\mathbb{N}}\Lambda(A_{n})$. The
random measure $\Lambda$ is infinitely divisible if $\Lambda(A)$ is infinitely
divisible for all Borel sets $A\subseteq{\mathbb{R}^{d}}$.

In this paper we assume that the Lévy basis $\Lambda$ is stationary and
isotropic. With $C(\lambda\dagger Y)=\log\mathbb{E}\mathrm{e}^{i\lambda Y}$
denoting the cumulant function for a random variable $Y$, this means that the
random variable $\Lambda(A)$ has Lévy-Khintchine representation

$C(\lambda\dagger\Lambda(A))=i\lambda a\lvert
A\rvert-\frac{1}{2}\lambda^{2}\theta\lvert
A\rvert+\int_{A\times\mathbb{R}}\bigl{(}\mathrm{e}^{i\lambda x}-1-i\lambda
x\boldsymbol{1}_{[-1,1]}(x)\bigr{)}F(\mathrm{d}u,\mathrm{d}x)$

for all Borel sets $A\subseteq{\mathbb{R}^{d}}$. Here $a\in\mathbb{R}$,
$\theta\geq 0$ and $F$ is the product measure $m\otimes\rho$ of Lebesgue
measure $m$ and a Lévy measure $\rho$. We furthermore assume that the Lévy
measure has a regularly varying right tail: Let $\mathcal{R}_{-\alpha}$ denote
the set of regularly varying functions (at infinity) of index
$-\alpha\in\mathbb{R}$, that is, $h\in\mathcal{R}_{-\alpha}$ if

$\frac{h(tx)}{h(x)}\to t^{-\alpha}\qquad\text{as }x\to\infty$

for all $t>0$. Note that this class of functions is closed under asymptotic
equivalence (at infinity). We say that a distribution $G$ is regularly varying
with index $\alpha>0$, writing $G\in\mathcal{RV}_{\alpha}$, if its tail
${\overline{G}}=1-G\in\mathcal{R}_{-\alpha}$ is a regularly varying function
of index $-\alpha$. We use similar notation for regularly varying random
variables, i.e. random variables with a regularly varying distribution.
Concerning the Lévy measure $\rho$, we thus assume that the right tail is a
regularly varying distribution of index $\alpha>0$, which formally reads
$\rho({}\cdot\cap(1,\infty))/\rho((1,\infty))\in\mathcal{RV}_{\alpha}$. Note
that regularly varying distributions are in particular subexponential, hence
$\mathcal{RV}_{\alpha}\subseteq\mathcal{S}$; see for instance [10, Appendix
A.3].

For the Lévy basis $\Lambda$ above, we consider the stationary Lévy-driven
field $(X_{v})_{v\in{\mathbb{R}^{d}}}$ defined by

$X_{v}=\int_{\mathbb{R}^{d}}f(v-u)\Lambda(\mathrm{d}u),$ (5)

where $f$ is a positive kernel function. It can be seen from [17, Theorem 2.7]
that the field is well-defined if only there is a $\gamma\in(0,1]$ such that
$\int_{\lvert y\rvert>1}\lvert y\rvert^{\gamma}\rho(\mathrm{d}y)$ is finite
and such that the bounded integration kernel $f:\mathbb{R}^{d}\to[0,\infty)$
satisfies $\int_{\mathbb{R}^{d}}f^{\gamma}(u)\mathrm{d}u<\infty$. Throughout
the paper we normalize $f$ and furthermore require that it is bounded by a
decreasing function only depending on $u$ through $\lvert u\rvert$.

According to [16, Theorem 2.6] we can (and will) choose a separable version,
also in the literature refereed to as a modification, of the random field
$(X_{v})_{v\in{\mathbb{R}^{d}}}$. Similarly we choose a separable version of
$(X_{v})_{v\in B}$ when we are dealing with a specific index set
$B\subseteq{\mathbb{R}^{d}}$. Likewise, all other random fields that will be
defined throughout the paper will be chosen to be separable versions.

As mentioned in the introduction, we will work under two different assumptions
on the Lévy basis and the kernel function. The assumptions differ in that one
is less restrictive with respect to the Lévy measure, and the other is less
restrictive with respect to the integration kernel. We first give the minimal
requirements on the pair $(\Lambda,f)$ before presenting the two sets of
assumptions. This minimal Assumption M, in particular, implies that the field
(5) is well-defined.

######  Assumption M.

The Lévy basis $\Lambda$ on ${\mathbb{R}^{d}}$ is stationary and isotropic
with a Lévy measure $\rho$ having a regularly varying right tail of index
$\alpha>0$. Moreover, there is $\gamma\in(0,\alpha)$ such that

$\int_{\lvert y\rvert>1}\lvert y\rvert^{\gamma}\rho(\mathrm{d}y)<\infty.$ (6)

The integration kernel $f:{\mathbb{R}^{d}}\to[0,\infty)$ satisfying $f\leq 1$
with $f(0)=1$ is lower semi-continuous and bounded from above by a decreasing
càdlàg function $g$, in the sense that $f(u)\leq g(\lvert u\rvert)$ for all
$u\in{\mathbb{R}^{d}}$. The function $g$ additionally satisfies

$\int_{\mathbb{R}^{d}}g^{\gamma}(\lvert u\rvert)\mathrm{d}u<\infty,$ (7)

where $\gamma\in(0,\alpha)\cap(0,1]$ satisfies (6).

Note that the integrability in (6) along the right tail of $\rho$ is already
given from the fact that $\rho$ is regularly varying. In fact,
$\int_{1}^{\infty}y^{\gamma}\rho(\mathrm{d}y)<\infty$ for all $\gamma<\alpha$
([10, Proposition A3.8]). Hence, the integrability (6) is simply a requirement
on the existence of moments of the finite measure
$\rho({}\cdot\cap(-\infty,-1))$. Furthermore, the integrability of
$g^{\gamma}$ and the fact that $g$ is decreasing implies that

$\int_{\mathbb{R}^{d}}\sup_{v\in B}g^{\gamma}(\lvert
v-u\rvert)\mathrm{d}u<\infty$

for all fixed sets $B\subseteq{\mathbb{R}^{d}}$. Since $f(\cdot)\leq
g(\lvert\cdot\rvert)$ and $\alpha>\gamma$, the claim also holds when
substituting $g^{\gamma}(\lvert v-u\rvert)$ with $g^{\alpha}(\lvert
v-u\rvert)$, $f^{\gamma}(v-u)$ or $f^{\alpha}(v-u)$. Note that the lower semi-
continuity of $f$ ensures that the supremum $\sup_{v\in B}$ can be replaced by
a suitable countable supremum, making the integrand measurable.

######  Example 1.

For any infinitely divisible distribution with a regularly varying right tail
there is asymptotic equivalence between the tail of the distribution and the
tail of the Lévy measure; see [9, Theorem 1]. In particular, the tail of
$\Lambda(A)$ will be regularly varying with index $\alpha$ if and only if the
Lévy measure $\rho$ is regularly varying with index $\alpha$.

######  Example 2.

Let $0<\alpha<2$. An $\alpha$-stable random field is obtained by letting
$\theta=0$ and defining

$\rho(\mathrm{d}x)=\big{(}p_{-}\lvert
x\rvert^{-(1+\alpha)}\boldsymbol{1}_{\\{x<0\\}}+p_{+}x^{-(1+\alpha)}\boldsymbol{1}_{\\{x>0\\}}\big{)}\mathrm{d}x,\qquad
p_{+},p_{-}\geq 0\ ,\ p_{+}+p_{-}>0,$

in the Lévy-Khintchine representation of $\Lambda$. Here $\rho$ has a
regularly varying right (and left) tail with index $\alpha$.

######  Example 3.

If the decreasing upper bound $g$ is a regularly varying function of index
$-(d+\epsilon)/\gamma$ for some $\gamma,\epsilon>0$, then (7) is satisfied.
This includes the simple case where $g(x)=c(1+x)^{-(d+\epsilon)/\gamma}$ is a
power function of order $-(d+\epsilon)/\gamma$.

######  Example 4.

If $h:[0,\infty)\to[0,\infty)$ is a lower semi-continuous function, which
satisfies $h\leq 1$, $h(0)=1$ and is bounded from above by a decreasing càdlàg
function $g:[0,\infty)\to[0,\infty)$ satisfying (7), then
$f:{\mathbb{R}^{d}}\to[0,\infty)$ defined by $f(u)=h(\lvert u\rvert)$ will
satisfy the requirement from Assumption M. With a kernel function of this
type, the random field defined by (5) will be isotropic. In particular, $f$
can be defined by $f(u)=g(\lvert u\rvert)$. Note that $g$ is indeed lower
semi-continuous, since it is decreasing and càdlàg.

For the first assumption we also require that $f$ is Hölder continuous.

######  Assumption 1.

The Lévy basis $\Lambda$ and integration kernel $f$ satisfy Assumption M.
Moreover, $f$ is Hölder continuous with some index $\zeta>0$. That is, there
is a constant $C$ such that

$\lvert f(u_{1})-f(u_{2})\rvert\leq C\lvert u_{1}-u_{2}\rvert^{\zeta}$

for all $u_{1},u_{2}\in{\mathbb{R}^{d}}$.

######  Example 5.

Let $A$ be a symmetric, positive definite matrix and define $f$ by

$f(u)=\exp(-u^{T}A^{-1}u),$

where $u^{T}$ is a row vector, and $u$ is a column vector. Then $f(0)=1$,
$f\leq 1$ and $f$ is Hölder continuous with index 1. Furthermore,
$f(\cdot)\leq g(\lvert\cdot\rvert)$ with $g(x)=\mathrm{e}^{-x^{2}/\lambda}$,
where $\lambda$ is the largest eigenvalue of $A$. In particular, defining
$f(u)=\mathrm{e}^{-\sigma\lvert u\rvert^{2}}$ yields an isotropic random
field, cf. Example 4.

The second assumption corresponds to the case where the Lévy basis is of
finite variation, by which we mean that the triple $(a,\theta,\rho)$ of its
Lévy-Khintchine representation satisfies that of a Lévy process of finite
variation.

######  Assumption 2.

The Lévy basis $\Lambda$ and integration kernel $f$ satisfy Assumption M.
Moreover, $\Lambda$ has Lévy-Khintchine representation with $\theta=0$ and a
Lévy measure $\rho$ satisfying

$\int_{\lvert y\rvert\leq 1}\lvert y\rvert\rho(\mathrm{d}y)<\infty.$ (8)

######  Example 6.

An $\alpha$-stable random field with $0<\alpha<1$ clearly satisfies the
condition in (8).

We are now ready to state the first main result of the paper concerning the
tail of $\sup X_{v}$.

######  Theorem 1.

Let $(X_{v})_{v\in{\mathbb{R}^{d}}}$ be a Lévy-driven field given by (5) where
the Lévy basis $\Lambda$ and the kernel function $f$ satisfy either Assumption
1 or Assumption 2. Let $B\subseteq{\mathbb{R}^{d}}$ be a fixed bounded set.
Then

$\mathbb{P}\bigl{(}\sup_{v\in
B}X_{v}>x\bigr{)}\sim\rho((x,\infty))\int_{{\mathbb{R}^{d}}}\sup_{v\in
B}f^{\alpha}(v-u)\mathrm{d}u$ (9)

as $x\to\infty$.

In fact, we show the more general tail result below. In this theorem, and in
the remainder of the paper, we use the notation
$y_{+}=y\boldsymbol{1}_{\\{y\geq 0\\}}$ and $y_{+}^{\gamma}=(y_{+})^{\gamma}$
for any $y\in\mathbb{R}$.

######  Theorem 2.

Let $(X_{v})_{v\in{\mathbb{R}^{d}}}$ be a Lévy-driven field given by (5) where
the Lévy basis $\Lambda$ and the kernel function $f$ satisfy either Assumption
1 or Assumption 2. Let $B\subseteq{\mathbb{R}^{d}}$ be a fixed bounded set,
and let $(Y_{v}^{1})_{v}$ be a field independent of $(X_{v})_{v}$ satisfying

$\mathbb{E}\bigl{(}\sup_{v\in B}Y_{v}^{1}\bigr{)}_{+}^{\beta}<\infty$ (10)

for some $\beta>\alpha$. Then

$\mathbb{P}\bigl{(}\sup_{v\in
B}(X_{v}+Y_{v}^{1})>x\bigr{)}\sim\rho((x,\infty))\int_{{\mathbb{R}^{d}}}\sup_{v\in
B}f^{\alpha}(v-u)\mathrm{d}u$

as $x\to\infty$.

Next we turn to the assumption on the expansion of the index sets $(C_{n})$ in
${\mathbb{R}^{d}}$ relating to the extremal result. For all $n\in\mathbb{N}$,
we require that $C_{n}$ is a $p$-convex set as defined below. In the
following, a compact convex set with non-empty interior is called a _convex
body_ ; see e.g. [25, Chapter 4].

######  Definition 1.

A set $C\subseteq\mathbb{R}^{d}$ is said to be $p$-convex, if it is connected
and has the form

$C=\bigcup_{i=1}^{p}\overline{C}_{i}\,,$

where $\overline{C}_{1},\ldots,\overline{C}_{p}$ are convex bodies in
$\mathbb{R}^{d}$.

We give the sufficient assumption on the index sets $(C_{n})$ used in Theorems
3 and 4 below. Due to stationarity of all fields involved, we can without loss
of generality in later results assume that $0\in C_{n}$ for all
$n\in\mathbb{N}$. Although not formulated in the assumption, this will be
assumed in the remainder of the paper. In the assumption, conditions on the
so-called _intrinsic volumes_ are stated. For $j=0,\dots,d$, the intrinsic
volumes $V_{j}(C)$ describe the geometry of the convex body $C$. For instance,
$V_{0}(C)=1$, $V_{1}(C)$ is proportional to the mean width, $V_{d-1}(C)$ is
half the surface area, and $V_{d}(C)=\lvert C\rvert$ equals the volume of $C$.
Furthermore, the functionals $V_{j}:\mathcal{K}\to\mathbb{R}$, where
$\mathcal{K}$ is the set of all convex bodies, satisfy some important
properties, among which we mention

  1. (i)

They are non-negative, i.e. $V_{j}(C)\geq 0$ for all $C\in\mathcal{K}$.

  2. (ii)

They are homogeneous, i.e. $V_{j}(\gamma C)=\gamma^{j}V_{j}(C)$ for all
$\gamma>0$.

  3. (iii)

They are monotone, i.e. $C\subseteq D$ implies $V_{j}(C)\leq V_{j}(D)$.

For a greater exposition of convex bodies and their intrinsic volumes we refer
to [25, Chapter 4].

######  Assumption 3.

The sequence $(C_{n})_{n\in\mathbb{N}}$ consists of $p$-convex bodies, where

$C_{n}=\bigcup_{i=1}^{p}C_{n,i}$

and $\lvert C_{n}\rvert\to\infty$ as $n\to\infty$. Furthermore,

$\frac{\sum_{i=1}^{p}V_{j}(C_{n,i})}{\lvert C_{n}\rvert^{j/d}}\quad\text{is
bounded in $n$ for each }j=1,\dots,d-1.$ (11)

######  Example 7.

Let $C=\cup_{i=1}^{p}\overline{C}_{i}$ be a $p$-convex set and define the
sequence $(C_{n})_{n\in\mathbb{N}}$ by

$C_{n}=r_{n}C=\bigcup_{i=1}^{p}r_{n}\overline{C}_{i},$

where $r_{n}\uparrow\infty$ as $n\to\infty$. Then $\lvert C_{n}\rvert=\lvert
r_{n}C\rvert=r_{n}^{d}\lvert C\rvert$ and
$V_{j}(r_{n}\overline{C}_{i})=r_{n}^{j}V_{j}(\overline{C}_{i})$ for
$j=0,\dots,d$. In particular,

$\frac{\sum_{i=1}^{p}V_{j}(r_{n}\overline{C}_{i})}{\lvert
C_{n}\rvert^{j/d}}=\frac{\sum_{i=1}^{p}V_{j}(\overline{C}_{i})}{\lvert
C\rvert^{j/d}}$

is constant in $n$ for all $j=1,\dots,d-1$. Thus the sequence
$(C_{n})_{n\in\mathbb{N}}$ satisfies Assumption 3.

Before presenting the last main results, we recall that the distributions in
$\mathcal{RV}_{\alpha}$ are exactly the distributions in the maximum domain of
attraction $\mathrm{MDA}(\Phi_{\alpha})$ of the Fréchet distribution
$\Phi_{\alpha}$, where
$\Phi_{\alpha}(x)=\exp(-x^{-\alpha})\boldsymbol{1}_{\\{x\geq 0\\}};$ see [10,
Chapter 3]. In other words, we know that there are norming constants
$(\tilde{a}_{n})_{n}$ with $0<\tilde{a}_{n}\to\infty$ such that

$n\rho((\tilde{a}_{n}x,\infty))\to x^{-\alpha}\rho((1,\infty))$

for all $x>0$. In the remainder of the paper, we let $(a_{n})_{n}$ be a
sequence of norming constants of $\rho$ relative to $\lvert C_{n}\rvert$, i.e.

$\lvert C_{n}\rvert\rho((a_{n}x,\infty))\to x^{-\alpha}\rho((1,\infty))$ (12)

for all $x>0$.

######  Theorem 3.

Let $(X_{v})_{v\in{\mathbb{R}^{d}}}$ be a Lévy-driven field given by (5) where
the Lévy basis $\Lambda$ and the kernel function $f$ satisfy either Assumption
1 or Assumption 2. Let $(C_{n})_{n\in\mathbb{N}}$ be a sequence of sets in
${\mathbb{R}^{d}}$ satisfying Assumption 3, and let $a_{n}$ be the norming
constants of the Lévy measure $\rho$ relative to $\lvert C_{n}\rvert$, i.e.
$\lim_{n}\lvert C_{n}\rvert\rho((a_{n}x,\infty))=x^{-\alpha}\rho((1,\infty))$
for all $x>0$. Then, as $n\to\infty$,

$\mathbb{P}\Bigl{(}a_{n}^{-1}\sup_{v\in C_{n}}X_{v}\leq
x\Bigr{)}\to\exp\Bigl{(}-x^{-\alpha}\rho((1,\infty))\Bigr{)}$

for all $x>0$.

The proof techniques used to show Theorem 3 almost immediately give us the
following more general result. In its formulation,
$B(t)=\\{u\in{\mathbb{R}^{d}}\>:\>\lvert u\rvert\leq t\\}$ denotes the closed
ball with radius $t$ and center in the origin $0\in{\mathbb{R}^{d}}$. This
notation will be used throughout the paper.

######  Theorem 4.

Let $(X_{v})_{v\in{\mathbb{R}^{d}}}$ be a Lévy-driven field given by (5) where
the Lévy basis $\Lambda$ and the kernel function $f$ satisfy either Assumption
1 or Assumption 2. Let $(C_{n})_{n\in\mathbb{N}}$ be a sequence of sets in
${\mathbb{R}^{d}}$ satisfying Assumption 3, and let $a_{n}$ be the norming
constants of the Lévy measure $\rho$ relative to $\lvert C_{n}\rvert$, i.e.
$\lim_{n}\lvert C_{n}\rvert\rho((a_{n}x,\infty))=x^{-\alpha}\rho((1,\infty))$
for all $x>0$. Furthermore, let $(Y_{v}^{1})_{v}$ and $(Y_{v}^{2})_{v}$ be
independent stationary fields independent of $(X_{v})_{v}$, where
$(Y_{v}^{1})$ is ergodic and satisfies

$\mathbb{E}\bigl{(}\sup_{v\in B(1)}Y_{v}^{1}\bigr{)}_{+}^{\beta_{1}}<\infty$
(13)

for some $\beta_{1}>\alpha$, and $(Y_{v}^{2})$ satisfies

$\mathbb{E}\bigl{(}\sup_{v\in B(1)}\lvert
Y_{v}^{2}\rvert\bigr{)}^{\beta_{2}}<\infty$ (14)

for some $\beta_{2}>\alpha$. Then, as $n\to\infty$,

$\mathbb{P}\Bigl{(}a_{n}^{-1}\sup_{v\in
C_{n}}\bigl{(}X_{v}+Y_{v}^{1}+Y_{v}^{2}\bigr{)}\leq
x\Bigr{)}\to\exp\Bigl{(}-x^{-\alpha}\rho((1,\infty))\Bigr{)}$

for all $x>0$.

##  3 Results for regularly varying distributions

In this section we list some useful results concerning regularly varying
distributions. By assumption, $\rho$ has a regularly varying right tail, and
we thus obtain by Karamata’s representation theorem [10, Theorem A3.3] that

$\rho((x,\infty))=c(x)\exp\Bigl{(}-\int_{1}^{x}\frac{\alpha(y)}{y}\mathrm{d}y\Bigr{)}\qquad\text{for
all }x\geq 1,$

where $c(x)\to c>0$ and $\alpha(x)\to\alpha$ as $x\to\infty$. Thus, for all
positive $\gamma<\alpha$ there exist constants $C>0$ and $x_{0}\geq 1$ such
that

$\frac{\rho((tx,\infty))}{\rho((x,\infty))}\leq Ct^{-\gamma}\qquad\text{for
all }t\geq 1,\ x\geq x_{0}.$ (15)

We need bounds similar to (15) for sums instead of products. The following
lemma describes this for arbitrary regularly varying distributions.

######  Lemma 1.

Let $G\in\mathcal{RV}_{\alpha}$ be a regularly varying distribution of index
$\alpha>0$, and let ${\overline{G}}$ be its tail. For all $\beta>\alpha$ there
are constants $K,C\geq 0$ and $x_{0}\geq 1$ such that

${\overline{G}}(x-y)\leq{\overline{G}}(x)C\bigl{(}K+y_{+}^{\beta}\bigr{)}$
(16)

for all $x\geq x_{0}$ and $y\in\mathbb{R}$.

###### Proof.

We show that there are constants $C^{\prime}$ and $K^{\prime}$ such that

${\overline{G}}(x-y)\leq{\overline{G}}(x)C^{\prime}\bigl{(}K^{\prime}+y_{+}\bigr{)}^{\beta}$
(17)

is satisfied for all $x\geq x_{0}$ and $y\in\mathbb{R}$. From this, (16)
easily follows.

Since $G\in\mathcal{RV}_{\alpha}$ and hence
${\overline{G}}\in\mathcal{R}_{-\alpha}$ it follows by Karamata’s
representation theorem for regularly varying functions that

${\overline{G}}(x)=a(x)\exp\Bigl{(}-\int_{1}^{x}\frac{\alpha(z)}{z}\mathrm{d}z\Bigr{)}\qquad\text{for
all }x\geq 1,$

for some functions $a(x)\to a>0$ and $\alpha(x)\to\alpha$ as $x\to\infty$. Fix
$\beta>\alpha$ and let $x_{0}\geq 1$ be given such that

$\alpha(x)<\beta,\qquad\text{and}\qquad\lvert a(x)-a\rvert\leq a/3$ (18)

for all $x\geq x_{0}$.

Now consider only $x\geq x_{0}$ and $y\geq 0$. If $y\leq x-x_{0}$ and hence
$x-y\geq x_{0}$, we find from (18) above that

$\displaystyle\frac{{\overline{G}}(x-y)}{{\overline{G}}(x)}$
$\displaystyle=\frac{a(x-y)}{a(x)}\exp\Bigl{(}\int_{x-y}^{x}\frac{\alpha(z)}{z}\mathrm{d}z\Bigr{)}$
$\displaystyle\leq 2\Bigl{(}\frac{x}{x-y}\Bigr{)}^{\beta}$ $\displaystyle\leq
2\Bigl{(}1+\frac{y}{x_{0}}\Bigr{)}^{\beta}.$

If on the other hand $y>x-x_{0}$, and thus $x-y<x_{0}$, we see that

$\displaystyle\frac{{\overline{G}}(x-y)}{{\overline{G}}(x)}\leq\frac{1}{{\overline{G}}(x)}$
$\displaystyle=\Bigl{[}a(x)\exp\Bigl{(}-\int_{1}^{x_{0}}\frac{\alpha(z)}{z}\mathrm{d}z\Bigr{)}\Bigr{]}^{-1}\exp\Bigl{(}\int_{x_{0}}^{x}\frac{\alpha(z)}{z}\mathrm{d}z\Bigr{)}$
$\displaystyle\leq C_{0}\Bigl{(}\frac{x}{x_{0}}\Bigr{)}^{\beta}$
$\displaystyle\leq C_{0}\Bigl{(}1+\frac{y}{x_{0}}\Bigr{)}^{\beta}$

where the constant $C_{0}$ can be chosen as

$C_{0}=\Bigl{[}\frac{2a}{3}\exp\Bigl{(}-\int_{1}^{x_{0}}\frac{\alpha(z)}{z}\mathrm{d}z\Bigr{)}\Bigr{]}^{-1}.$

Choosing $K^{\prime}=x_{0}$ and
$C^{\prime}=(K^{\prime})^{-\beta}\cdot\max\\{2,C_{0}\\}$ shows (17) for $y\geq
0$.

For $y<0$ the claim (17) reads
${\overline{G}}(x-y)\leq{\overline{G}}(x)C^{\prime}(K^{\prime})^{\beta}$,
which is clearly true since $x\mapsto{\overline{G}}(x)$ is decreasing and
$C^{\prime}(K^{\prime})^{\beta}\geq 2$. ∎

In the proof of Theorem 4 we need to find a sequence $d_{n}$, which tends to
infinity at a slower rate than the norming constants $a_{n}$ of $\rho$, i.e.
the sequence satisfying (12).

######  Lemma 2.

Let $\beta>\alpha$ where $\alpha$ is the regularly varying index of $\rho$.
There is a sequence $d_{n}\to\infty$ of order $d_{n}=o(a_{n})$ such that

$\lim_{n\to\infty}\frac{\overline{H}(d_{n})}{\rho((a_{n},\infty))}=0$

for all distributions $H$ on $[0,\infty)$ satisfying $\int
x^{\beta}H(\mathrm{d}x)<\infty$.

###### Proof.

Since $\rho$ is regularly varying of index $\alpha>0$, there is a slowly
varying function $L$ such that $\rho((x,\infty))=x^{-\alpha}L(x)$. Let
$\epsilon=\beta-\alpha>0$ and define $d_{n}=a_{n}^{1-\epsilon/(2\beta)}$. Then
$d_{n}\to\infty$ and $d_{n}=o(a_{n})$ as $n\to\infty$. Noticing that
$a_{n}^{\epsilon/2}L(a_{n})\to\infty$ and $d_{n}^{\beta}\overline{H}(d_{n})\to
0$ concludes the proof. ∎

##  4 Boundedness and tail behavior

In this and the following sections we will need a decomposition of $(X_{v})$
into independent fields. For this we write $\Lambda=\Lambda_{1}+\Lambda_{2}$
as the independent sum of two Lévy bases with Lévy-Khintchine representations

$\displaystyle C(\lambda\dagger\Lambda_{1}(A))$
$\displaystyle=\int_{A\times\mathbb{R}}\bigl{(}\mathrm{e}^{i\lambda
x}-1\bigr{)}F_{(1,\infty)}(\mathrm{d}u,\mathrm{d}x)$ $\displaystyle
C(\lambda\dagger\Lambda_{2}(A))$ $\displaystyle=i\lambda a\lvert
A\rvert-\frac{1}{2}\lambda^{2}\theta\lvert
A\rvert+\int_{A\times\mathbb{R}}\bigl{(}\mathrm{e}^{i\lambda x}-1-i\lambda
x\boldsymbol{1}_{[-1,1]}(x)\bigr{)}F_{(-\infty,1]}(\mathrm{d}u,\mathrm{d}x)\,,$

respectively. Here, $F_{D}=m\otimes\rho_{D}$, where $\rho_{D}$ is the
restriction of $\rho$ to $D\subseteq\mathbb{R}$. Similarly, we decompose the
field $(X_{v})_{v}$ into a sum of two independent random fields
$X_{v}=Z_{v}+Y_{v}$ for all ${v\in{\mathbb{R}^{d}}}$, where

$\displaystyle
Z_{v}=\int_{\mathbb{R}^{d}}f(v-u)\Lambda_{1}(\mathrm{d}u)\qquad\text{and}$
(19) $\displaystyle
Y_{v}=\int_{\mathbb{R}^{d}}f(v-u)\Lambda_{2}(\mathrm{d}u).$ (20)

In this section we will make use of a further decomposition of $\Lambda_{2}$
into $\Lambda_{2}=\Lambda_{2,C}+\Lambda_{2,-}$, where

$\displaystyle C(\lambda\dagger\Lambda_{2,-}(A))$
$\displaystyle=\int_{A\times\mathbb{R}}\bigl{(}\mathrm{e}^{i\lambda
x}-1\bigr{)}F_{(-\infty,-1)}(\mathrm{d}u,\mathrm{d}x),$ $\displaystyle
C(\lambda\dagger\Lambda_{2,C}(A))$ $\displaystyle=i\lambda a\lvert
A\rvert-\frac{1}{2}\lambda^{2}\theta\lvert
A\rvert+\int_{A\times\mathbb{R}}\bigl{(}\mathrm{e}^{i\lambda x}-1-i\lambda
x\boldsymbol{1}_{[-1,1]}(x)\bigr{)}F_{[-1,1]}(\mathrm{d}u,\mathrm{d}x).$

We let $Y_{v}=Y_{v}^{C}+Y_{v}^{-}$ be the corresponding independent
decomposition of $(Y_{v})_{v}$.

It will be essential in the subsequent proofs that the sample paths of
$(X_{v})_{v\in B}$ are bounded, when $B$ is a fixed bounded set. For this we
will study the different parts of the decomposition separately.

######  Lemma 3.

Under Assumption 1 the field $(Y_{v}^{C})_{v\in B}$ has a continuous version.
In particular, it holds that $\mathbb{P}(\sup_{v\in B}\lvert
Y_{v}^{C}\rvert<\infty)=1$.

###### Proof.

Let $\Lambda_{2,C}^{\prime}$ be the so-called spot variable of
$\Lambda_{2,C}$, i.e. a random variable equivalent in distribution to
$\Lambda_{2,C}(S)$ for sets $S\subseteq{\mathbb{R}^{d}}$ with Lebesgue measure
$1$. Then all moments of $\Lambda_{2,C}^{\prime}$ are finite, and consequently
all moments of $Y^{C}_{v}$ are finite. Following the lines of the proof of
[28, Theorem 5.1] with a Lipschitz assumption replaced by the Hölder
assumption in Assumption 1, it is immediately seen that there exist finite
constants $C^{\prime}_{n}$, chosen independently of $r$ and $v\in B$, and
natural numbers $n^{\prime}\geq n/2$, such that

$\mathbb{E}[(Y_{v+r}-Y_{v})^{n}]\leq C^{\prime}_{n}\lvert
r\rvert^{\zeta{n^{\prime}}}$

with the equality $n^{\prime}=n/2$ whenever $n$ is even. Using the fact that
$\lvert r\rvert\leq\mathrm{diam}(B)$, we find finite $C^{\prime}\geq 0$ and
$\eta>n>4d/\zeta$ such that

$\mathbb{E}\left\lvert Y^{C}_{v+r}-Y^{C}_{v}\right\rvert^{n}\leq
C^{\prime}_{n}\lvert r\rvert^{2d}\lvert r\rvert^{\zeta
n/2-2d}\leq\frac{C^{\prime}\lvert r\rvert^{2d}}{\big{\lvert}\log\lvert
r\rvert\big{\rvert}^{1+\eta}}$

for all $v\in B$. From a corollary to [2, Theorem 3.2.5] we conclude that
$(Y_{v})_{v\in B}$ has a continuous version on $B$. ∎

It remains to show that Assumption 1 implies that $(Z_{v})_{v\in B}$ and
$(Y^{-}_{v})_{v\in B}$ are both bounded. For this, the representation in
Theorem 5 below will be useful. This theorem is an adapted version of [23,
Theorem 3.4.3] and in the formulation we apply a similar (when possible)
notation. We say that $(\Gamma_{n})$ is a sequence of standard Poisson
arrivals, if

$\Gamma_{n}=e_{1}+\dots+e_{n},$

where $(e_{n})$ are independent and all follow an exponential distribution
with mean 1.

The theorem is formulated in terms of $(X_{v})_{v\in B}$ as defined under the
minimal Assumption M. It will thus be applicable to $(Z_{v})_{v\in B}$ and
$(Y_{v}^{-})_{v\in B}$ defined under Assumptions 1 and 2.

######  Theorem 5.

Assume, in addition to Assumption M, that the support of the Lévy measure
$\rho$ is a subset of $(0,\infty)$, that
$\int_{0}^{1}x\rho(\mathrm{d}x)<\infty$, and that $\Lambda(A)$ has Lévy-
Khintchine representation

$C(\lambda\dagger\Lambda(A))=\int_{A\times\mathbb{R}}\bigl{(}\mathrm{e}^{i\lambda
x}-1\bigr{)}F(\mathrm{d}u,\mathrm{d}x)$

with $F=m\otimes\rho$. Let $\mu$ be a probability measure on $\mathbb{R}^{d}$
that is equivalent with Lebesgue measure $m$. Let

$r(v)=\frac{\mathrm{d}m}{\mathrm{d}\mu}(v)$

for $v\in\mathbb{R}^{d}$ and define

$\rho_{r}(v,\cdot)=r(v)\rho(\cdot)$

for each $v\in\mathbb{R}^{d}$. Furthermore, define the generalized inverse of
$x\mapsto\rho_{r}(v,(x,\infty))$ as

$G(x,v)=\inf\\{y>0\>:\>\rho_{r}(v,(y,\infty))\leq x\\}.$

Let $(U_{n})$ be a sequence of independent and identically distributed random
variables with common distribution $\mu$ and let $(\Gamma_{n})$ be a sequence
of standard Poisson arrivals on $(0,\infty)$ independent of $(U_{n})$. Then
$(\tilde{X}_{v})_{v\in B}$ defined by

$\tilde{X}_{v}=\sum_{n=1}^{\infty}G(\Gamma_{n},U_{n})f(v-U_{n})$ (21)

for all $v\in B$ is a well-defined random field, which is equal in its finite-
dimensional distributions to $(X_{v})_{v\in B}$

###### Proof.

The proof follows the lines of the proof of [23, Theorem 3.4.3] closely,
utilizing the fact that both $G$ and $f$ are non-negative and, with the
notation $[\\![\cdot]\\!]$ from [23], that both

$\int_{{\mathbb{R}^{d}}}\int_{\mathbb{R}}[\\![f(v-u)x]\\!]\,\rho(\mathrm{d}x)\mathrm{d}u\quad\text{and}\quad\int_{{\mathbb{R}^{d}}}\int_{\mathbb{R}}[\\![f(v-u)x]\\!]^{2}\,\rho(\mathrm{d}x)\mathrm{d}u$

are bounded from above by

$\int_{{\mathbb{R}^{d}}}\int_{0<x\leq
1}f(v-u)x\,\rho(\mathrm{d}x)\mathrm{d}u+2\int_{{\mathbb{R}^{d}}}\int_{x>1}f(v-u)^{\gamma}x^{\gamma}\,\rho(\mathrm{d}x)\mathrm{d}u,$

which is finite by assumption. ∎

######  Corollary 1.

The field $(\tilde{X}_{v})_{v\in B}$ defined in (21) is bounded on a set with
probability 1. If furthermore, the kernel function $f$ is continuous, then the
field is continuous with probability 1.

###### Proof.

The $n$th term in (21) is non-negative and bounded (uniformly in $v\in B$)
from above by $G(\Gamma_{n},U_{n})\sup_{u\in B}f(u-U_{n})$. Since the series

$\sum_{n=1}^{\infty}G(\Gamma_{n},U_{n})\sup_{u\in B}f(u-U_{n})$ (22)

converges almost surely by [23, Theorem 3.4.1], the boundedness follows. If,
additionally, $f$ is continuous then each term in (21) is continuous, and the
convergence of the series is uniform due to (22). Thereby the continuity of
$(\tilde{X}_{v})_{v\in B}$ follows. ∎

The corollary almost immediately gives the desired result for $(Z_{v})_{v\in
B}$ and $(Y^{-}_{v})_{v\in B}$.

######  Lemma 4.

Under Assumption 1 the fields $(Z_{v})$ and $(Y^{-}_{v})$ both have continuous
versions. In particular, they are almost surely bounded, i.e.

$\mathbb{P}(\sup_{v\in B}\lvert Z_{v}\rvert<\infty)=\mathbb{P}(\sup_{v\in
B}\lvert Y_{v}^{-}\rvert<\infty)=1.$

###### Proof.

We give the argument for $(Z_{v})$. The proof for $(Y_{v}^{-})$ is, except for
a change in notation in Theorem 5 and Corollary 1, identical. First note that
$(Z_{v})_{v\in B}$ satisfies the conditions of Theorem 5. Recall that we have
chosen a separable version of $(Z_{v})_{v\in B}$ and let $T\subset B$ be the
corresponding countable separating dense subset. By Corollary 1 and the
equality of distributions given in Theorem 5 the field $(Z_{v})_{v\in B}$ is
uniformly continuous on $T$ with probability 1. In particular, it has a
continuous version on $B$. ∎

######  Lemma 5.

Under Assumption 2 the field $(X_{v})_{v\in B}$ has a version that is almost
surely bounded.

###### Proof.

We can decompose $\Lambda=\Lambda^{+}+\Lambda^{-}+\tilde{a}m$, where
$\tilde{a}m$ is the scaled Lebesgue measure for some constant $\tilde{a}$, and
$\Lambda^{+}$ and $\Lambda^{-}$ are independent, given such that

$\displaystyle C(\lambda\dagger\Lambda^{+}(A))$
$\displaystyle=\int_{A\times\mathbb{R}}\bigl{(}\mathrm{e}^{i\lambda
x}-1\bigr{)}F_{(0,\infty)}(\mathrm{d}u,\mathrm{d}x)$ $\displaystyle
C(\lambda\dagger\Lambda^{-}(A))$
$\displaystyle=\int_{A\times\mathbb{R}}\bigl{(}\mathrm{e}^{i\lambda
x}-1\bigr{)}F_{(-\infty,0)}(\mathrm{d}u,\mathrm{d}x).$

respectively. We let $(X_{v}^{+})$ and $(X_{v}^{-})$ be the random fields
obtained by integrating with respect to $\Lambda^{+}$ and $\Lambda^{-}$. It
suffices to show that each of $(X_{v}^{+})_{v\in B}$ and $(X_{v}^{-})_{v\in
B}$ have versions with bounded sample paths. We focus on $(X_{v}^{+})_{v\in
B}$, since the proof for $(X_{v}^{-})_{v\in B}$ is identical. Note that
$(X_{v}^{+})_{v\in B}$ satisfies the conditions of Theorem 5 such that, by
Corollary 1, there exists $(\tilde{X}_{v}^{+})_{v\in B}$ that is almost surely
bounded and has the same finite dimensional distributions as
$(X_{v}^{+})_{v\in B}$. Recall that we have chosen a separable version of
$(X_{v}^{+})_{v\in B}$ and let $T\subset B$ be the corresponding countable
separating dense subset. Now, the distributions of $(X_{v}^{+})_{v\in T}$ and
$(\tilde{X}_{v}^{+})_{v\in T}$ are identical, which concludes the proof. ∎

###### Proof of Theorem 1.

Fix the bounded index set $B\subseteq{\mathbb{R}^{d}}$. Let $m$ denote
Lebesgue measure, and define the function $H$ by

$H(x)=m\otimes\rho\bigl{(}\\{(u,z)\in{\mathbb{R}^{d}}\times\mathbb{R}\>:\>\sup_{v\in
B}zf(v-u)>x\\}\bigr{)}$

for $x>0$. For convenience, define $s_{u}=(\sup_{v\in B}f(v-u))^{-1}\geq 1$
for all $u\in{\mathbb{R}^{d}}$. With this notation at hand, we see that

$\displaystyle\frac{H(x)}{\rho((x,\infty))}=\int_{\mathbb{R}^{d}}\frac{\rho((s_{u}x,\infty))}{\rho((x,\infty))}\mathrm{d}u,$

and since $\rho$ has a regularly varying right tail of index $\alpha$, we
obtain the convergence

$\frac{\rho((s_{u}x,\infty))}{\rho((x,\infty))}\to
s_{u}^{-\alpha},\qquad\text{as }x\to\infty,$

for all $u\in{\mathbb{R}^{d}}$. Now choose $\gamma<\alpha$ according to (7)
satisfying that $g^{\gamma}$ is integrable. This implies in particular that
$f^{\gamma}$ is integrable, and also $u\mapsto s_{u}^{-\gamma}$ is so. Since
$s_{u}\geq 1$ for all $u\in{\mathbb{R}^{d}}$ we find by dominated convergence,
using the bound (15), that

$\frac{H(x)}{\rho((x,\infty))}\to\int_{\mathbb{R}^{d}}\bigl{(}\sup_{v\in
B}f(v-u)\bigr{)}^{\alpha}\mathrm{d}u=\int_{\mathbb{R}^{d}}\sup_{v\in
B}f^{\alpha}(v-u)\mathrm{d}u<\infty$

as $x\to\infty$. Therefore, the distribution
$1-\min\\{H,1\\}\in\mathcal{RV}_{\alpha}$, and thus it is especially in the
class of subexponential distributions. Recall that $(X_{v})_{v\in B}$ is
chosen separable and let $T\subset B$ be the corresponding countable
separating dense subset. By Lemmas 3–5, the field $(X_{v})$ is almost surely
bounded on $B$,

$\mathbb{P}\bigl{(}\sup_{v\in B}\lvert
X_{v}\rvert<\infty\bigr{)}=\mathbb{P}\bigl{(}\sup_{v\in T}\lvert
X_{v}\rvert<\infty\bigr{)}=1,$

and, using that $\sup_{v\in B}X_{v}$ and $\sup_{v\in T}X_{v}$ coincide, we
conclude by [22, Theorem 3.1] that

$\mathbb{P}\bigl{(}\sup_{v\in B}X_{v}>x\bigr{)}\sim
H(x)\sim\rho((x,\infty))\int_{\mathbb{R}^{d}}\sup_{v\in
B}f^{\alpha}(v-u)\mathrm{d}u$

as $x\to\infty$. ∎

###### Proof of Theorem 2.

Fix the bounded index set $B\subseteq{\mathbb{R}^{d}}$. Let $F$ denote the
distribution of $\sup_{v\in B}X_{v}$, which, by Theorem 1 and the fact that
$\rho$ is regularly varying, satisfies $F\in\mathcal{RV}_{\alpha}$. In
particular, $F$ is subexponential and hence

$\frac{\overline{F}(x-y)}{\overline{F}(x)}\to 1$ (23)

for all $y\in\mathbb{R}$ as $x\to\infty$. Let $\pi^{1}$ denote the
distribution of $(Y_{v}^{1})_{v}$, and let $y^{*}=\sup_{v\in B}y_{v}$ for a
deterministic field $(y_{v})_{v}$. Then

$\displaystyle\frac{\mathbb{P}(\sup_{v\in
B}(X_{v}+Y_{v}^{1})>x)}{\overline{F}(x)}$
$\displaystyle\leq\frac{\mathbb{P}(\sup_{v\in B}X_{v}>x-\sup_{v\in
B}Y_{v}^{1})}{\overline{F}(x)}$
$\displaystyle=\int\frac{\overline{F}(x-y^{*})}{\overline{F}(x)}\pi^{1}(\mathrm{d}y),$

with the integrand tending to $1$ by (23) above. Now fix $\beta>\alpha$ such
that (10) is satisfied. By Lemma 1, the integrand has an $x$-independent and
integrable upper bound, and we thus obtain by dominated convergence and (9)
that

$\limsup_{x\to\infty}\frac{\mathbb{P}(\sup_{v\in
B}(X_{v}+Y_{v}^{1})>x)}{\rho((x,\infty))}\leq\int_{\mathbb{R}^{d}}\sup_{v\in
B}f^{\alpha}(v-u)\mathrm{d}u.$

Since $(\inf_{v\in B}Y_{v}^{1})_{+}$ also has $\beta$-moment, we similarly
find that

$\liminf_{x\to\infty}\frac{\mathbb{P}(\sup_{v\in
B}(X_{v}+Y_{v}^{1})>x)}{\rho((x,\infty))}\geq\int_{\mathbb{R}^{d}}\sup_{v\in
B}f^{\alpha}(v-u)\mathrm{d}u,$

proving the claim. ∎

##  5 The geometry

The purpose of this section is to introduce the geometry necessary to the
proof of the extremal results Theorems 3 and 4 above. In particular, we
construct certain boxes which will be used to approximate the continuous index
sets $(C_{n})$, and we show their limiting behavior. Recall by Assumption 3
that the index sets $(C_{n})$ are $p$-convex bodies with relatively bounded
intrinsic volumes as in (11).

We first introduce some new notation which will be used throughout the
remainder of the paper. For two sets $A,B\subseteq\mathbb{R}^{d}$, we define
their Minkowski sum by $A\oplus B=\\{a+b\mid a\in A,b\in B\\}$. When referring
to (discrete or continuous) cubes with side-length equal to $s>0$, we mean a
box where all side-lengths equal $s$. Moreover, we say that $C_{r}(u)$ is the
closed $r$-cube with corner $u\in{\mathbb{R}^{d}}$ if

$C_{r}(u)=u+[0,r]^{d}$

for $r>0$. Lastly, we let $\mathrm{dist}(u,A)$ be the distance from
$u\in{\mathbb{R}^{d}}$ to the set $A\subseteq{\mathbb{R}^{d}}$.

The following corollary, which is shown in [27], is a consequence of the
famous Steiner Theorem from convex geometry stating that the volume of the
Minkowski sum of a convex body in ${\mathbb{R}^{d}}$ and a ball $B(r)$ of
radius $r$ is a polynomial of degree $d$ in $r$.

######  Corollary 2 ([27, Corollary 1]).

Let $C\subseteq\mathbb{R}^{d}$ be a convex body with boundary $\partial C$ and
let $r>0$. Then

$\sum_{j=0}^{d-1}\omega_{d-j}V_{j}(C)r^{d-j}\leq\lvert\partial C\oplus
B(r)\rvert\leq 2\sum_{j=0}^{d-1}\omega_{d-j}V_{j}(C)r^{d-j},$

where $V_{j}(C)$ is the $j$th intrinsic volume of $C$ and $\omega_{j}$ is the
volume of the $j$-dimensional unit ball.

For the sequence of index sets $(C_{n})_{n\in\mathbb{N}}$ satisfying
Assumption 3 we will for each $k,L\in\mathbb{N}$ define
$t_{n,k,L}\in\mathbb{N}$ by

$t_{n,k,L}=L\cdot\left\lfloor\sqrt[d]{\lvert C_{n}\rvert/k}\right\rfloor.$

The integer $t_{n,k,L}$ will serve as the side-length of a sequence of cubes
approximating the index sets $C_{n}$. Formally, for each
$z=(z_{1},\ldots,z_{d})\in\mathbb{Z}^{d}$ and $n$ large enough relative to
$k$, we define the cube $I_{z}^{n,k,L}\subseteq{\mathbb{R}^{d}}$ as

$\displaystyle I_{z}^{n,k,L}$ $\displaystyle=z\,t_{n,k,L}+[0,t_{n,k,L})^{d}$
$\displaystyle=\bigtimes_{i=1}^{d}[z_{i}\,t_{n,k,L},(z_{i}+1)t_{n,k,L}).$

Furthermore, we let $P_{n,k,L}$ be the set of indices $z\in\mathbb{Z}^{d}$ for
which $I_{z}^{n,k,L}$ is contained in $C_{n}$, and we let $Q_{n,k,L}$ be the
set of indices $z$ for which $I_{z}^{n,k,L}$ intersects $C_{n}$:

$P_{n,k,L}=\\{z\in{\mathbb{Z}^{d}}\>:\>I_{z}^{n,k,L}\subseteq
C_{n}\\}\quad\text{and}\quad
Q_{n,k,L}=\\{z\in{\mathbb{Z}^{d}}\>:\>I_{z}^{n,k,L}\cap
C_{n}\neq\emptyset\\}.$

We let the number of such inner and outer approximating boxes be denoted by
$p_{n,k,L}=\lvert P_{n,k,L}\rvert$ and $q_{n,k,L}=\lvert Q_{n,k,L}\rvert$,
respectively. Note that, by construction, $p_{n,k,L}\leq k/L^{d}$ and
$q_{n,k,L}\geq k/L^{d}$ for values of $n$ large enough relative to $k$. When
proving our results we essentially divide $I_{z}^{n,k,L}$ into cubes of side-
length $L$, $C_{L}(v)$ for $v\in(L\mathbb{Z})^{d}$. To this end, define the
grid points of $I_{z}^{n,k,L}$ by
$J_{z}^{n,k,L}=I_{z}^{n,k,L}\cap(L\mathbb{Z})^{d}$, and note that $\lvert
J_{z}^{n,k,L}\rvert=(t_{n,k,L})^{d}/L^{d}\sim\lvert C_{n}\rvert/k$ as
$n\to\infty$. Lastly, we define a set of grid points which, when continuously
filled with cubes $C_{L}(v)$, approximate $C_{n}$ from the inside and outside,
respectively:

$D_{n,k,L}^{-}=\bigcup_{z\in P_{n,k,L}}J_{z}^{n,k,L}\quad\text{and}\quad
D_{n,k,L}^{+}=\bigcup_{z\in Q_{n,k,L}}J_{z}^{n,k,L},$

which then satisfy

$\bigcup_{z\in D_{n,k,L}^{-}}C_{L}(z)\subseteq C_{n}\subseteq\bigcup_{z\in
D_{n,k,L}^{+}}C_{L}(z).$ (24)

######  Theorem 6.

Let $(C_{n})_{n\in\mathbb{N}}$ satisfy Assumption 3. Then,

  1. (i)

for all $L\in\mathbb{N}$ the sequences $p_{n,k,L}$ and $q_{n,k,L}$, defined
above, satisfy that

$\liminf_{n\to\infty}p_{n,k,L}\sim\frac{k}{L^{d}}\qquad\text{and}\qquad\limsup_{n\to\infty}q_{n,k,L}\sim\frac{k}{L^{d}}$

as $k\to\infty$,

  2. (ii)

for each $k$, $L$ and $n$ with $n$ large enough relative to $k$, it holds that
$D_{n,k,L}^{+}\subseteq K_{n,k,L}$, where $K_{n,k,L}$ is the cube

$K_{n,k,L}=\bigcup_{z\in N_{k,L}}J_{z}^{n,k,L},$

and $N_{k,L}$ is on the form
$N_{k,L}=[-c_{k,L}\>,\>c_{k,L}]^{d}\cap\mathbb{Z}^{d}$ for some
$c_{k,L}<\infty$,

  3. (iii)

for all $L\in\mathbb{N}$ there exists $c_{L}<\infty$ such that for all
$n\in\mathbb{N}$

$D_{n,k,L}^{+}\subseteq K_{n,L}=\big{[}-c_{L}\cdot\lvert
C_{n}\rvert^{1/d}\>,\>c_{L}\cdot\lvert
C_{n}\rvert^{1/d}\big{]}^{d}\cap(L\mathbb{Z})^{d}.$

We note that the discrete index set $N_{k,L}$ in (ii) above does not depend on
$n$. In particular, for any fixed $k$ and $L$, all $D_{n,k,L}^{+}$ are
contained in the same finite collection of (increasing) sets of grid points
$J_{z}^{n,k,L}$.

###### Proof.

For each $n,k,L\in\mathbb{N}$ define $\tilde{t}_{n,k,L}=L\,\bigl{(}\lvert
C_{n}\rvert/k\bigr{)}^{1/d}$. Turning to Corollary 2, we obtain

$\lvert\partial C_{n}\oplus
B(\tilde{t}_{n,k,L})\rvert\leq\sum_{i=1}^{p}\lvert\partial C_{n,i}\oplus
B(\tilde{t}_{n,k,L})\rvert\leq
2\sum_{j=0}^{d-1}\omega_{d-j}\Bigl{(}\sum_{i=1}^{p}V_{j}(C_{n,i})\Bigr{)}\tilde{t}_{n,k,L}^{d-j},$

which clearly implies that

$\frac{L^{d}}{k}\frac{1}{\tilde{t}_{n,k,L}^{d}}\lvert\partial C_{n}\oplus
B(\tilde{t}_{n,k,L})\rvert\leq
2\sum_{j=0}^{d-1}\omega_{d-j}\frac{\sum_{i=1}^{p}V_{j}(C_{n,i})}{\lvert
C_{n}\rvert^{j/d}}\Bigl{(}\frac{L}{\sqrt[d]{k}}\Bigr{)}^{d-j}.$

By (11) of Assumption 3 we conclude the convergence

$\limsup_{n\to\infty}\frac{L^{d}}{k}\frac{1}{\tilde{t}_{n,k,L}^{d}}\lvert\partial
C_{n}\oplus B(\tilde{t}_{n,k,L})\rvert\to 0$

as $k\to\infty$. In fact, replacing $L$ by $r\cdot L$ for any $r\in\mathbb{N}$
and realizing that $\tilde{t}_{n,k,r\cdot L}=r\,\tilde{t}_{n,k,L}$ show that
also

$\limsup_{n\to\infty}\frac{L^{d}}{k}\frac{1}{\tilde{t}_{n,k,L}^{d}}\lvert\partial
C_{n}\oplus B(r\,\tilde{t}_{n,k,L})\rvert\to 0$

as $k\to\infty$ for all $r\in\mathbb{N}$. Since
$t_{n,k,L}\leq\tilde{t}_{n,k,L}$, and $t_{n,k,L}\sim\tilde{t}_{n,k,L}$ as
$n\to\infty$, we also have

$\limsup_{n\to\infty}\frac{L^{d}}{k}\frac{1}{t_{n,k,L}^{d}}\lvert\partial
C_{n}\oplus B(r\,t_{n,k,L})\rvert\to 0$ (25)

as $k\to\infty$ for all $r\in\mathbb{N}$. Since $t_{n,k,L}$ denotes the side-
length of the cubes $I_{z}^{n,k,L}$, we can find an integer $r\in\mathbb{N}$
independent of $n,k,L$ such that $I_{z}^{n,k,L}\subseteq\partial C_{n}\oplus
B(r\,t_{n,k,L})$ whenever $I_{z}^{n,k,L}\cap\partial C_{n}\neq\emptyset$.
Using that $\lvert I_{z}^{n,k,L}\rvert=t_{n,k,L}^{d}$, we then find

$\displaystyle\frac{L^{d}}{k}(q_{n,k,L}-p_{n,k,L})$
$\displaystyle\leq\frac{L^{d}}{k}\frac{1}{t_{n,k,L}^{d}}\Big{\lvert}\bigcup_{I_{z}^{n,k,L}\cap\partial
C_{n}\neq\emptyset}I_{z}^{n,k,L}\Big{\rvert}$
$\displaystyle\leq\frac{L^{d}}{k}\frac{1}{t_{n,k,L}^{d}}\lvert\partial
C_{n}\oplus B(r\,t_{n,k,L})\rvert.$

Together with (25) and the fact that $p_{n,k,L}\leq k/L^{d}\leq q_{n,k,L}$ for
$n$ large enough relative to $k$, this proves statement (i).

We now show statement (ii). For each fixed $k$ and $L$, the sequence
$(q_{n,k,L})_{n}$ is bounded by a constant $c_{k,L}$ for $n$ large enough
relative to $k$. Furthermore, as the origin $0\in C_{n}$, it must also satisfy
$0\in Q_{n,k,L}$. By assumption, $C_{n}$ is connected, and thus $Q_{n,k,L}$
consists of at most $c_{k,L}$ points that are pairwise neighbors. In
particular,

$Q_{n,k,L}\subseteq[-c_{k,L}\>,\>c_{k,L}]^{d}\cap{\mathbb{Z}^{d}},$

which implies the desired result. Finally, (iii) follows from (ii). ∎

Before moving on to proving our remaining main results, we recall a lemma from
[27] which will be useful throughout the paper:

######  Lemma 6 ([27, Lemma 1]).

Let $C$ be a convex body and let $r\geq 0$ be fixed. For any decreasing
function $g:[0,\infty)\to[0,\infty)$ satisfying
$\int_{0}^{\infty}g(x)x^{d-1}\mathrm{d}x<\infty$, it holds that

$\displaystyle\int_{(C\oplus B(r))^{c}}\sup_{v\in C}g(\lvert
v-u\rvert)\mathrm{d}u$
$\displaystyle=\sum_{j=0}^{d-1}\mu_{j}V_{j}(C)\int_{r}^{\infty}g(x)x^{d-j-1}\mathrm{d}x\qquad\text{and}$
(26) $\displaystyle\int_{C}\sup_{v\in(C\oplus B(r))^{c}}g(\lvert
v-u\rvert)\mathrm{d}u$
$\displaystyle\leq\sum_{j=0}^{d-1}\mu_{j}V_{j}(C)\int_{r}^{\infty}g(x)x^{d-j-1}\mathrm{d}x$
(27)

for constants $\mu_{j}$, $j=0,\dots,d-1$, independent of $C$.

##  6 Extremal results

In this section we prove the main extreme value results Theorems 3 and 4
assuming either Assumption 1 or Assumption 2. Throughout this section we work
under either of the assumptions, which in particular implies that the minimal
Assumption M is satisfied.

In proving the extreme value results, we define a field $(Z_{v}^{(t)})$ with
distribution which, when $t\to\infty$, approximates the distribution of
$(Z_{v})$ defined in (19). For fixed $t>0$ we let
$f^{(t)}(\cdot)=f(\cdot)\boldsymbol{1}_{\\{\lvert\cdot\rvert<t\\}}$ and define
the field

$Z_{v}^{(t)}=\int_{\mathbb{R}^{d}}f^{(t)}(v-u)\Lambda_{1}(\mathrm{d}u)=\int_{\\{\lvert
v-u\rvert<t\\}}f(v-u)\Lambda_{1}(\mathrm{d}u).$ (28)

Note that $Z_{v}^{(t)}$ and $Z_{v^{\prime}}^{(t)}$ are independent for $\lvert
v-v^{\prime}\rvert>2t$, and furthermore $Z_{v}^{(t)}\leq Z_{v}$ for any $t$.
We will explicitly use this to bound the asymptotic distribution function of
$\sup_{C_{n}}Z_{v}$ with that of $\sup_{C_{n}}Z_{v}^{(t)}$, which is more
easily managed due to independence. In proving our main results we will only
need $t$ to take a fixed finite value and the value $\infty$. Many of the
lemmas below will be formulated in terms of any finite or infinite choice of
$t$, with $t=\infty$ corresponding to the field $Z_{v}^{(\infty)}=Z_{v}$. As
it turns out, the final result will not depend on the fixed finite $t$.

######  Lemma 7.

For all $0<t\leq\infty$,

$\int_{\mathbb{R}^{d}}\sup_{v\in
C_{L}(0)}\bigl{(}f^{(t)}(v-u)\bigr{)}^{\alpha}\mathrm{d}u\sim L^{d}$

as $L\to\infty$.

###### Proof.

Since $f\leq 1$ with $f(0)=1$ and $\lvert C_{L}(0)\rvert=L^{d}$, it is seen
that the claim follows once we show that

$\int_{(C_{L}(0))^{c}}\sup_{v\in
C_{L}(0)}\bigl{(}f^{(t)}(v-u)\bigr{)}^{\alpha}\mathrm{d}u=o(L^{d})$

as $L\to\infty$. Since $f^{(t)}(\cdot)$ is increasing in $t$ and bounded by
$g(\lvert\cdot\rvert)$, this is satisfied for all $t$ if

$\int_{(C_{L}(0))^{c}}\sup_{v\in C_{L}(0)}g^{\alpha}(\lvert
v-u\rvert)\mathrm{d}u=o(L^{d})$ (29)

as $L\to\infty$. We now show this claim.

Since the cube $C_{L}(0)\subseteq{\mathbb{R}^{d}}$ is a convex body and
$g^{\alpha}$ is decreasing, we find by Lemma 6, equation (26), and the
homogeneity of the intrinsic volumes that there are constants $\mu_{j}$
independent of $C_{L}(0)$ such that

$\displaystyle\int_{(C_{L}(0))^{c}}\sup_{v\in C_{L}(0)}g^{\alpha}(\lvert
v-u\rvert)\mathrm{d}u$
$\displaystyle=\sum_{j=0}^{d-1}\mu_{j}V_{j}(C_{L}(0))\int_{0}^{\infty}g^{\alpha}(x)x^{d-j-1}\mathrm{d}x$
$\displaystyle=\sum_{j=0}^{d-1}\mu_{j}V_{j}(C_{1}(0))L^{j}\int_{0}^{\infty}g^{\alpha}(x)x^{d-j-1}\mathrm{d}x.$
(30)

By assumption, $g^{\alpha}(\lvert\cdot\rvert)$ is integrable, or equivalently,

$\int_{0}^{\infty}g^{\alpha}(x)x^{d-1}\mathrm{d}x<\infty,$

and we conclude that the integral in (30) is finite for all $j=0,\dots,d-1$.
Thus, dividing (30) by $L^{d}$ and letting $L\to\infty$ show that (29) is
satisfied. This concludes the proof. ∎

The lemma below follows by arguments similar to that of Theorem 1 realizing
that $Z_{v}^{(t)}$ is almost surely bounded on bounded sets.

######  Lemma 8.

Let $B\subseteq{\mathbb{R}^{d}}$ be a fixed bounded set. Then, for all
$0<t\leq\infty$,

$\displaystyle\mathbb{P}\bigl{(}\sup_{v\in
B}Z_{v}^{(t)}>x\bigr{)}\sim\rho((x,\infty))\int_{{\mathbb{R}^{d}}}\sup_{v\in
B}\bigl{(}f^{(t)}(v-u)\bigr{)}^{\alpha}\mathrm{d}u$

as $x\to\infty$. In particular, $\sup_{v\in
B}Z_{v}^{(t)}\in\mathcal{RV}_{\alpha}$.

Since $\rho$ is regularly varying and in particular in the maximum domain of
attraction of the Fréchet distribution (i.e. it satisfies (12)), Lemma 8 in
particular implies that

$\lvert C_{n}\rvert\mathbb{P}\bigl{(}\sup_{v\in
B}Z_{v}^{(t)}>a_{n}x\bigr{)}\to
x^{-\alpha}\rho((1,\infty))\int_{{\mathbb{R}^{d}}}\sup_{v\in
B}\bigl{(}f^{(t)}(v-u)\bigr{)}^{\alpha}\mathrm{d}u$ (31)

for any fixed set $B\subseteq{\mathbb{R}^{d}}$ and any $t>0$.

To ease notation in the remainder of the paper, we define $\tau_{L}^{(t)}$ as

$\tau_{L}^{(t)}=x^{-\alpha}\rho((1,\infty))\int_{{\mathbb{R}^{d}}}\sup_{v\in
C_{L}(0)}\bigl{(}f^{(t)}(v-u)\bigr{)}^{\alpha}\mathrm{d}u$

for any $L\in\mathbb{N}$, $t>0$ and fixed $x>0$. For $t=\infty$ we write
$\tau_{L}=\tau_{L}^{(\infty)}$. Note that, by Lemma 7, $\tau_{L}^{(t)}$
satisfies

$\frac{\tau_{L}^{(t)}}{L^{d}}\to x^{-\alpha}\rho((1,\infty))$ (32)

as $L\to\infty$.

The lemma below will be used repeatedly in this section. It ensures in
particular that the field $(Y_{v})_{v}$ has moments of all orders and thus is
of minor importance when determining the extremal behavior of the Lévy-driven
field $(X_{v})_{v}$.

######  Lemma 9.

Let the field $(Y_{v})_{v\in{\mathbb{R}^{d}}}$ be given by (20). Then

$\mathbb{E}\exp\bigl{(}\epsilon\sup_{v\in C_{L}(0)}Y_{v}\bigr{)}<\infty$ (33)

for all $\epsilon>0$ and all $L\in\mathbb{N}$.

###### Proof.

Write $Y_{v}=Y_{v}^{C}+Y_{v}^{-}$ as the independent decomposition given in
Section 4. Utilizing the fact that $(Y_{v}^{-})$ is a non-positive field, it
is clear that (33) follows once we show that

$\mathbb{E}\exp\bigl{(}\epsilon\sup_{v\in C_{L}(0)}Y_{v}^{C}\bigr{)}<\infty.$
(34)

for all $\epsilon>0$ and all $L\in\mathbb{N}$. Let $T\subset C_{L}(0)$ be the
countable separating dense subset associated to the separable field
$(Y_{v}^{C})_{v\in C_{L}(0)}$. By considerations as in [19] and [28], the
countable field $(Y_{v}^{C})_{v\in T}$ is infinitely divisible with a
characteristic function given as in [6, Eq. (1.1)]. Its Lévy measure $\nu$
defined on $\mathbb{R}^{T}$ is given as follows: Let
$H:{\mathbb{R}^{d}}\times[-1,1]\to\mathbb{R}^{T}$ be given as

$H(u,x)=(xf(v-u))_{v\in T},$

and let $\nu=(m\otimes\rho_{[-1,1]})\circ H^{-1}$ be the image-measure of
$m\otimes\rho_{[-1,1]}$. Here $m$ denotes Lebesgue measure and $\rho_{[-1,1]}$
is the Lévy measure of the basis $\Lambda_{2,C}$ defining $Y_{v}^{C}$. By
Lemmas 3 and 5 we find that

$\mathbb{P}(\sup_{v\in C_{L}(0)}\lvert
Y_{v}^{C}\rvert<\infty)=\mathbb{P}(\sup_{v\in T}\lvert
Y_{v}^{C}\rvert<\infty)=1,$

and, by construction, $\nu(\\{z\in\mathbb{R}^{T}\>:\>\sup_{T}\lvert
z_{v}\rvert>1\\})=0$. By an application of [6, Lemma 2.1] we conclude that
$\mathbb{E}\exp(\epsilon\sup_{v\in C_{L}(0)}\lvert Y_{v}^{C}\rvert)<\infty$
for all $\epsilon$, which clearly implies (34). ∎

The following theorem, which shows that the field
$(Y_{v})_{v\in{\mathbb{R}^{d}}}$ is ergodic, is exactly Corollary 4 of [27].
For a greater exposition of spatial ergodicity, we refer to their Section 5.
In the theorem and in the remainder of the paper, we write $J_{z}$ for
$J_{z}^{n,k,L}$ to avoid too heavy notation. Furthermore, we recall the set
$N_{k,L}$ initially defined in Theorem 6(ii).

######  Theorem 7 ([27, Corollary 5]).

Let the field $(Y_{v})_{v\in{\mathbb{R}^{d}}}$ be given by (20), and let $h$
be a function satisfying $\mathbb{E}\lvert h((Y_{u})_{u\in
C_{L}(0)})\rvert<\infty$. For all $z\in N_{k,L}$ it then holds that

$\frac{1}{\lvert J_{z}\rvert}\sum_{v\in J_{z}}h((Y_{u+v})_{u\in
C_{L}(0)})\to\mathbb{E}h((Y_{u})_{u\in C_{L}(0)})$

almost surely as $n\to\infty$. The result also holds true if $J_{z}$ is
replaced with a subset of $J_{z}$ in the shape of a box, which increases in
size asymptotically as $J_{z}$.

From now on we let $x>0$ be fixed, and we let the sequence $(a_{n})$ be the
norming constants of $\rho$ satisfying (12).

######  Lemma 10.

Let $(Z_{v}^{(t)})_{v}$ and $(Y_{v})_{v}$ be given by (28) and (20),
respectively. Then, for almost all realizations $(y_{v})_{v}$ of
$(Y_{v})_{v}$, it holds for all $z\in N_{k,L}$ and for all $0<t\leq\infty$
that

$\frac{\lvert C_{n}\rvert}{\lvert J_{z}\rvert}\sum_{v\in
J_{z}}\mathbb{P}\bigl{(}\sup_{u\in
C_{L}(v)}(Z_{u}^{(t)}+y_{u})>a_{n}x\bigr{)}\to\tau_{L}^{(t)}$ (35)

as $n\to\infty$. The result also holds true if $J_{z}$ is replaced with a
subset of $J_{z}$ in the shape of a box, which increases in size
asymptotically as $J_{z}$.

###### Proof.

We only show the convergence (35) as the expression for $J_{z}$ replaced by an
asymptotically size-equivalent box follows identically.

Throughout this proof we fix $\beta>\alpha$ such that

$\mathbb{E}\bigl{(}\sup_{v\in C_{L}(0)}Y_{v}\bigr{)}_{+}^{\beta}<\infty,$ (36)

which is possible due to Lemma 9.

Let $F_{L,t}$ denote the distribution of $\sup_{u\in C_{L}(v)}Z_{u}^{(t)}$ for
$t>0$, and note that $F_{L,t}\in\mathcal{RV}_{\alpha}$ by Lemma 8. By (31),

$\lvert C_{n}\rvert\overline{F}_{L,t}(a_{n}x)\to\tau_{L}^{(t)}$

as $n\to\infty$, where $\overline{F}_{L,t}$ is the tail of $F_{L,t}$. Since in
particular $F_{L,t}\in\mathcal{S}$, the convergence

$\frac{\overline{F}_{L,t}(a_{n}x-y)}{\overline{F}_{L,t}(a_{n}x)}\to 1$

as $n\to\infty$ is uniform for all $\lvert y\rvert\leq N$, for all
$N\in\mathbb{N}$; see e.g. [15, Definition 2.1]. Hence, with
$y^{*}(v)=\sup_{C_{L}(v)}y_{u}$, we find by an application of Lemma 1 and
Theorem 7, utilizing (36), that

$\displaystyle\frac{\lvert C_{n}\rvert}{\lvert J_{z}\rvert}\sum_{v\in
J_{z}}\overline{F}_{L,t}(a_{n}x-y^{*}(v))$ $\displaystyle\leq\frac{\lvert
C_{n}\rvert\overline{F}_{L,t}(a_{n}x)}{\lvert J_{z}\rvert}\sum_{v\in
J_{z}}\Bigl{(}\frac{\overline{F}_{L,t}(a_{n}x-y^{*}(v))}{\overline{F}_{L,t}(a_{n}x)}\boldsymbol{1}_{\\{\lvert
y^{*}(v)\rvert\leq N\\}}$ $\displaystyle\hskip
99.58464pt+C\bigl{(}K+(y^{*}(v))_{+}^{\beta}\bigr{)}\boldsymbol{1}_{\\{\lvert
y^{*}(v)\rvert>N\\}}\Bigr{)}$
$\displaystyle\to\tau_{L}^{(t)}\mathbb{E}\Bigl{(}\boldsymbol{1}_{\\{\lvert\sup_{C_{L}(0)}Y_{v}\rvert\leq
N\\}}+C\bigl{(}K+(\sup_{C_{L}(0)}Y_{v})_{+}^{\beta}\bigr{)}\boldsymbol{1}_{\\{\lvert\sup_{C_{L}(0)}Y_{v}\rvert>N\\}}\Bigr{)}$

as $n\to\infty$, almost surely. Letting $N\to\infty$ shows by monotone
convergence that

$\limsup_{n\to\infty}\frac{\lvert C_{n}\rvert}{\lvert J_{z}\rvert}\sum_{v\in
J_{z}}\overline{F}_{L,t}(a_{n}x-y^{*}(v))\leq\tau_{L}^{(t)}$

almost surely. By similar arguments we find that also

$\liminf_{n\to\infty}\frac{\lvert C_{n}\rvert}{\lvert J_{z}\rvert}\sum_{v\in
J_{z}}\overline{F}_{L,t}(a_{n}x-\overline{y}(v))\geq\tau_{L}^{(t)},$

where we used the notation $\overline{y}(v)=\inf_{C_{L}(v)}y_{u}$. As

$\overline{F}_{L,t}(a_{n}x-\overline{y}(v))\leq\mathbb{P}\bigl{(}\sup_{u\in
C_{L}(v)}(Z_{u}^{(t)}+y_{u})>a_{n}x\bigr{)}\leq\overline{F}_{L,t}(a_{n}x-y^{*}(v)),$

the convergence (35) follows. ∎

We say that two subsets $A,B$ of ${\mathbb{Z}^{d}}$ or ${\mathbb{R}^{d}}$ are
$r$-separated if $\mathrm{dist}(A,B)=\inf\\{\lvert a-b\rvert\>:\>a\in A,b\in
B\\}\geq r$ and there are two disjoint sets
$A^{\prime},B^{\prime}\subseteq{\mathbb{R}^{d}}$, both of which are connected,
such that $A\subseteq A^{\prime}$ and $B\subseteq B^{\prime}$. In particular
in the coming lemmas, we consider subsets of $(L\mathbb{Z})^{d}$ and require
that these are $\gamma_{n}$-separated, where, even though we do not explicitly
state it, we assume that $\gamma_{n}\geq L$.

Before proceeding, we define the last piece of convenient notation. For a set
$A\subseteq(L\mathbb{Z})^{d}$ and a deterministic field $(y_{v})_{v}$ we let

$M_{L,y}^{(t)}(A)=\max_{z\in A}\sup_{u\in C_{L}(z)}(Z^{(t)}_{u}+y_{u}),$

where we simply write $M_{L,y}$ for $M_{L,y}^{(\infty)}$, that is, when
considering the field $(Z_{v})$.

######  Lemma 11.

Let $(Z_{v})_{v}$ and $(Y_{v})_{v}$ be given by (19) and (20), respectively.
There is a sequence $\gamma_{n}=o(\sqrt[d]{\lvert C_{n}\rvert})$ such that for
all $\gamma_{n}$-separated sets $A,B\subseteq
K_{n,L}\subseteq(L\mathbb{Z})^{d}$, where at least one is a box, it holds that

$\big{\lvert}\mathbb{P}(M_{L,y}(A\cup B)\leq a_{n}x)-\mathbb{P}(M_{L,y}(A)\leq
a_{n}x)\mathbb{P}(M_{L,y}(B)\leq a_{n}x)\big{\rvert}\leq\alpha_{y,n,L},$

where $\alpha_{y,n,L}\to 0$ as $n\to\infty$ for all $L\in\mathbb{N}$ and
almost all realizations $(y_{v})_{v}$ of $(Y_{v})_{v}$.

######  Proof.

We choose $\gamma_{n}/2=\lvert C_{n}\rvert^{1/d-\delta}$ for some
$0<\delta<\min\\{\frac{1}{d\alpha},\frac{1}{2d}\\}$. In particular,
$\gamma_{n}=o(\sqrt[d]{\lvert C_{n}\rvert})$ as $n\to\infty$. Let the sets $A$
and $B$ be given as in the lemma and define

$\mathcal{A}=\bigcup_{v\in
A}C_{L}(v)\qquad\text{and}\qquad\mathcal{B}=\bigcup_{v\in
B}C_{L}(v)\qquad\text{and}\qquad\mathcal{K}_{n,L}=\bigcup_{v\in
K_{n,L}}C_{L}(v).$

Throughout the proof, we assume that $A$ is a box, and thus $\mathcal{A}$ is a
continuous box and in particular a convex body. Recall that $B(r)$ denotes the
closed ball in ${\mathbb{R}^{d}}$ of radius $r\geq 0$ with center
$0\in{\mathbb{R}^{d}}$. Define $\mathcal{A}_{n}=\mathcal{A}\oplus
B(\gamma_{n}/2)$ and note that
$\mathcal{B}\subseteq\mathcal{B}_{n}=\bigl{(}\mathcal{A}\oplus
B(\gamma_{n})\bigr{)}^{c}=\bigl{(}\mathcal{A}_{n}\oplus
B(\gamma_{n}/2)\bigr{)}^{c}$, since $\mathcal{A}$ and $\mathcal{B}$ are
$\gamma_{n}$-separated (strictly speaking they are only
$(\gamma_{n}-L)$-separated, however, this is equivalent as $n\to\infty$).

For all $v\in\mathcal{A}$ let

$Z_{v}^{A}=\int_{\mathcal{A}_{n}}f(v-u)\Lambda_{1}(\mathrm{d}u),\qquad\text{and}\qquad\overline{Z}_{v}^{A}=\int_{\mathcal{A}_{n}^{c}}f(v-u)\Lambda_{1}(\mathrm{d}u).$

Similarly, for all $v\in\mathcal{B}$,

$Z_{v}^{B}=\int_{\mathcal{A}_{n}^{c}}f(v-u)\Lambda_{1}(\mathrm{d}u),\qquad\text{and}\qquad\overline{Z}_{v}^{B}=\int_{\mathcal{A}_{n}}f(v-u)\Lambda_{1}(\mathrm{d}u).$

Since $\Lambda_{1}$ is a positive measure all $Z_{v}^{A}$,
$\overline{Z}_{v}^{A}$, $Z_{v}^{B}$ and $\overline{Z}_{v}^{B}$ are non-
negative, and we have

$\sup_{v\in\mathcal{A}}\overline{Z}^{A}_{v}\leq\int_{\mathcal{A}_{n}^{c}}\sup_{v\in\mathcal{A}}f(v-u)\Lambda_{1}(\mathrm{d}u)$

and

$\sup_{v\in\mathcal{B}}\overline{Z}^{B}_{v}\leq\int_{\mathcal{A}_{n}}\sup_{v\in\mathcal{B}}f(v-u)\Lambda_{1}(\mathrm{d}u)\,.$

If $\alpha<1$, we choose $\gamma<\alpha<\beta$ such that
$\frac{\gamma}{\beta}>\frac{d-1}{d}$ and such that the upper bound $g$ has
$\gamma$-moment (in accordance with Assumption M). If $\alpha\geq 1$, we let
$\gamma=1$ and choose $\beta>\alpha$ such that $\frac{1}{\beta}>d\delta$
(recall that $\frac{1}{\alpha}>d\delta$). Let $\epsilon_{n}=\lvert
C_{n}\rvert^{1/\beta}$, which clearly satisfies $\epsilon_{n}\to\infty$, and
define the events $S_{n}^{A}$ and $S_{n}^{B}$ by

$S^{A}_{n}=\bigl{(}\sup_{v\in\mathcal{A}}\overline{Z}^{A}_{v}\leq\epsilon_{n}\bigr{)}\qquad\text{and}\qquad
S^{B}_{n}=\bigl{(}\sup_{v\in\mathcal{B}}\overline{Z}^{B}_{v}\leq\epsilon_{n}\bigr{)}.$

First, we show that the probability $\mathbb{P}((S^{A}_{n})^{c})$ has an
$A$-independent upper bound tending to $0$ as $n\to\infty$. For a fixed $n$,
we define the sequence $(r_{m})_{m\geq 1}$ by $r_{1}=\gamma_{n}/2$ and
recursively for $m>1$ by

$\big{\lvert}(\mathcal{A}\oplus B(r_{m+1}))\setminus(\mathcal{A}\oplus
B(r_{m}))\big{\rvert}=1.$

For each $m\geq 1$ define the sets

$E_{m}=(\mathcal{A}\oplus B(r_{m+1}))\setminus(\mathcal{A}\oplus B(r_{m}))$

of unit volume, and note that their union equals $\mathcal{A}_{n}^{c}$ by
construction. Utilizing that $\Lambda_{1}$ is a positive measure, that $g$ is
decreasing and that $g(\lvert\cdot\rvert)$ constitutes an upper bound to
$f(\cdot)$, we find

$\int_{\mathcal{A}_{n}^{c}}\sup_{v\in\mathcal{A}}f(v-u)\Lambda_{1}(\mathrm{d}u)\leq\sum_{m=1}^{\infty}g(r_{m})\Lambda_{1}(E_{m}).$

Let $\Lambda_{1}^{\prime}$ be the spot variable of the Lévy basis
$\Lambda_{1}$, that is, a random variable equivalent in distribution to
$\Lambda_{1}(S)$ with $S\subseteq{\mathbb{R}^{d}}$ satisfying $\lvert
S\rvert=1$. Using Markov’s inequality we now find that

$\mathbb{P}((S^{A}_{n})^{c})\leq\frac{1}{\epsilon_{n}^{\gamma}}\mathbb{E}\Bigl{(}\sum_{m=1}^{\infty}g(r_{m})^{\gamma}\Lambda_{1}(E_{m})^{\gamma}\Bigr{)}=\frac{1}{\epsilon_{n}^{\gamma}}\mathbb{E}((\Lambda_{1}^{\prime})^{\gamma})\sum_{m=1}^{\infty}g(r_{m})^{\gamma}.$
(37)

Note that the underlying Lévy measure $\rho_{1}$ has finite $\gamma$-moment
(see e.g. [10, Proposition A3.8]), and thus also $\Lambda_{1}^{\prime}$ has
finite $\gamma$-moment (cf. [24, Theorem 25.3]). Showing that
$\mathbb{P}((S^{A}_{n})^{c})\to 0$ uniformly in $A$ therefore amounts to
showing that the sum above is of order $o(\epsilon_{n}^{\gamma})$ uniformly in
$A$. Clearly, it is enough to consider the sum starting at index $m=2$, which
is more convenient. Using the construction of the sets $E_{m}$ and the fact
that $g$ is decreasing, Lemma 6 and the construction of $\mathcal{K}_{n,L}$
implies the existence of $n$\- and $A$-independent constants $\mu_{j}$ and
$\tilde{\mu}_{j}$ such that

$\displaystyle\sum_{m=2}^{\infty}g(r_{m})^{\gamma}$
$\displaystyle=\sum_{m=2}^{\infty}\int_{E_{m-1}}g(r_{m})^{\gamma}\mathrm{d}u$
(38)
$\displaystyle\leq\sum_{m=1}^{\infty}\int_{E_{m}}\sup_{v\in\mathcal{A}}g(\lvert
v-u\rvert)^{\gamma}\mathrm{d}u$
$\displaystyle=\int_{\mathcal{A}_{n}^{c}}\sup_{v\in\mathcal{A}}g(\lvert
v-u\rvert)^{\gamma}\mathrm{d}u$
$\displaystyle=\sum_{j=0}^{d-1}\mu_{j}V_{j}(\mathcal{A})\int_{\gamma_{n}/2}^{\infty}g(x)^{\gamma}x^{d-j-1}\mathrm{d}x$
$\displaystyle\leq\sum_{j=0}^{d-1}\mu_{j}V_{j}(\mathcal{K}_{n,L})\int_{\gamma_{n}/2}^{\infty}g(x)^{\gamma}x^{d-j-1}\mathrm{d}x$
$\displaystyle\leq\sum_{j=0}^{d-1}\tilde{\mu}_{j}\lvert
C_{n}\rvert^{j/d}\int_{\gamma_{n}/2}^{\infty}g(x)^{\gamma}x^{d-j-1}\mathrm{d}x.$

For each $j=0,\dots,d-1$ we have

$\displaystyle\lvert
C_{n}\rvert^{j/d}\int_{\gamma_{n}/2}^{\infty}g(x)^{\gamma}x^{d-j-1}\,\mathrm{d}x$
(39) $\displaystyle\leq\lvert
C_{n}\rvert^{(d-1)/d}\int_{\gamma_{n}/2}^{\infty}g(x)^{\gamma}\,\mathrm{d}x+\int_{\gamma_{n}/2}^{\infty}g(x)^{\gamma}x^{d-1}\,\mathrm{d}x.$

The second integral on the right hand side clearly tends to $0$, and in
particular it is of order $o(\epsilon_{n}^{\gamma})$ as $n\to\infty$. In the
case $\alpha<1$, the integrability of $g^{\gamma}$ and the construction of
$\epsilon_{n}$, gives that the first term on the right hand side is of order
$o(\epsilon_{n}^{\gamma})$ as $n\to\infty$. In the case $\alpha\geq 1$, where
$\gamma=1$, we have for the first term that

$\displaystyle\frac{1}{\epsilon_{n}}\lvert
C_{n}\rvert^{(d-1)/d}\int_{\gamma_{n}/2}^{\infty}g(x)\,\mathrm{d}x$
$\displaystyle\leq\lvert
C_{n}\rvert^{d\delta-1/\beta}\Bigl{(}\frac{\gamma_{n}}{2}\Bigr{)}^{d-1}\int_{\gamma_{n}/2}^{\infty}g(x)\,\mathrm{d}x$
$\displaystyle\leq\lvert
C_{n}\rvert^{d\delta-1/\beta}\int_{\gamma_{n}/2}^{\infty}g(x)x^{d-1}\,\mathrm{d}x$

which tends to 0, since we have chosen $\beta,\delta$ with $1/\beta>d\delta$.
Thus, combining (37)–(39), we have shown for all $\alpha>0$ that
$\mathbb{P}((S_{n}^{A})^{c})$ has an upper bound independent of $A$ that tends
to 0 as $n\to\infty$.

The procedure for evaluating $\mathbb{P}((S_{n}^{B})^{c})$ is similar: Let $n$
be fixed and define for each $r>0$ the set

$\mathcal{A}_{n}^{-r}=\mathcal{A}_{n}\setminus(\partial\mathcal{A}_{n}\oplus
B(r)),$

recalling that $\partial\mathcal{A}_{n}$ denotes the boundary of
$\mathcal{A}_{n}$. Hence, $\mathcal{A}_{n}^{-r}$ is the set $\mathcal{A}_{n}$
without the strip of thickness $r$ closest to its boundary. Note that
$\mathcal{A}_{n}^{-r}$ is decreasing in $r$ and that it will be empty for $r$
large enough. Define for all $m\geq 0$ the sequence $s_{m}=r_{m}-\gamma_{n}/2$
with $(r_{m})$ as above, and define for $m\geq 1$ the sets

$F_{m}=\mathcal{A}_{n}^{-s_{m}}\setminus\mathcal{A}_{n}^{-s_{m+1}}.$

As a consequence of the proof of Lemma 6, equation (27) (see [27]), we obtain
that

$\displaystyle\lvert F_{m}\rvert$
$\displaystyle=\lvert\mathcal{A}_{n}^{-s_{m}}\rvert-\lvert\mathcal{A}_{n}^{-s_{m+1}}\rvert$
$\displaystyle\leq\lvert\mathcal{A}_{n}\oplus
B(s_{m+1})\rvert-\lvert\mathcal{A}_{n}\oplus B(s_{m})\rvert$
$\displaystyle=\lvert\mathcal{A}\oplus
B(r_{m+1})\rvert-\lvert\mathcal{A}\oplus B(r_{m})\rvert$ $\displaystyle=\lvert
E_{m}\rvert,$

with $E_{m}$ defined as above. Since $\Lambda_{1}$ is non-negative, we e.g.
have that
$\mathbb{E}(\Lambda_{1}(F_{m})^{\gamma})\leq\mathbb{E}(\Lambda_{1}(E_{m})^{\gamma})$,
and then, similarly to (37), we find

$\mathbb{P}((S^{B}_{n})^{c})\leq\frac{1}{\epsilon_{n}^{\gamma}}\mathbb{E}((\Lambda_{1}^{\prime})^{\gamma})\sum_{m=1}^{\infty}g(r_{m})^{\gamma}.$

We can proceed as above to find an upper bound that tends to 0 and is
independent of $B$.

Identically to arguments applied in the proof of [27, Lemma 6] we can derive
that

$\displaystyle\big{\lvert}\mathbb{P}(M_{L,y}(A\cup B)\leq
a_{n}x)-\mathbb{P}(M_{L,y}(A)\leq a_{n}x)\mathbb{P}(M_{L,y}(B)\leq
a_{n}x)\big{\rvert}$ (40) $\displaystyle\leq 2\Bigl{(}\sum_{v\in
K_{n,L}}\mathbb{P}(M_{L,y}(\\{v\\})>a_{n}x-\epsilon_{n})-\sum_{v\in
K_{n,L}}\mathbb{P}(M_{L,y}(\\{v\\})>a_{n}x+\epsilon_{n})\Bigr{)}$
$\displaystyle\phantom{=}+2\Bigl{(}\mathbb{P}\bigl{(}(S^{A}_{n})^{c}\bigr{)}+\mathbb{P}\bigl{(}(S^{B}_{n})^{c}\bigr{)}\Bigr{)}.$

We have already seen that the second term above tends to 0 as $n\to\infty$.
Since $a_{n}=\lvert C_{n}\rvert^{1/\alpha}S(\lvert C_{n}\rvert)$ for a slowly
varying function $S$ (cf. [10, p. 131]), we have $\epsilon_{n}=o(a_{n})$. From
this it is not difficult to see that
$\rho((a_{n}x,\infty))\sim\rho((a_{n}x\pm\epsilon_{n},\infty))$ as
$n\to\infty$. Therefore, realizing that $\lvert
C_{n}\rvert\mathbb{P}(M_{L,y}(\\{v\\})>a_{n}x+\epsilon_{n})$ and $\lvert
C_{n}\rvert\mathbb{P}(M_{L,y}(\\{v\\})>a_{n}x-\epsilon_{n})$ have the same
limit for fixed $v$, the considerations that led to (35) also show that the
two sums in the first term of (40) have the same limit as $n\to\infty$. This
completes the proof. ∎

In the remainder of the paper, we consider the sequence $\gamma_{n}$ given in
Lemma 11. The following generalization follows easily by induction.

######  Lemma 12.

Let $(Z_{v})_{v}$ and $(Y_{v})_{v}$ be given by (19) and (20), respectively,
and let $(y_{v})_{v}$ be a realization of $(Y_{v})_{v}$. Let for
$r\in\mathbb{N}$ the boxes $A_{1},\dots,A_{r}\subseteq K_{n,L}$ be pairwise
$\gamma_{n}$-separated boxes. Then

$\Big{\lvert}\mathbb{P}\Bigl{(}\bigcap_{i=1}^{r}\\{M_{L,y}(A_{i})\leq
a_{n}x\\}\Bigr{)}-\prod_{i=1}^{r}\mathbb{P}(M_{L,y}(A_{i})\leq
a_{n}x)\Big{\rvert}\leq(r-1)\alpha_{y,n,L}.$

By construction we have $\gamma_{n}<t_{n,k,L}$ for $n$ sufficiently large
relative to fixed $k$ and $L$. For each $z\in N_{k,L}$ we divide
$J_{z}=J_{z}^{n,k,L}\subseteq(L\mathbb{Z})^{d}$ into two disjoint subsets,
$H_{z}=H_{z}^{n,k,L}$ and $H_{z}^{*}=H_{z}^{*,n,k,L}$, where

$H_{z}^{n,k,L}=\bigl{\\{}u\in J_{z}\>:\>z_{j}t_{n,k,L}\leq
u_{j}<(z_{j}+1)t_{n,k,L}-\gamma_{n},\>\text{for all }j=1,\dots,d\bigr{\\}}$

is a box constructed such that it is $\gamma_{n}$-separated from
$H_{u}^{n,k,L}$ for all $v\neq u\in N_{k,L}$. From now on we simply write
$H_{z}$ and $H_{z}^{*}=J_{z}\setminus H_{z}$ to ease notation, but bare in
mind the dependence on $n,k,L$. In the proof of Lemma 14 we explicitly use the
fact that $H_{z}^{*}$ equals the union of a set of overlapping boxes: Defining
$B_{z,1}^{*},\dots,B_{z,d}^{*}$ by

$B_{z,j}^{*}=\bigl{\\{}u\in J_{z}\>:\>(z_{j}+1)t_{n,k,L}-\gamma_{n}\leq
u_{j}<(z_{j}+1)t_{n,k,L}\bigr{\\}}$

for all $j=1,\dots,d$, it is easily seen that their union equals $H_{z}^{*}$.
Note that they are of size $\lvert B_{z,j}^{*}\rvert\sim
t_{n,k,L}^{d-1}\gamma_{n}/L^{d}$ as $n\to\infty$.

######  Lemma 13.

Let $(Z_{v})_{v}$ and $(Y_{v})_{v}$ be given by (19) and (20), respectively,
and let $(y_{v})_{v}$ be a realization of $(Y_{v})_{v}$. Then it holds that

$\displaystyle\Big{\lvert}\mathbb{P}(M_{L,y}(D_{n,k,L}^{-})\leq
a_{n}x)-\prod_{z\in P_{n,k,L}}\mathbb{P}(M_{L,y}(J_{z})\leq
a_{n}x)\Big{\rvert}$ (41) $\displaystyle\leq 2\sum_{z\in
P_{n,k,L}}\mathbb{P}(M_{L,y}(H_{z})\leq
a_{n}x<M_{L,y}(H_{z}^{*}))+(p_{n,k,L}-1)\alpha_{y,n,L},$

and similarly

$\displaystyle\Big{\lvert}\mathbb{P}(M_{L,y}(D_{n,k,L}^{+})\leq
a_{n}x)-\prod_{z\in Q_{n,k,L}}\mathbb{P}(M_{L,y}(J_{z})\leq
a_{n}x)\Big{\rvert}$ (42) $\displaystyle\leq 2\sum_{z\in
Q_{n,k,L}}\mathbb{P}(M_{L,y}(H_{z})\leq
a_{n}x<M_{L,y}(H_{z}^{*}))+(q_{n,k,L}-1)\alpha_{y,n,L}.$

###### Proof.

As $H_{z}$ is a subset of $J_{z}$, we find that

$\displaystyle 0$ $\displaystyle\leq\mathbb{P}\Bigl{(}\bigcap_{z\in
P_{n,k,L}}\\{M_{L,y}(H_{z})\leq
a_{n}x\\}\Bigr{)}-\mathbb{P}\bigl{(}M_{L,y}(D_{n,k,L}^{-})\leq a_{n}x\bigr{)}$
$\displaystyle\leq\mathbb{P}\Bigl{(}\bigcup_{z\in
P_{n,k,L}}\\{M_{L,y}(H_{z})\leq a_{n}x<M_{L,y}(H_{z}^{*})\\}\Bigr{)}$
$\displaystyle\leq\sum_{z\in P_{n,k,L}}\mathbb{P}\bigl{(}M_{L,y}(H_{z})\leq
a_{n}x<M_{L,y}(H_{z}^{*})\bigr{)}.$

By construction of $H_{z}$, an application of Lemma 12 shows that

$\Big{\lvert}\mathbb{P}\Bigl{(}\bigcap_{z\in P_{n,k,L}}\\{M_{L,y}(H_{z})\leq
a_{n}x\\}\Bigr{)}-\prod_{z\in P_{n,k,L}}\mathbb{P}\bigl{(}M_{L,y}(H_{z})\leq
a_{n}x\bigr{)}\Big{\rvert}\leq(p_{n,k,L}-1)\alpha_{y,n,L},$

which, combined with the fact that

$\displaystyle 0$ $\displaystyle\leq\prod_{z\in
P_{n,k,L}}\mathbb{P}\bigl{(}M_{L,y}(H_{z})\leq a_{n}x\bigr{)}-\prod_{z\in
P_{n,k,L}}\mathbb{P}\bigl{(}M_{L,y}(J_{z})\leq a_{n}x\bigr{)}$
$\displaystyle\leq\sum_{z\in
P_{n,k,L}}\Bigl{(}\mathbb{P}\bigl{(}M_{L,y}(H_{z})\leq
a_{n}x\bigr{)}-\mathbb{P}\bigl{(}M_{L,y}(J_{z})\leq a_{n}x\bigr{)}\Bigr{)}$
$\displaystyle=\sum_{z\in P_{n,k,L}}\mathbb{P}\bigl{(}M_{L,y}(H_{z})\leq
a_{n}x<M_{L,y}(H_{z}^{*})\bigr{)},$

concludes (41). The claim (42) follows similarly. ∎

######  Lemma 14.

Let $(Z_{v})_{v}$ and $(Y_{v})_{v}$ be given by (19) and (20), respectively.
Then, for almost all realizations $(y_{v})_{v}$ of $(Y_{v})_{v}$, the
following is satisfied

$\displaystyle\Big{(}\liminf_{n\to\infty}\min_{z\in
N_{k,L}}\mathbb{P}(M_{L,y}(J_{z})\leq a_{n}x)\Big{)}^{\tilde{q}_{k,L}}$
$\displaystyle\leq\liminf_{n\to\infty}\mathbb{P}\bigl{(}\sup_{v\in
C_{n}}(Z_{v}+y_{v})\leq a_{n}x\bigr{)}$ (43)
$\displaystyle\leq\limsup_{n\to\infty}\mathbb{P}\bigl{(}\sup_{v\in
C_{n}}(Z_{v}+y_{v})\leq a_{n}x\bigr{)}$
$\displaystyle\leq\Big{(}\limsup_{n\to\infty}\max_{z\in
N_{k,L}}\mathbb{P}(M_{L,y}(J_{z})\leq a_{n}x)\Big{)}^{\tilde{p}_{k,L}},$

where $\tilde{p}_{k,L}=\liminf_{n}p_{n,k,L}$ and
$\tilde{q}_{k,L}=\limsup_{n}q_{n,k,L}$.

###### Proof.

Let $R_{n,k,L}^{p}\geq 0$ and $R_{n,k,L}^{q}\geq 0$ denote the upper bounds in
(41) and (42), respectively. For all $z\in N_{k,L}$, we have

$\displaystyle\mathbb{P}(M_{L,y}(H_{z})\leq a_{n}x<M_{L,y}(H_{z}^{*}))$
$\displaystyle\leq\mathbb{P}(M_{L,y}(H_{z}^{*})>a_{n}x)$
$\displaystyle\leq\sum_{j=1}^{d}\mathbb{P}(M_{L,y}(B_{z,j}^{*})>a_{n}x)$
$\displaystyle\leq\sum_{j=1}^{d}\sum_{v\in
B_{z,j}^{*}}\mathbb{P}\bigl{(}\sup_{u\in
C_{L}(v)}(Z_{u}+y_{u})>a_{n}x\bigr{)},$

where we recall that $\lvert B_{z,j}^{*}\rvert\sim
t_{n,k,L}^{d-1}\gamma_{n}/L^{d}$ as $n\to\infty$. In particular, $\lvert
B_{z,j}^{*}\rvert=o(\lvert J_{z}\rvert)$ as $n\to\infty$ for all
$j=1,\dots,d$, and hence $J_{z}\setminus B_{z,j}^{*}$ is a box, which
increases in size asymptotically as $J_{z}$. Since the limit appearing in
Lemma 10 is finite and $\lvert J_{z}\rvert,\lvert J_{z}\setminus
B_{z,j}^{*}\rvert$ and $\lvert C_{n}\rvert$ are asymptotically of the same
order, we conclude by Lemma 10 that

$\displaystyle\mathbb{P}(M_{L,y}(H_{z})\leq a_{n}x<M_{L,y}(H_{z}^{*}))$
$\displaystyle\leq\sum_{j=1}^{d}\sum_{v\in J_{z}}\mathbb{P}\bigl{(}\sup_{u\in
C_{L}(v)}(Z_{u}+y_{u})>a_{n}x\bigr{)}$
$\displaystyle\qquad-\sum_{j=1}^{d}\sum_{v\in J_{z}\setminus
B_{z,j}^{*}}\mathbb{P}\bigl{(}\sup_{u\in
C_{L}(v)}(Z_{u}+y_{u})>a_{n}x\bigr{)}$ $\displaystyle\to 0$

almost surely for all $z\in N_{k,L}$. By Lemma 11 we thus obtain that

$\lim_{n\to\infty}R_{n,k,L}^{p}=\lim_{n\to\infty}R_{n,k,L}^{q}=0$

almost surely. Turning to (24) and using Lemma 13 show that

$\displaystyle\liminf_{n\to\infty}\prod_{z\in
Q_{n,k,L}}\mathbb{P}(M_{L,y}(J_{z})\leq a_{n}x)$
$\displaystyle\leq\liminf_{n\to\infty}\mathbb{P}\bigl{(}\sup_{v\in
C_{n}}(Z_{v}+y_{v})\leq a_{n}x\bigr{)}$
$\displaystyle\leq\limsup_{n\to\infty}\mathbb{P}\bigl{(}\sup_{v\in
C_{n}}(Z_{v}+y_{v})\leq a_{n}x\bigr{)}$
$\displaystyle\leq\limsup_{n\to\infty}\prod_{z\in
P_{n,k,L}}\mathbb{P}(M_{L,y}(J_{z})\leq a_{n}x).$

As all factors in the products are probabilities and thus lie in the interval
$[0,1]$, it is easily seen that (43) is satisfied. ∎

In the following lemma, the summation over $\\{v<v^{\prime}\in J_{z}\\}$
indicates the double sum of points in $v\in J_{z}$ and subsequent points
$v^{\prime}\in J_{z}$ falling strictly after $v$ under some underlying
enumeration. This notation will be used in the remainder of the paper.

######  Lemma 15.

Let $(Z_{v}^{(t)})_{v}$ and $(Y_{v})_{v}$ be given by (28) and (20),
respectively. For all $0<t<\infty$ there is a sequence of functions $g_{L}$
satisfying

$\limsup_{k\to\infty}k\,g_{L}(k)=o(L^{d})$ (44)

as $L\to\infty$ such that

$\limsup_{n\to\infty}\sum_{v<v^{\prime}\in J_{z}}\mathbb{P}\bigl{(}\sup_{u\in
C_{L}(v)}(Z_{u}^{(t)}+y_{u})>a_{n}x,\sup_{u\in
C_{L}(v^{\prime})}(Z_{u}^{(t)}+y_{u})>a_{n}x\bigr{)}\leq g_{L}(k)$ (45)

for all $z\in N_{k,L}$ and almost all realizations $(y_{v})_{v}$ of
$(Y_{v})_{v}$.

###### Proof.

Throughout this proof we fix $\beta>\alpha$ such that

$\mathbb{E}\bigl{(}\sup_{v\in C_{L}(0)}Y_{v}\bigr{)}_{+}^{\beta}<\infty,$

which is possible due to Lemma 9. Furthermore, let
$F_{L,t}\in\mathcal{RV}_{\alpha}$ denote the regularly varying distribution of
$\sup_{u\in C_{L}(v)}Z_{u}^{(t)}$ for $t>0$, and recall that

$\lvert C_{n}\rvert\overline{F}_{L,t}(a_{n}x)\to\tau_{L}^{(t)}$ (46)

as $n\to\infty$.

Fix $L>2t$. By construction, $\sup_{C_{L}(v)}Z_{u}^{(t)}$ and
$\sup_{C_{L}(v^{\prime})}Z_{u}^{(t)}$ are independent for all non-neighbors
$v,v^{\prime}\in J_{z}$ (recall that $J_{z}\subseteq(L\mathbb{Z})^{d}$), that
is, for all $v,v^{\prime}\in J_{z}$ such that $\lvert
v-v^{\prime}\rvert>L\sqrt{d}$. Writing $y^{*}(v)=\sup_{u\in C_{L}(v)}y_{u}$
and turning to Lemma 1, we find constants $C,K$ such that

$\displaystyle\sum_{\begin{subarray}{c}v<v^{\prime}\in J_{z}\\\ \lvert
v-v^{\prime}\rvert>L\sqrt{d}\end{subarray}}\mathbb{P}\bigl{(}\sup_{u\in
C_{L}(v)}(Z_{u}^{(t)}+y_{u})>a_{n}x,\sup_{u\in
C_{L}(v^{\prime})}(Z_{u}^{(t)}+y_{u})>a_{n}x\bigr{)}$
$\displaystyle\leq\sum_{v<v^{\prime}\in
J_{z}}\overline{F}_{L,t}(a_{n}x-y^{*}(v))\overline{F}_{L,t}(a_{n}x-y^{*}(v^{\prime}))$
$\displaystyle\leq\bigl{(}\overline{F}_{L,t}(a_{n}x)\bigr{)}^{2}\Bigl{(}\sum_{v\in
J_{z}}C\bigl{(}K+(y^{*}(v))_{+}^{\beta}\bigr{)}\Bigr{)}^{2}$

for sufficiently large $n$ and all $z\in N_{k,L}$. Since $\lvert
C_{n}\rvert/k\sim\lvert J_{z}\rvert$ as $n\to\infty$, we conclude by Theorem 7
and (46) that

$\displaystyle\limsup_{n\to\infty}\,\bigl{(}\overline{F}_{L,t}(a_{n}x)\bigr{)}^{2}\Bigl{(}\sum_{v\in
J_{z}}C\bigl{(}K+(y^{*}(v))_{+}^{\beta}\bigr{)}\Bigr{)}^{2}$
$\displaystyle=\frac{1}{k^{2}}\Bigl{(}\tau_{L}^{(t)}C\bigl{(}K+\mathbb{E}\bigl{(}\sup_{v\in
C_{L}(0)}Y_{v}\bigr{)}_{+}^{\beta}\bigr{)}\Bigr{)}^{2}$

almost surely. This is independent of $z$ and of order $o(k^{-1})$ as
$k\to\infty$ for all $L\in\mathbb{N}$. This shows (44) and (45) for the terms
in the sum with indices more than $L\sqrt{d}$ apart.

For notational convenience, define

$R_{L}=\int_{\mathbb{R}^{d}}\sup_{v\in
C_{L}(0)}\bigl{(}f^{(t)}(v-u)\bigr{)}^{\alpha}\mathrm{d}u.$

Now consider $v,v^{\prime}\in J_{z}$ such that $|v-v^{\prime}|\leq L\sqrt{d}$.
Due to Lemma 8 we have for all fixed $v\neq v^{\prime}$ that

$\displaystyle\lim_{n\to\infty}\frac{1}{\overline{F}_{L,t}(a_{n}x)}\mathbb{P}\bigl{(}\sup_{u\in
C_{L}(v)}Z_{u}^{(t)}>a_{n}x-y,\sup_{u\in
C_{L}(v^{\prime})}Z_{u}^{(t)}>a_{n}x-y\bigr{)}$ $\displaystyle\to
2-R_{L}^{-1}\int_{{\mathbb{R}^{d}}}\sup_{s\in C_{L}(v)\cup
C_{L}(v^{\prime})}\bigl{(}f^{(t)}(s-u)\bigr{)}^{\alpha}\mathrm{d}u$

uniformly for $\lvert y\rvert\leq N$, for all $N\in\mathbb{N}$. Define
$y^{*}(v,v^{\prime})=\max\\{y^{*}(v),y^{*}(v^{\prime})\\}$ and similarly for
$(Y_{v})_{v}$ and note that

$\mathbb{E}(Y^{*}(v,v^{\prime}))_{+}^{\beta}\leq 2\,\mathbb{E}(\sup_{v\in
C_{L}(0)}Y_{v})_{+}^{\beta}<\infty.$ (47)

Combining the uniform convergence above with Theorem 7, Lemma 1, equation (46)
and the fact that $\lvert C_{n}\rvert/k\sim\lvert J_{z}\rvert$ then yield

$\displaystyle\limsup_{n\to\infty}\sum_{\begin{subarray}{c}v<v^{\prime}\in
J_{z}\\\ \lvert v-v^{\prime}\rvert\leq
L\sqrt{d}\end{subarray}}\mathbb{P}\bigl{(}\sup_{u\in
C_{L}(v)}(Z_{u}^{(t)}+y_{u})>a_{n}x,\sup_{u\in
C_{L}(v^{\prime})}(Z_{u}^{(t)}+y_{u})>a_{n}x\bigr{)}$
$\displaystyle\leq\limsup_{n\to\infty}\sum_{\begin{subarray}{c}v<v^{\prime}\in
J_{z}\\\ \lvert v-v^{\prime}\rvert\leq
L\sqrt{d}\end{subarray}}\mathbb{P}\bigl{(}\sup_{u\in
C_{L}(v)}Z_{u}^{(t)}>a_{n}x-y^{*}(v,v^{\prime}),\sup_{u\in
C_{L}(v^{\prime})}Z_{u}^{(t)}>a_{n}x-y^{*}(v,v^{\prime})\bigr{)}$
$\displaystyle\leq\frac{\tau_{L}^{(t)}}{k}\sum_{\begin{subarray}{c}v\in(L\mathbb{Z})^{d}\\\
0<\lvert v\rvert\leq
L\sqrt{d}\end{subarray}}\mathbb{E}\bigl{(}\boldsymbol{1}_{\\{\lvert
Y^{*}(0,v)\rvert\leq
N\\}}\bigr{)}\Bigl{(}2-R_{L}^{-1}\int_{{\mathbb{R}^{d}}}\sup_{s\in
C_{L}(v)\cup
C_{L}(0)}\bigl{(}f^{(t)}(s-u)\bigr{)}^{\alpha}\mathrm{d}u\Bigr{)}$
$\displaystyle\quad+\frac{\tau_{L}^{(t)}}{k}\sum_{\begin{subarray}{c}v\in(L\mathbb{Z})^{d}\\\
0<\lvert v\rvert\leq
L\sqrt{d}\end{subarray}}\mathbb{E}\Bigl{(}C\bigl{(}K+Y^{*}(0,v)_{+}^{\beta}\bigr{)}\,\boldsymbol{1}_{\\{\lvert
Y^{*}(0,v)\rvert>N\\}}\Bigr{)}$

for all $N\in\mathbb{N}$. Since this is independent of $z$, and due to the
fact that there are only finitely many terms in the sum, we conclude (45) by
letting $N\to\infty$ using a dominated convergence argument justified by (47):

$\displaystyle\limsup_{n\to\infty}\sum_{\begin{subarray}{c}v<v^{\prime}\in
J_{z}\\\ \lvert v-v^{\prime}\rvert\leq
L\sqrt{d}\end{subarray}}\mathbb{P}\bigl{(}\sup_{u\in
C_{L}(v)}(Z_{u}^{(t)}+y_{u})>a_{n}x,\sup_{u\in
C_{L}(v^{\prime})}(Z_{u}^{(t)}+y_{u})>a_{n}x\bigr{)}$
$\displaystyle\leq\frac{\tau_{L}^{(t)}}{k}\sum_{\begin{subarray}{c}v\in(L\mathbb{Z})^{d}\\\
0<\lvert v\rvert\leq
L\sqrt{d}\end{subarray}}\Bigl{(}2-R_{L}^{-1}\int_{{\mathbb{R}^{d}}}\sup_{s\in
C_{L}(v)\cup
C_{L}(0)}\bigl{(}f^{(t)}(s-u)\bigr{)}^{\alpha}\mathrm{d}u\Bigr{)}.$

The proof is finished once we show that this upper bound satisfies (44). Since
$\tau_{L}^{(t)}$ is asymptotically equivalent to $L^{d}$ by (32), this is the
case if only the ($k$-independent) sum is of order $o(1)$ as $L\to\infty$. As
the number of terms in the sum is fixed, it is a matter of showing that the
summands tend to $0$. As $f(0)=1$ and $f\geq 0$ it is easily seen that

$2L^{d}\leq\int_{{\mathbb{R}^{d}}}\sup_{s\in C_{L}(v)\cup
C_{L}(0)}\bigl{(}f^{(t)}(s-u)\bigr{)}^{\alpha}\mathrm{d}u\leq 2R_{L}$

for all $0\neq v$, and the claim follows since $R_{L}\sim L^{d}$ by Lemma 7. ∎

###### Proof of Theorem 3.

We continue with the notation used throughout this section. In particular,

$\tau_{L}^{(t)}=x^{-\alpha}\rho((1,\infty))\int_{{\mathbb{R}^{d}}}\sup_{v\in
C_{L}(0)}\bigl{(}f^{(t)}(v-u)\bigr{)}^{\alpha}\mathrm{d}u$

for all fixed $x>0$. Now consider $J_{z}$ for $z\in N_{k,L}$. Recalling the
definition of $M_{L,y}$ and $M_{L,y}^{(t)}$ and the fact that
$M_{L,y}^{(t)}\leq M_{L,y}$, it can easily be seen that

$\displaystyle\sum_{v\in
J_{z}}\mathbb{P}(M_{L,y}^{(t)}(\\{v\\})>a_{n}x)-\sum_{v<v^{\prime}\in
J_{z}}\mathbb{P}(M_{L,y}^{(t)}(\\{v\\})>a_{n}x,M_{L,y}^{(t)}(\\{v^{\prime}\\})>a_{n}x)$
$\displaystyle\leq\mathbb{P}(M_{L,y}(J_{z})>a_{n}x)$
$\displaystyle\leq\sum_{v\in J_{z}}\mathbb{P}(M_{L,y}(\\{v\\})>a_{n}x).$

We now obtain from Lemma 14 that

$\displaystyle\Bigl{(}\liminf_{n\to\infty}\min_{z\in
N_{k,L}}\Bigl{(}1-\sum_{v\in J_{z}}\mathbb{P}\bigl{(}\sup_{u\in
C_{L}(v)}(Z_{u}+y_{u})>a_{n}x\bigr{)}\Bigr{)}\Bigr{)}^{\tilde{q}_{k,L}}$ (48)
$\displaystyle\leq\liminf_{n\to\infty}\mathbb{P}\bigl{(}\sup_{v\in
C_{n}}(Z_{v}+y_{v})\leq a_{n}x\bigr{)}$
$\displaystyle\leq\limsup_{n\to\infty}\mathbb{P}\bigl{(}\sup_{v\in
C_{n}}(Z_{v}+y_{v})\leq a_{n}x\bigr{)}$
$\displaystyle\leq\Bigl{(}\limsup_{n\to\infty}\max_{z\in
N_{k,L}}\Bigl{(}1-\sum_{v\in J_{z}}\mathbb{P}\bigl{(}\sup_{u\in
C_{L}(v)}(Z_{u}^{(t)}+y_{u})>a_{n}x\bigr{)}+S_{n,k,L}(z)\Bigr{)}\Bigr{)}^{\tilde{p}_{k,L}},$

where

$S_{n,k,L}(z)=\sum_{v<v^{\prime}\in J_{z}}\mathbb{P}\bigl{(}\sup_{u\in
C_{L}(v)}(Z_{u}^{(t)}+y_{u})>a_{n}x,\sup_{u\in
C_{L}(v^{\prime})}(Z_{u}^{(t)}+y_{u})>a_{n}x\bigr{)},$

and $\tilde{q}_{k,L}=\limsup_{n}q_{n,k,L}$ and
$\tilde{p}_{k,L}=\liminf_{n}p_{n,k,L}$. By Lemma 15 there is a sequence of
functions $g_{L}(k)$ satisfying

$\limsup_{k\to\infty}k\,g_{L}(k)=o(L^{d})$

as $L\to\infty$, such that $\limsup_{n}S_{n,k,L}(z)\leq g_{L}(k)$ uniformly in
$z$. Since $\lvert C_{n}\rvert/k\sim\lvert J_{z}\rvert$, we find by Lemma 10
and (48) that

$\displaystyle\Bigl{(}1-\frac{\tau_{L}}{k}\Bigr{)}^{\tilde{q}_{k,L}}$
$\displaystyle\leq\liminf_{n\to\infty}\mathbb{P}\bigl{(}\sup_{v\in
C_{n}}(Z_{v}+y_{v})\leq a_{n}x\bigr{)}$
$\displaystyle\leq\limsup_{n\to\infty}\mathbb{P}\bigl{(}\sup_{v\in
C_{n}}(Z_{v}+y_{v})\leq a_{n}x\bigr{)}$
$\displaystyle\leq\Bigl{(}1-\frac{\tau_{L}^{(t)}}{k}+g_{L}(k)\Bigr{)}^{\tilde{p}_{k,L}}$

almost surely. Since

$\tilde{p}_{k,L}\sim\tilde{q}_{k,L}\sim\frac{k}{L^{d}}$

as $k\to\infty$, we use the equivalence $\log(1-y)\sim-y$ ($y\to 0$) to
obtain, as $k\to\infty$,

$\displaystyle\exp\Bigl{(}-\frac{\tau_{L}}{L^{d}}\Bigr{)}$
$\displaystyle\leq\liminf_{n\to\infty}\mathbb{P}\bigl{(}\sup_{v\in
C_{n}}(Z_{v}+y_{v})\leq a_{n}x\bigr{)}$
$\displaystyle\leq\limsup_{n\to\infty}\mathbb{P}\bigl{(}\sup_{v\in
C_{n}}(Z_{v}+y_{v})\leq a_{n}x\bigr{)}$
$\displaystyle\leq\exp\Bigl{(}-\frac{\tau_{L}^{(t)}}{L^{d}}+o(1)\Bigr{)}.$

Here the remainder $o(1)$ is with respect to the limit $L\to\infty$. Secondly,
taking the limit $L\to\infty$ and using the equivalence (32) show that

$\lim_{n\to\infty}\mathbb{P}\bigl{(}\sup_{v\in C_{n}}(Z_{v}+y_{v})\leq
a_{n}x\bigr{)}=\exp\Bigl{(}-x^{-\alpha}\rho((1,\infty))\Bigr{)}$

almost surely.

Let $\pi$ denote the distribution of the field $(Y_{v})_{v}$. Then, by
independence and dominated convergence,

$\displaystyle\mathbb{P}\bigl{(}\sup_{v\in C_{n}}X_{v}\leq a_{n}x\bigr{)}$
$\displaystyle=\int\mathbb{P}\bigl{(}\sup_{v\in C_{n}}(Z_{v}+y_{v})\leq
a_{n}x\bigr{)}\pi(\mathrm{d}y)\to\exp\Bigl{(}-x^{-\alpha}\rho((1,\infty))\Bigr{)}$

as $n\to\infty$. ∎

###### Proof of Theorem 4.

Following the proofs in this section, it can easily be seen that the claim in
Theorem 3 can be extended as follows:

$\mathbb{P}\Bigl{(}\sup_{v\in C_{n}}\bigl{(}X_{v}+Y_{v}^{1}\bigr{)}\leq
a_{n}x\pm d_{n}\Bigr{)}\to\exp\Bigl{(}-x^{-\alpha}\rho((1,\infty))\Bigr{)}$
(49)

for all $x>0$, where $(Y_{v}^{1})_{v}$ is the sufficiently light-tailed and
ergodic field given by (13) in the present theorem, and $d_{n}$ is a sequence
of order $o(a_{n})$ tending to $\infty$.

Consider the stationary field $(Y_{v}^{2})_{v}$ and note that the
integrability assumed in (14) is equivalent to

$\displaystyle\mathbb{E}\bigl{(}\sup_{v\in
C_{1}(0)}Y_{v}^{2}\bigr{)}_{+}^{\beta_{2}}<\infty,\qquad\text{and}$
$\displaystyle\mathbb{E}\bigl{(}\inf_{v\in
C_{1}(0)}Y_{v}^{2}\bigr{)}_{-}^{\beta_{2}}<\infty,$

where $y_{-}^{\beta_{2}}=(-y\boldsymbol{1}_{\\{y\leq 0\\}})^{\beta_{2}}$, and
$C_{1}(0)$ denotes the unit cube in ${\mathbb{R}^{d}}$. With this
$\beta_{2}>\alpha$ in mind, choose the sequence $d_{n}$ according to Lemma 2.
Then, due to Lemma 2 and the fact that $d_{n}\to\infty$,

$\lvert C_{n}\rvert\mathbb{P}\bigl{(}\sup_{v\in
C_{1}(0)}Y_{v}^{2}>d_{n}\bigr{)}\sim\frac{\rho((1,\infty))\mathbb{P}\bigl{(}(\sup_{v\in
C_{1}(0)}Y_{v}^{2})_{+}>d_{n}\bigr{)}}{\rho((a_{n},\infty))}\to 0$ (50)

as $n\to\infty$, and similarly

$\lvert C_{n}\rvert\mathbb{P}\bigl{(}\inf_{v\in
C_{1}(0)}Y_{v}^{2}<-d_{n}\bigr{)}\sim\frac{\rho((1,\infty))\mathbb{P}\bigl{(}(\inf_{v\in
C_{1}(0)}Y_{v}^{2})_{-}>d_{n}\bigr{)}}{\rho((a_{n},\infty))}\to 0.$ (51)

Now fix $k=L=1$ and recall the discrete set
$K_{n}=K_{n,1}\subseteq{\mathbb{Z}^{d}}$ from Theorem 6, which in particular
satisfies that

$C_{n}\subseteq\bigcup_{z\in K_{n}}C_{1}(z)$

and $\lvert K_{n}\rvert\leq c\cdot\lvert C_{n}\rvert$ for some finite,
$n$-independent $c$. Turning to (50) and the stationarity of $(Y_{v}^{2})$ we
thus see that

$\displaystyle\mathbb{P}\bigl{(}\sup_{v\in C_{n}}Y_{v}^{2}>d_{n}\bigr{)}$
$\displaystyle\leq\mathbb{P}\bigl{(}\max_{z\in K_{n}}\sup_{v\in
C_{1}(z)}Y_{v}^{2}>d_{n}\bigr{)}$ (52) $\displaystyle\leq c\cdot\lvert
C_{n}\rvert\mathbb{P}\bigl{(}\sup_{v\in C_{1}(0)}Y_{v}^{2}>d_{n}\bigr{)}\to 0$

as $n\to\infty$. Similarly, by (51),

$\mathbb{P}\bigl{(}\inf_{v\in C_{n}}Y_{v}^{2}<-d_{n}\bigr{)}\to 0$ (53)

as $n\to\infty$. Since, by independence of the fields involved,

$\displaystyle\mathbb{P}\bigl{(}\sup_{v\in C_{n}}(X_{v}+Y_{v}^{1})\leq
a_{n}x-d_{n}\bigr{)}\mathbb{P}\bigl{(}\sup_{v\in C_{n}}Y_{v}^{2}\leq
d_{n}\bigr{)}$ $\displaystyle\leq\mathbb{P}\bigl{(}\sup_{v\in
C_{n}}(X_{v}+Y_{v}^{1}+Y_{v}^{2})\leq a_{n}x\bigr{)}$
$\displaystyle\leq\mathbb{P}\bigl{(}\sup_{v\in C_{n}}(X_{v}+Y_{v}^{1})\leq
a_{n}x+d_{n}\bigr{)}\mathbb{P}\bigl{(}\inf_{v\in C_{n}}Y_{v}^{2}\geq-
d_{n}\bigr{)}+\mathbb{P}\bigl{(}\inf_{v\in C_{n}}Y_{v}^{2}<-d_{n}\bigr{)},$

the claim follows by (49) in combination with (52) and (53). ∎

## References

  * Adler and Taylor [2007] Adler, R., Taylor, J.E., 2007\.  Random Fields and Geometry.  Monographs in mathematics, Springer Science.
  * Adler [1981] Adler, R.J., 1981.  The Geometry of Random Fields.  John Wiley & Sons, New York.
  * Barndorff-Nielsen and Schmiegel [2004] Barndorff-Nielsen, O.E., Schmiegel, J., 2004\.  Lévy-based tempo-spatial modelling with applications to turbulence.  Uspekhi Mat. Nauk 159, 63–90.
  * Barndorff-Nielsen and Shephard [2001] Barndorff-Nielsen, O.E., Shephard, N., 2001\.  Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics.  J. R. Stat. Soc. Ser. B 63, 167–241.
  * Benth and Pircalabu [2018] Benth, F.E., Pircalabu, A., 2018\.  A non-Gaussian Ornstein-Uhlenbeck model for pricing wind power futures.  Appl. math. finance 25, 36–65.
  * Braverman and Samorodnitsky [1995] Braverman, M., Samorodnitsky, G., 1995\.  Functionals of infinitely divisible stochastic processes with exponential tails.  Stoch. Process. Their Appl. 56, 207–231.
  * Cline [1986] Cline, D.B.H., 1986.  Convolution tails, product tails and domains of attraction.  Probab. Theory Relat. Fields 72, 529–557.
  * Cline [1987] Cline, D.B.H., 1987.  Convolutions of distributions with exponential and subexponential tails.  J. Aust. Math. Soc. Ser. A 43, 347–365.
  * Embrechts et al. [1979] Embrechts, P., Goldie, C.M., Veraverbeke, N., 1979.  Subexponentiality and infinite divisibility.  Z. Wahrscheinlichkeitstheorie verw. Gebiete 49, 335–347.
  * Embrechts et al. [1997] Embrechts, P., Klüppelberg, C., Mikosch, T., 1997.  Modelling Extremal Events for Insurance and Finance.  Applications of mathematics, New York: Springer.
  * Fasen [2005] Fasen, V., 2005.  Extremes of regularly varying Lévy-driven mixed moving average processes.  Adv. Appl. Probab 37, 993–1014.
  * Jakubowski and Soja-Kukieła [2019] Jakubowski, A., Soja-Kukieła, N., 2019\.  Managing local dependencies in asymptotic theory for maxima of stationary random fields.  Extremes 22, 293–315.
  * Jónsdóttir et al. [2013] Jónsdóttir, K.Y., Rønn-Nielsen, A., Mouridsen, K., Jensen, E. B. V., 2013\.  Lévy-based modelling in brain imaging.  Scand. J. Stat. 40, 511–529.
  * Leadbetter et al. [1983] Leadbetter, M.R., Lindgren, G., Rootzen, H., 1983.  Extremes and Related Properties of Random Sequences and Processes.  Springer Series in Statistics, Springer Verlag.
  * Pakes [2004] Pakes, A.G., 2004.  Convolution equivalence and infinite divisibility.  J. Appl. Probab. 41, 407–424.
  * Potthoff [2009] Potthoff, J., 2009.  Sample properties of random fields. I. Separability and measurability.  Commun. Stoch. Analysis 3, 143–153.
  * Rajput and Rosinski [1989] Rajput, B.S., Rosinski, J., 1989\.  Spectral representations of infinitely divisible processes.  Probab. Theory Relat. Fields 82, 451–487.
  * Resnick [2008] Resnick, S.I., 2008.  Extreme Values, Regular Variation and Point Processes.  Springer Series in Operations Research and Financial Engineering, Springer, New York.
  * Rønn-Nielsen and Jensen [2016] Rønn-Nielsen, A., Jensen, E., 2016\.  Tail asymptotics for the supremum of an infinitely divisible field with convolution equivalent Lévy measure.  J. Appl. Probab. 53, 244–261.
  * Rønn-Nielsen and Jensen [2019] Rønn-Nielsen, A., Jensen, E.B.V., 2019\.  Central limit theorem for mean and variogram estimators in Lévy-based models.  J. Appl. Probab. 56, 209–222.
  * Rootzen [1978] Rootzen, H., 1978.  Extremes of moving averages of stable processes.  Ann. Probab. 6, 847–869.
  * Rosinski and Samorodnitsky [1993] Rosinski, J., Samorodnitsky, G., 1993\.  Distributions of subadditive functionals of sample paths of infinitely divisible processes.  Ann. Probab. 21, 996–1014.
  * Samorodnitsky [2016] Samorodnitsky, G., 2016.  Stochastic Processes and Long Range Dependence.  Springer Series in Operations Research and Financial Engineering, Springer, Switzerland.
  * Sato [1999] Sato, K., 1999.  Lévy Processes and Infinitely Divisible Distributions.  Cambridge Studies in Advanced Mathematics, Cambridge University Press.
  * Schneider [1993] Schneider, R., 1993.  Convex Bodies: The Bruun–Minkowski Theory.  Cambridge University Press.
  * Soja-Kukieła [2019] Soja-Kukieła, N., 2019.  On maxima of stationary fields.  J. Appl. Probab. 56, 1217–1230.
  * Stehr and Rønn-Nielsen [2021a] Stehr, M., Rønn-Nielsen, A., 2021a.  Extreme value theory for spatial random fields – with applications to a Lévy-driven field _Submitted_.
  * Stehr and Rønn-Nielsen [2021b] Stehr, M., Rønn-Nielsen, A., 2021b.  Tail asymptotics of an infinitely divisible space-time model with convolution equivalent Lévy measure.  J. Appl. Probab. 58, 42–67.  doi:10.1017/jpr.2020.73.