2. Preliminaries and initial assumptions

We define a Lévy basis to be an infinitely divisible and independently scattered random measure. Then the random measure M on ${\mathbb{R}}^{d+1}$ is independently scattered, such that for all disjoint Borel sets $(A_n)_{n\in\mathbb{N}} \subseteq {\mathbb{R}}^{d+1}$ the random variables $(M(A_n))_{n\in\mathbb{N}}$ are independent and furthermore satisfy $M(\cup_{n\in \mathbb{N}} A_n)=\sum_{n\in \mathbb{N}} M(A_n)$. Furthermore, M(A) is infinitely divisible for all Borel sets $A\subseteq {\mathbb{R}}^{d+1}$.

Moreover, in this paper we assume M to be a stationary and isotropic Lévy basis on ${\mathbb{R}}^{d+1}$. With $m(\!\cdot\!)$ denoting the Lebesgue measure, and $C(\lambda \dagger Y ) = \log {\mathbb E} \mathrm{e}^{{\mathrm{i}}\lambda Y}$ the cumulant function for a random variable Y, this means that the random variable M(A) has Lévy–Khinchin representation

(2.1) \begin{equation}C(\lambda \dagger M(A))= {\mathrm{i}} \lambda a m(A)- \dfrac{1}{2} \lambda^2 \theta m(A) +\int_{A \times {\mathbb{R}}} ( \mathrm{e}^{{\mathrm{i}}\lambda x} - 1 - {\mathrm{i}}\lambda x \textbf{1}_{[-1,1]}(x) ) F(\mathrm{d}u, \mathrm{d} x),\end{equation}

where $a\in {\mathbb{R}}$, $\theta \ge 0$, and F is the product measure $m \otimes \rho$ of the Lebesgue measure and a Lévy measure $\rho$. The notion of the so-called spot variable M ^′ will be useful. It is a random variable equivalent in distribution to M(A) when $m(A) = 1$.

We assume that the Lévy basis M has a convolution equivalent Lévy measure $\rho$ with index $\beta>0$, by which we formally mean that the probability measure $\rho_1$, the normalized restriction of $\rho$ to $(1,\infty)$, is in $\mathcal{S}_{\beta}$. This means that $\rho_1\in \mathcal{L}_{\beta}$, the class of probability measures with an exponential right tail $\beta$, that is,

(2.2) \begin{equation}\dfrac{\rho_1((x-y,\infty))}{\rho_1((x,\infty))}\to \mathrm{e}^{\beta y}\quad \text{as } x \to\infty ,\end{equation}

for all $y\in {\mathbb{R}}$, and that it furthermore satisfies the convolution property

(2.3) \begin{equation}\dfrac{(\rho_1*\rho_1)((x,\infty))}{\rho_1((x,\infty))}\to 2 \int_{\mathbb{R}} \mathrm{e}^{\beta y} \rho_1(\mathrm{d} y) < \infty\quad \text{as } x \to\infty ,\end{equation}

where $*$ denotes convolution. To ease notation, we write $\rho \in \mathcal{S}_{\beta}$ when $\rho_1\in\mathcal{S}_{\beta}$.

For later reference, we list the mentioned properties as part of Assumption 2.1 below.

We write the tail of $\rho$ as $\rho((x,\infty)) = L(x) \exp({-}\beta x)$, so for all $y\in{\mathbb{R}}$, (2.2) implies that

(2.4) \begin{equation}\dfrac{L(x-y)}{L(x)}\to 1\quad \text{as } x \to\infty .\end{equation}

Equation (2.4) implies that the mapping $x\mapsto L(\log(x))$ is slowly varying. A consequence (see formula (3.6) in [Reference Rønn-Nielsen and Jensen20]) is that for all $\gamma >0$ there exist $x_0>0$ and $C_0>0$ such that

(2.5) \begin{equation}\dfrac{L(\alpha x)}{L(x)} \le C_0 \exp((\alpha-1)\gamma x)\quad \text{for all }x\ge x_0,\ \alpha \ge 1 .\end{equation}

Before we define the kernel function f and consequently the field X, we introduce a continuity property called t-càdlàg, which is of importance in this paper. Under assumptions on the basis M and the integration kernel f (appearing below), the entire field $X=(X_{v,t})$ exhibits this regularity, thus ensuring that the supremum of such fields on bounded sets behaves nicely; see e.g. the proof of Lemma 3.2 in the supplementary material [Reference Stehr and Rønn-Nielsen27]. Moreover, this continuity property will be explicitly used in the proofs of Section 4.

Definition 2.1. (t-càdlàg.) A field $(y_{v,t})_{{(v,t)}}$ is t-càdlàg if, for all (v, t), it satisfies

(2.6) \begin{equation}\lim_{(u,s)\to (v,t^-)} y_{u,s} \ \text{exists in }{\mathbb{R}}\quad\text{and}\quad\lim_{(u,s)\to (v,t^+)} y_{u,s} = y_{v,t} .\end{equation}

In defining the field $X=(X_{v,t})_{(v,t)\in B^{\prime}\times T^{\prime}}$ below, we make the following assumptions on the Lévy basis M and the integration kernel f. The assumptions are stronger than needed to ensure the existence of the integral (2.10) defining $X_{v,t}$. By [Reference Rajput and Rosinski18, Theorem 2.7] the integral is well-defined if only the stationary and isotropic basis has a Lévy measure $\rho$ satisfying $\int_{|y|>1} \lvert{y}\rvert \rho ( \mathrm{d} y) < \infty$, and if the bounded integration kernel f is integrable in the sense of (2.8). However, we make the stronger assumptions below as these also guarantee the existence of a t-càdlàg version of X; see Theorem 5.3.

Assumption 2.1. The Lévy basis M on ${\mathbb{R}}^{d+1}$ is stationary and isotropic with a Lévy measure $\rho\in \mathcal{S}_{\beta}$, $\beta > 0$. Moreover, $\rho$ satisfies

(2.7) \begin{equation}\int_{|y|>1} |y|^k \rho ( \mathrm{d} y) < \infty\quad {for\ all\ k\in \mathbb{N}.}\end{equation}

Note that the integrability along the right tail is already given from the exponential tail property, and since $\rho$ is a Lévy measure it also satisfies $\int_{[-1,1]}y^2 \rho( \mathrm{d} y) < \infty$. Also, by [Reference Sato24, Theorem 25.3], (2.7) is equivalent to finite moments ${\mathbb E}|M'|^k<\infty$ of the spot variable. This is explicitly used when showing that there is a t-càdlàg version of X. Here we use a result from [Reference Adler1] which requires finite moments of a certain high order. In Sections 2, 3, and 4, it is assumed that Assumption 2.1 is satisfied.

Assumption 2.2. The kernel $f\colon[0,\infty) \times {\mathbb{R}} \to [0,\infty)$ is bounded, it satisfies $f(x,y)=0$ for all $x\in [0,\infty)$ and $y<0$, it is integrable in the sense that

(2.8) \begin{equation}\int_{\mathbb{R}^d} \int_{\mathbb{R}} f(\lvert{u}\rvert,s) \,\mathrm{d} s \, \mathrm{d} u < \infty ,\end{equation}

and it is Lipschitz-continuous on $[0,\infty)\times [0,\infty)$, that is, there exists $C_L\in(0,\infty)$ such that

(2.9) \begin{equation}|f(x_1,y_1)-f(x_2,y_2)| \le C_L |(x_1,y_1)-(x_2,y_2)|\end{equation}

for all $(x_1,y_1),(x_2,y_2)\in[0,\infty)\times[0,\infty)$.

Let $B\subseteq {\mathbb{R}^d}$ be a compact set with strictly positive Lebesgue measure, and consider [0, T] for deterministic $0<T<\infty$. For $r,\ell \ge 0$ fixed, define the expanded sets $B^{\prime}=B\oplus C_r(0)=\{x+y \colon x\in B, |y|\le r\}$ and $T^{\prime}=[0,T+\ell]$. Here $C_r(u)\subseteq {\mathbb{R}^d}$ is the d-dimensional closed ball with radius r and centre in $u\in {\mathbb{R}^d}$. Under Assumptions 2.1 and 2.2 we define the random field $X=(X_{v,t})_{(v,t)\in B^{\prime}\times T^{\prime}}$ by

(2.10) \begin{equation}X_{v,t} = \int_{{\mathbb{R}^d}\times {\mathbb{R}}} f(\lvert{v-u}\rvert, t-s) M(\mathrm{d}u,\mathrm{d}s) .\end{equation}

Note that alternatively we can write

\begin{equation*}X_{v,t} = \int_{{\mathbb{R}^d} \times ({-}\infty,t]} f(\lvert{v-u}\rvert, t-s) M(\mathrm{d}u,\mathrm{d}s)\end{equation*}

due to the assumptions on f. Thus X has a causal structure in the time direction in the sense that $X_{v,t}$ only depends on M restricted to the subset ${\mathbb{R}^d} \times ({-}\infty,t]$.

We are ultimately interested in extremal probabilities of the form

(2.11) \begin{equation}\mathbb{P} ( \Psi((X_{v,t})_{(v,t)\in B^{\prime}\times T^{\prime}}) > x ),\end{equation}

where $\Psi\colon {\mathbb{R}}^{B^{\prime}\times T^{\prime}} \to {\mathbb{R}}$ is a functional satisfying some assumptions that will be given in Section 3. For notational convenience we usually write $\Psi(y_{v,t})$ when applying $\Psi$ to a field $(y_{v,t})_{(v,t)\in B^{\prime}\times T^{\prime}}$, but when it is necessary to clarify the indices of the field we write it out in full. For the type of functionals $\Psi$ we shall consider, it will be convenient to make some further assumptions on the kernel. The following Assumption 2.3 clearly implies Assumption 2.2 above. In Sections 2, 3, and 4, Assumption 2.3 is assumed to be satisfied.

Assumption 2.3. The kernel $f\colon[0,\infty)\times {\mathbb{R}} \to [0,\infty)$ satisfies $f(0,0)=1$ and $f(x,y)=0$ for all $x\in[0,\infty)$ and $y<0$. Moreover,

(2.12) \begin{equation}\int_{\mathbb{R}^d} \int_{\mathbb{R}} \sup_{v\in B^{\prime}} \sup_{t\in T^{\prime}} f(\lvert{v-u}\rvert, t-s) \,\mathrm{d}s \,\mathrm{d}u < \infty,\end{equation}

and f is Lipschitz on $[0,\infty)\times [0,\infty)$, i.e. it satisfies (2.9).

It turns out that the infinite divisibility of M is inherited to the field X. We shall spend the remainder of this section establishing this property and use it to obtain a useful representation of the field as an independent sum of a compound Poisson term and a term with a lighter tail than exponentials. The procedure is inspired by a similar technique used in [Reference Rønn-Nielsen and Jensen19], [Reference Rønn-Nielsen and Jensen20], and [Reference Rosinski and Samorodnitsky23]. Here we present the procedure in full to introduce all relevant notation.

The cumulant function of $X_{v,t}$ takes the form (see [Reference Rajput and Rosinski18, Theorem 2.7])

\begin{align*}C(\lambda \dagger X_{v,t})& = {\mathrm{i}} \lambda a \int_{\mathbb{R}^d} \int_{\mathbb{R}} f(\lvert{v-u}\rvert, t-s) \,\mathrm{d}s \,\mathrm{d}u- \dfrac12 \theta \lambda^2 \int_{\mathbb{R}^d} \int_{\mathbb{R}} f(\lvert{v-u}\rvert, t-s)^2 \,\mathrm{d}s \,\mathrm{d}u \\[5pt] &\quad\, + \int_{{\mathbb{R}^d}}\int_{\mathbb{R}} \int_{\mathbb{R}}( {\mathrm{e}}^{{\mathrm{i}}\, f(\lvert{v-u}\rvert, t-s) \lambda z} - 1 - {\mathrm{i}} f(\lvert{v-u}\rvert, t-s)\lambda z 1_{[-1,1]}(z) ) \rho(\mathrm{d} z) \,\mathrm{d}s \,\mathrm{d}u .\end{align*}

A similar expression can be obtained for any finite linear combination of $X_{v,t}$ for varying v, t by replacing f with a relevant linear combination of f for varying v, t. Thus all finite-dimensional distributions of $(X_{v,t})_{(v,t)\in B^{\prime}\times T'}$ are infinitely divisible, and consequently any countably indexed field $(X_{v,t})$ is infinitely divisible. Define the countable set $\mathbb{K}=(B^{\prime}\times T^{\prime}) \cap \mathbb{Q}^{d+1}$, and let $\nu=(m \otimes m \otimes \rho)\circ H^{-1}$ be the measure on $\big({\mathbb{R}}^{\mathbb{K}}, \mathcal{B}\big({\mathbb{R}}^{\mathbb{K}}\big)\big)$ defined as the image-measure of H on $ m \otimes m \otimes \rho $, where $H\colon{{\mathbb{R}^d} \times {\mathbb{R}}}\times{\mathbb{R}} \to {\mathbb{R}}^{\mathbb{K}}$ is given by

\begin{equation*}H(u,s,z) = (z\,f(\lvert{v-u}\rvert, t-s))_{(v,t)\in \mathbb{K}} .\end{equation*}

Then direct manipulations show that $\nu$ is the Lévy measure of $(X_{v,t})_{(v,t)\in \mathbb{K}}$, and furthermore the Lévy–Khinchin representation is

\begin{align*}C(\beta \dagger (X_{v,t})_{(v,t)\in \mathbb{K}}) & = {\mathrm{i}} \sum_{{(v,t)}} \beta_{v,t}a_{v,t}- \dfrac12 \theta \int_{\mathbb{R}^d} \int_{\mathbb{R}} \biggl( \sum_{{(v,t)}} \beta_{v,t}\,f(\lvert{v-u}\rvert, t-s) \biggr)^2 \,\mathrm{d}s \,\mathrm{d}u \\* &\quad\,+ \int_{{\mathbb{R}}^\mathbb{K}} \biggl( \mathrm{e}^{{\mathrm{i}}\sum_{{(v,t)}}\beta_{v,t} z_{v,t}} - 1 - {\mathrm{i}}\sum_{{(v,t)}}\beta_{v,t}z_{v,t} 1_{[-1,1]^{\mathbb{K}}}(z) \biggr) \nu( \mathrm{d} z)\end{align*}

for suitable $(a_{v,t})_{(v,t)\in \mathbb{K}} \in{\mathbb{R}}^{\mathbb{K}}$. Here $\beta\in{\mathbb{R}}^{\mathbb{K}}$ with $\beta_{v,t}\neq 0$ for at most finitely many ${(v,t)\in \mathbb{K}}$. From the infinite divisibility, $(X_{v,t})_{(v,t)\in \mathbb{K}}$ can be represented as the independent sum

\begin{equation*}X_{v,t} = X_{v,t}^1 + X_{v,t}^2 .\end{equation*}

The field $(X_{v,t}^1)_{(v,t)\in \mathbb{K}}$ is a compound Poisson sum

\begin{equation*}X_{v,t}^1 = \sum_{n=1}^N V_{v,t}^n ,\end{equation*}

where N is Poisson-distributed with intensity $\nu(A)<\infty$ and

\begin{equation*}A = \biggl\{z\in{\mathbb{R}}^\mathbb{K} \colon \sup_{(v,t)\in \mathbb{K}} z_{v,t} > 1 \biggr\} .\end{equation*}

The finiteness of $\nu(A)$ follows from arguments similar to those of [Reference Rønn-Nielsen and Jensen19, Lemma A.1] using (2.12). The fields $((V_{v,t}^n)_{(v,t)\in \mathbb{K}})_{n\in\mathbb{N}}$ are independent and identically distributed (i.i.d.) with common distribution $\nu_1 = \nu_A / \nu(A)$, i.e. the normalization of the restriction of $\nu$ to A. Also, $(X_{v,t}^2)_{(v,t)\in \mathbb{K}}$ is infinitely divisible and has Lévy measure $\nu_{A^c}$, the restriction of $\nu$ to $A^c$.

It will be essential that there exist extensions of the fields $(X_{v,t}^1)$ and $(X_{v,t}^2)$ to $B^{\prime}\times T^{\prime}$ with t-càdlàg sample paths. In law, each of the fields $(V_{v,t}^n)$ can be represented by $(Z\,f(|v-U|,t-S))_{(v,t)\in \mathbb{K}}$, where $(U,S,Z)\in {{\mathbb{R}^d} \times {\mathbb{R}}}\times{\mathbb{R}}$ has distribution $F_1$, the normalized restriction of F to the set

\begin{equation*}H^{-1}(A) = \biggl\{(u,s,z)\in {{\mathbb{R}^d} \times {\mathbb{R}}}\times{\mathbb{R}} \colon \sup_{(v,t)\in \mathbb{K}} z\,f(\lvert{v-u}\rvert, t-s) >1\biggr\} .\end{equation*}

Hence a t-càdlàg extension $(V_{v,t})_{(v,t)\in B^{\prime}\times T^{\prime}}$ clearly exists, and it is represented by $(Z\,f(|v-U|,t-S))_{(v,t)\in B^{\prime}\times T^{\prime}}$. As $X^1$ is a finite sum of such fields it also has an extension to $B^{\prime}\times T^{\prime}$ which is t-càdlàg. As mentioned above and shown in Theorem 5.3, the entire field $(X_{v,t})_{(v,t)\in B^{\prime}\times T^{\prime}}$ has a version with t-càdlàg sample paths, and hence $X^2$ also has an extension with such paths.

3. Functional assumptions and main theorem

In this section we introduce assumptions on $\Psi$ and related functionals, and we derive the main theorem on the asymptotic behaviour of the extremal probability $\mathbb{P}(\Psi(X_{v,t}) > x)$ as $x\to\infty$. As the proofs of some of the results follow the same ideas as in [Reference Rønn-Nielsen and Jensen19] and [Reference Rønn-Nielsen and Jensen20], we refer to the supplementary material for these [Reference Stehr and Rønn-Nielsen27].

Throughout this section we shall assume the following.

Proof. Statement (i) is seen as follows. Let x, u, s be fixed, and assume for contradiction the existence of $\varepsilon > 0$ such that $\psi_{x,u,s,f}(y_{v,t}) + \varepsilon = \psi_{x,u,s,f}(y_{v,t} + z_{v,t}) .$ Now choose a such that $a-\varepsilon \le \psi_{x,u,s,f}(y_{v,t}) < a ,$ and therefore $\psi_{x,u,s,f}(y_{v,t} + z_{v,t}) \ge a .$ However, appealing to Assumptions 3.1 (ii) and 3.1 (iii) we also conclude that

\begin{equation*}x < \Psi(a f(\lvert{v-u}\rvert, t-s) + y_{v,t} ) \le \Psi(a f(\lvert{v-u}\rvert, t-s) + y_{v,t} + z_{v,t}) ,\end{equation*}

so also $\psi_{x,u,s,f}(y_{v,t} + z_{v,t}) < a$, a contradiction.

Parts (ii) and (iii) are seen using Assumptions 3.1 (i) and 3.1 (iii).

Before giving two examples of functionals easily seen to satisfy Assumption 3.1, we introduce some notation. Let $D \subseteq C_r(0) \subseteq {\mathbb{R}^d}$ be a fixed spatial object, and for all rotations $R\in SO(d)$ and translations $v\in {\mathbb{R}^d}$, define $D^R(v)=RD + v$. Similarly, let $D(v)=D + v$. Furthermore, let $I(t)=[t,t+\ell]$ for all $t\ge 0$. In Example 3.1 below we assume that the set D in fact has radius $r/2\ge 0$, by which we mean there exists $\alpha \in \mathbb{S}^{d-1}$ such that $\{-\alpha r/2, \alpha r/2\} \subseteq D \subseteq C_{r/2}(0)$.

Example 3.1. Suppose we are interested in the probability that there exist a time point t, a translation $v_0$, and a rotation R of a given set D such that the field exceeds the level x on the entire set $\{t\} \times D^R(v_0)$. More formally, we assume that $D \subseteq C_{r/2}(0) \subseteq {\mathbb{R}^d}$ has radius $r/2$ and study the probability

\begin{equation*}\mathbb{P} (\text{there exist } t\in[0,T], v_0\in B, R\in SO(d) \colon X_{v,t} > x\text{ for all } v\in D^R(v_0)) .\end{equation*}

To put this within the more general framework introduced in (2.11), we define $\Psi$ by

\begin{equation*}\Psi(y_{v,t}) = \sup_{t\in [0,T]}\sup_{v_0\in B} \sup_{R\in SO(d)}\inf_{v\in D^R(v_0)} y_{v,t} .\end{equation*}

Consequently (obtained by straightforward manipulations),

\begin{equation*}\psi_{x,u,s,f}(y_{v,t}) = \inf_{t\in [0,T]}\inf_{v_0\in B} \inf_{R\in SO(d)}\sup_{v\in D^R(v_0)} \dfrac{x-y_{v,t}}{f(\lvert{v-u}\rvert, t-s)} .\end{equation*}

Note that the probability above can be reformulated in terms of the excursion set $A_{x,t}$, defined in (1.2), as

\begin{equation*}\mathbb{P} (\text{there exist } t\in[0,T], v_0\in B, R\in SO(d) \colon D^R(v_0)\subseteq A_{x,t} ) .\end{equation*}

Example 3.2. Suppose we are interested in the probability that there is a time interval and a location and rotation of the fixed spatial object D, in which the average of the field exceeds the level x. For this, let $D \subseteq C_r(0) \subseteq {\mathbb{R}^d}$ be given and consider the probability

\begin{equation*}\mathbb{P} \biggl(\text{there exist } t_0\in[0,T], v_0\in B, R\in SO(d) \colon \dfrac{1}{K} \int_{D^R(v_0)} \int_{I(t_0)}X_{v,t} \,\mathrm{d} t \,\mathrm{d} v > x\biggr) ,\end{equation*}

where $K=\int_D \int_0^\ell 1\,\mathrm{d} t \,\mathrm{d} v$. The set D can both be of full dimension in ${\mathbb{R}^d}$ and a subset of some lower-dimensional subspace. In either case, $\mathrm{d} v$ refers to the relevant version of the Lebesgue measure. The special cases of

\begin{equation*}\mathbb{P} \biggl(\text{there exist } t\in[0,T], v_0\in B, R\in SO(d) \colon \dfrac{1}{K} \int_{D^R(v_0)} X_{v,t} \,\mathrm{d} v > x\biggr) ,\end{equation*}

with a time point instead of an interval (and K defined appropriately), and

\begin{equation*}\mathbb{P} \biggl(\text{there exist } t_0\in[0,T], v\in B \colon \dfrac{1}{K} \int_{I(t_0)}X_{v,t} \,\mathrm{d} t > x\biggr) ,\end{equation*}

with a single spatial point, will be covered by the general formulation of the example, simply by defining $\int_{I(t_0)}X_{v,t} \,\mathrm{d} t = X_{v,t_0}$ when $\ell=0$ and $\int_{D^R(v_0)} X_{v,t} \,\mathrm{d} v = X_{v_0,t}$ when $D=\{0\}$ and hence $D^R(v_0)=\{v_0\}$. In the same spirit, the special case of

\begin{equation*}\mathbb{P} (\text{there exist } t\in[0,T], v\in B \colon X_{v,t} > x)\end{equation*}

corresponds to letting $\ell=0$ and $D=\{0\}$. Note that this probability could be formulated alternatively as

\begin{equation*}\mathbb{P} \biggl(\sup_{t\in [0,T]}\sup_{v\in B}X_{v,t} > x\biggr) .\end{equation*}

To put this example into the framework of functionals, we define

\begin{equation*}\Psi(y_{v,t}) = \sup_{t_0\in [0,T]} \sup_{v_0\in B} \sup_{R\in SO(d)} \dfrac{1}{K}\int_{D^R(v_0)} \int_{I(t_0)} {y_{v,t}} \, \mathrm{d} t \,\mathrm{d} v,\end{equation*}

leading to

\begin{equation*}\psi_{x,u,s,f}(y_{v,t}) = \inf_{t_0\in [0,T]}\inf_{v_0\in B} \inf_{R\in SO(d)} \dfrac{x-{{K}^{-1}}\int_{D^R(v_0)} \int_{I(t_0)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v}{{{K}^{-1}}\int_{D^R(v_0)} \int_{I(t_0)} f(\lvert{v-u}\rvert, t-s) \, \mathrm{d} t \,\mathrm{d} v} .\end{equation*}

For the further arguments to hold it will be important that $\psi_{x,u,s,f}$ converges in a particular way as $x\to\infty$. The following assumption is satisfied under further case-specific assumptions on the kernel f in each of the Examples 3.1 and 3.2, as illustrated in Section 4.

Assumption 3.2. With the functionals $\Psi$ and $\psi_{x,u,s,f}$ as in Assumption 3.1, there exists c such that

(3.1) \begin{equation}c = \Psi((f(\lvert{v-u}\rvert, t-s))_{(v,t)})\end{equation}

for all ${(u,s)\in B\times [0,T]}$ and $\Psi((f(\lvert{v-u}\rvert, t-s))_{(v,t)})<c$ for all ${(u,s)}\not\in B\times [0,T]$. Furthermore, for all ${(u,s)\in B\times [0,T]}$ there is a functional $\lambda_{u,s}\colon{\mathbb{R}}^{B^{\prime}\times T^{\prime}}\to{\mathbb{R}}$, such that

(3.2) \begin{equation}\psi_{x,u,s,f}(y_{v,t}) - \dfrac{x}{c} + \lambda_{u,s}(y_{v,t}) \to 0\end{equation}

as $x\to\infty$, holds for all t-càdlàg fields $(y_{v,t})_{(v,t)\in B^{\prime}\times T^{\prime}}$.

The following proposition is easily seen from Assumption 3.1 and Proposition 3.1.

In the remainder of this section it is assumed that Assumption 3.2 is also satisfied.

The first step in proving the asymptotic behaviour of the extremal probability $\mathbb{P}(\Psi(X_{v,t})>x)$ is to consider the asymptotic behaviour of extremal sets of a single jump-field

\begin{equation*}V=(V_{v,t})_{{(v,t)\in B^{\prime}\times T^{\prime}}}\end{equation*}

with distribution $\nu_1$.

Theorem 3.1. Let $(V_{v,t})_{{(v,t)\in B^{\prime}\times T^{\prime}}}$ have distribution $\nu_1$ and let $(y_{v,t})_{{(v,t)\in B^{\prime}\times T^{\prime}}}$ be t-càdlàg. As $x\to\infty$, it holds that

(3.3) \begin{align}\dfrac{\mathbb{P}(\Psi(V_{v,t} + y_{v,t}) > x)}{L(x/c) \exp({-}\beta x/c)}& \to\dfrac{1}{\nu(A)} \int_B\int_0^T \exp(\beta \lambda_{u,s}(y_{v,t})) \,\mathrm{d}s \,\mathrm{d}u .\end{align}

Proof. For sufficiently large $x>0$ we find

(3.4) \begin{align} &\nu(A) \, \mathbb{P}(\Psi(V_{v,t} + y_{v,t}) > x) \notag \\*& \quad= F ( \{ (u,s,z)\in {\mathbb{R}^d} \times {\mathbb{R}} \times {\mathbb{R}}_+ \colon \Psi (z\,f(\lvert{v-u}\rvert, t-s) + y_{v,t} )> x \}) \notag \\*&\quad= F ( \{ (u,s,z)\in {\mathbb{R}^d} \times {\mathbb{R}} \times {\mathbb{R}}_+ \colon \psi_{x,u,s,f}(y_{v,t})<z \}) \notag \\& \quad = \int_{B\times [0,T]} L(\psi_{x,u,s,f}(y_{v,t})) \exp({-}\beta \psi_{x,u,s,f}(y_{v,t}) ) m(\mathrm{d}u,\mathrm{d}s) \notag \\ & \quad \quad\, + \int_{(B\times [0,T])^c} L(\psi_{x,u,s,f}(y_{v,t})) \exp({-}\beta \psi_{x,u,s,f}(y_{v,t}) ) m(\mathrm{d}u,\mathrm{d}s) .\end{align}

First we show that the latter integral is of order ${\mathrm{o}}(L(x/c)\exp({-}\beta x/c))$ as $x\rightarrow \infty$. Let $y^*=\sup_{(v,t)\in B^{\prime}\times T^{\prime}} y_{v,t}$. Using Proposition 3.1 (iii) and the fact that $x\mapsto L(x)\exp({-}\beta x)$ is decreasing, we obtain that the second integral in (3.4) is bounded from above by

\begin{equation*} \int_{(B\times [0,T])^c}L\biggl(\dfrac{x-y^*}{\Psi(f(\lvert{v-u}\rvert, t-s))}\biggr) \exp\biggl({-}\beta \dfrac{x-y^*}{\Psi(f(\lvert{v-u}\rvert, t-s))} \biggr) m(\mathrm{d}u,\mathrm{d}s).\end{equation*}

Let $h(u, s;\, x)$ denote the integrand. For all $(u,s)\in (B\times [0,T])^c $ we have $\Psi( f(\lvert{v-u}\rvert,\break t-s)) < c$. In combination with (2.4) and (2.5), this implies the existence of $\gamma >0$ and $C>0$ such that

\begin{equation*}\dfrac{h(u,s;\,x)}{L(x/c)\exp({-}\beta x/c)}\le C \exp({-}\gamma x)\end{equation*}

for sufficiently large x. Thus $h(u,s;\,x)$ is of order ${\mathrm{o}}(L(x/c)\exp({-}\beta x/c))$ at infinity. By dominated convergence, the integral is also of order ${\mathrm{o}}(L(x/c)\exp({-}\beta x/c))$ if we can find an integrable function $g\colon{{\mathbb{R}^d} \times {\mathbb{R}}}\to{\mathbb{R}}$ such that

\begin{equation*}\dfrac{h(u,s;\,x)}{L(x/c)\exp({-}\beta x/c)} \le g(u,s)\end{equation*}

for all $(u,s)\in{{\mathbb{R}^d} \times {\mathbb{R}}}$. Returning to (2.5) we see that for all $0<\gamma<\beta/c$ there exist $C>0$ and $x_0>y^*$ such that

(3.5) \begin{equation}\dfrac{h(u,s;\,x)}{L(x/c)\exp({-}\beta x/c)}\le C \exp\biggl({-}(x_0-y^*)(\beta-\gamma c)\biggl(\dfrac{1}{\Psi(f(\lvert{v-u}\rvert, t-s))} -\dfrac{1}{c}\biggr)\biggr)\end{equation}

for all $x\ge x_0$. Independent of (u, s) there is a constant $\tilde C$ such that the right-hand side of (3.5) is bounded by

\begin{equation*}\tilde C \, \Psi( f(\lvert{v-u}\rvert, t-s))\le \tilde C \sup_{{(v,t)\in B^{\prime}\times T^{\prime}}} f(\lvert{v-u}\rvert, t-s),\end{equation*}

where we used Assumptions 3.1 (i) and 3.1 (ii). By Assumption 2.3, this is integrable.

It remains to show that the first integral in (3.4) has the desired mode of convergence. From (3.2), the representation of L, and the fact that $\rho$ has an exponential tail, we have that, for any ${(u,s)\in B\times [0,T]}$,

\begin{equation*}\dfrac{L (\psi_{x,u,s,f}(y_{v,t})) \exp({-}\beta \psi_{x,u,s,f}(y_{v,t}) )}{L ({{x}/{c}}) \exp ({-}\beta {{x}/{c}})}\to \exp ( \beta \lambda_{u,s}(y_{v,t}))\end{equation*}

as $x\to \infty$. Since $x\mapsto L(x)\exp({-}\beta x)$ is decreasing, we find using Proposition 3.1 (iii) that, for sufficiently large x,

\begin{equation*}\dfrac{L(\psi_{x,u,s,f}(y_{v,t})) \exp({-}\beta \psi_{x,u,s,f}(y_{v,t}) )}{L({{x}/{c}}) \exp({-}\beta {{x}/{c}})}\le \dfrac{L({{(x-y^*)}/{c}}) \exp({-}\beta ({{(x-y^*)}/{c}}))}{L({{x}/{c}}) \exp({-}\beta {{x}/{c}})} \le C \exp(\beta y^*/c)\end{equation*}

for any ${(u,s)\in B\times [0,T]}$, where, according to (2.4), C is such that

\begin{equation*}\frac{L((x-y^*)/c)}{L(x/c)}\leq C.\end{equation*}

As $B\times [0,T]$ is compact, the upper bound is integrable over $B\times [0,T]$, and (3.3) then follows by dominated convergence.

The next step is to extend the relation (3.3) to an asymptotic result for $\mathbb{P}(\Psi(V_{v,t}^1 + \cdots + V_{v,t}^n + y_{v,t}) > x)$, where, for $i=1,\ldots,n$, $V^i$ are i.i.d. with common distribution $\nu_1$. Here it will be useful to recall that each $V^i$ can be represented by $(Z^i f(|v-U^i|,t-S^i))_{(v,t)\in B^{\prime}\times T^{\prime}}$, where $(U^i,S^i,Z^i)$ has distribution $F_1$. Before being able to extend (3.3), we need a final assumption on the existence of a function $\phi$ ensuring sufficient integrability properties.

For the assumption we need some notation representing a deterministic version of the sum $V_{v,t}^1 + \dots + V_{v,t}^n$. Thus, for each $i=1,\ldots,n$, let the field $(y_{v,t}^i)_{(v,t)\in B'\times T^{\prime}}$ be given by

\begin{equation*}y_{v,t}^i = z^i f\big(\big|v-u^i\big|, t-s^i\big) ,\end{equation*}

where all $z^i \ge 0$, $u^i\in {\mathbb{R}^d}$, and $s^i\in{\mathbb{R}}$.

Assumption 3.3. There exists a Lebesgue-integrable function $\phi\colon{\mathbb{R}^d}\times {\mathbb{R}} \to [0,\infty)$ such that

(3.6) \begin{equation}\phi(u,s)\begin{cases}= c & \textit{for $ {(u,s)\in B^{\prime}\times T^{\prime}}$,} \\ < c & \textit{for $ (u,s)\not\in B^{\prime}\times T^{\prime} $,}\end{cases}\end{equation}

where $c>0$ is the constant defined in (3.1).

The function $\phi$ satisfies

(3.7) \begin{equation} \Psi \biggl( \sum_{i=1}^n y_{v,t}^i \biggr) \le \sum_{i=1}^n z^i \phi\big(u^i, s^i\big) ,\end{equation}

and

(3.8) \begin{equation}\sup_{s\in[0,T]} \sup_{u\in B} \lambda_{u,s}\biggl(\sum_{i=1}^n y_{v,t}^i\biggr)\le \dfrac{1}{c}\sum_{i=1}^n z^i \phi\big(u^i,s^i\big) .\end{equation}

The definition of $\phi$ ensures that the tail of $Z\phi(U,S)$ is asymptotically equivalent to $\rho((x/c,\infty))$ and hence $Z\phi(U,S)$ is convolution equivalent with index $\beta/c$; see Lemma 3.1 below. Equation (3.7) then provides a convolution equivalent upper bound of the extremal probability for the functional $\Psi$ of a sum of jump-fields. Finally, finiteness of relevant exponential moments of $\lambda_{u,s}$ applied to jump-fields is ensured by (3.8). This result is seen in Theorem 3.2 below.

When showing the convolution equivalence of $Z\phi(U,S)$, we use the integrability of $\phi$, although the weaker assumption that $\exp({-}\gamma/\phi(u,s))$ is integrable for some $\gamma >0$ is sufficient; see the proof of Lemma 3.1 in the supplementary material [Reference Stehr and Rønn-Nielsen27]. However, as seen in Section 4, in practice $\phi(u,s)$ is often bounded by $\sup_{(v,t)} f(\lvert{v-u}\rvert, t-s)$ and hence its integrability follows from that of $\sup_{(v,t)} f(\lvert{v-u}\rvert, t-s)$.

In the remainder of this section it is also assumed that Assumption 3.3 is satisfied. The proof of Lemma 3.1 below follows by arguments similar to the proof of Theorem 3.1 above, but for completeness the proof can be found in the supplementary material [Reference Stehr and Rønn-Nielsen27].

Lemma 3.1. Let (U, S, Z) have distribution $F_1$. Then, as $x\to\infty$,

(3.9) \begin{equation}\dfrac{\mathbb{P}(Z\phi(U,S) > x)}{L(x/c) \exp(\!-\beta x/c)}\to\dfrac{1}{\nu(A)} m(B^{\prime}\times T^{\prime}) .\end{equation}

In particular, the distribution of $Z \phi(U,S)$ is convolution equivalent with index $\beta/c$ and

\begin{equation*} {\mathbb E} \biggl[ \exp\biggl( \dfrac{\beta}{c} Z\phi(U,S)\biggr) \biggr] < \infty .\end{equation*}

As mentioned, the convolution equivalence of $Z \phi(U,S)$ is translated into a convolution equivalent upper bound for the extremal probability of a sum of jump-fields. Here (3.7) is applied together with the relation

\begin{equation*}\overline{ F^{*n}}(x) \sim n \overline F(x) \biggl(\int \mathrm{e}^{\beta y} F(\mathrm{d} y) \biggr)^{n-1},\quad x\to\infty,\end{equation*}

when F is a convolution equivalent distribution with index $\beta$, $F^{*n}$ is its n-fold convolution, and $\overline F$ is its tail. For this relation see e.g. [Reference Cline9, Corollary 2.11].

In Theorem 3.2 below, a similar convolution equivalence for the sum of jump-fields is obtained.

Theorem 3.2. Let $V^1,V^2,\dots$ be i.i.d. fields with common distribution $\nu_1$, and assume that $(y_{v,t})_{{(v,t)\in B^{\prime}\times T^{\prime}}}$ is t-càdlàg. For all $n\in\mathbb{N}$ it holds that

\begin{align*} &\dfrac{\mathbb{P}(\Psi(V_{v,t}^1 + \dots + V_{v,t}^n + y_{v,t}) > x)}{\mathbb{P}(\Psi(V_{v,t}^1) > x)} \\* &\quad \to\dfrac{n}{m(B\times [0,T])} \int_B \int_0^T {\mathbb E} \big[\exp\big( \beta \lambda_{u,s}\big(V_{v,t}^1 + \dots + V_{v,t}^{n-1} + y_{v,t}\big) \big) \big] \,\mathrm{d}s \,\mathrm{d}u\end{align*}

as $x \to \infty$.

Recall that the field $X^1$ is defined as the compound Poisson sum with i.i.d. jump-fields $V^1,V^2,\dots$ and an independent Poisson-distributed variable N with intensity $\nu(A)<\infty$. The following result on the extremal behaviour of $X^1$ follows from Theorem 3.2 by conditioning on the value of N.

Theorem 3.3. For each $(u,s)\in B\times [0,T]$,

\begin{equation*} {\mathbb E} \big[\exp \big(\beta \lambda_{u,s}\big(X_{v,t}^1\big) \big) \big] < \infty. \end{equation*}

For a field $(y_{v,t})_{{(v,t)\in B^{\prime}\times T^{\prime}}}$ satisfying (2.6),

\begin{equation*}\dfrac{\mathbb{P}\big(\Psi\big(X_{v,t}^1+y_{v,t}\big) > x \big)}{L(x/c)\exp({-}\beta x/c)}\to \int_B\int_0^T {\mathbb E} \big[\exp \big( \beta \lambda_{u,s}\big(X_{v,t}^1 + y_{v,t}\big) \big) \big] \,\mathrm{d}s \,\mathrm{d}u\end{equation*}

as $x\to\infty$.

Now recall that we write the field $X=(X_{v,t})_{(v,t)\in B^{\prime}\times T^{\prime}}$ defined in (2.10) as the independent sum $X=X^1+X^2$, where $X^1$ is the compound Poisson sum of fields with distribution $\nu_1$. Also, the fields in the decomposition can be assumed to be t-càdlàg. One can show that the tail of $\sup_{{(v,t)}} X_{v,t}^2$ is lighter than that of $\Psi(X_{v,t}^1)$, which is equivalent to the tail of $\rho$ by Theorem 3.3. Combining this fact with [Reference Pakes16, Lemma 2.1], an argument based on independence and dominated convergence can be used to conclude Theorem 3.4 below from Theorem 3.3.

Theorem 3.4. Let the field X be given by (2.10), where the Lévy basis M satisfies Assumption 2.1 and the kernel function f satisfies Assumption 2.3. Let the functionals $\Psi$ and $\lambda_{u,s}$ satisfy Assumptions 3.1 and 3.2, respectively. Then

\begin{equation*}\lim_{x\to\infty} \dfrac{\mathbb{P}( \Psi(X_{v,t}) > x)}{\rho((x/c,\infty))}= \int_B\int_0^T {\mathbb E} [ \exp(\beta \lambda_{u,s}(X_{v,t}) )]\,\mathrm{d}s \,\mathrm{d}u.\end{equation*}

4. Example results

In this section we return to Examples 3.1 and 3.2 to show versions of Theorem 3.4 when $\Psi$ is specifically given as in the examples. We make further assumptions on the kernel f that guarantee Assumptions 3.2 and 3.3.

In the setting of Example 3.1 we assume the following.

Assumption 4.1. The kernel $f\colon[0,\infty)\times {\mathbb{R}} \to [0,\infty)$ is decreasing in both coordinates on $[0,\infty) \times [0,\infty)$, and it is strictly decreasing in the point $(r/2,0)$ in the sense that

(4.1) \begin{equation}f(x,y)<f(r/2,0)\quad \textit{for all }(x,y) \in ( [r/2,\infty) \times [0,\infty) ) \setminus \{ (r/2,0) \} .\end{equation}

Moreover, the derivative

\begin{equation*}f_1(x) = \dfrac{\partial f}{\partial x}(x,0)\end{equation*}

exists for all $x \ge 0$, and there is a function g such that

(4.2) \begin{equation}g(x) = f_1(r/2)(x-r/2) + f(r/2,0)\end{equation}

for all $x\in [0,r]$, where also $f(x,0) \le g(x) $ for all $x\in [0,r]$.

Such a g exists in particular when f is concave on [0, r]. The following lemma shows that Assumption 3.2 is satisfied when the kernel satisfies Assumption 4.1.

Lemma 4.1. If $\Psi$ and $\psi_{x,u,s,f}$ are given as in Example 3.1 and f satisfies Assumption 4.1, then Assumption 3.2 is satisfied with $c=f(r/2,0)$. Furthermore, for a t-càdlàg field $y=(y_{v,t})_{(v,t)\in B^{\prime} \times [0,T]}$, the functional $\lambda_{u,s}$ takes the form

\begin{equation*}\lambda_{u,s}((y_{v,t})_{(v,t)\in B^{\prime} \times [0,T]}) = \lambda_u( (y_{v,s})_{v\in B^{\prime}})\end{equation*}

for a functional $\lambda_u\colon{\mathbb{R}}^{B^{\prime}} \to {\mathbb{R}}$.

Proof. From [Reference Rønn-Nielsen and Jensen20, Lemma 3.1] we have, for fixed $s\in [0,T]$ and for all $u\in B$, a functional $\lambda_{u}$ such that

(4.3) \begin{equation}\inf_{v_0\in B} \inf_{R\in SO(d)} \sup_{v\in D^R(v_0)} \dfrac{x - y_{v,s}}{f(\lvert{v-u}\rvert,0)}- \dfrac{x}{f(r/2,0)} + \lambda_{u}((y_{v,s})_{v\in B^{\prime}})\to 0\end{equation}

as $x\to\infty$. With $\lambda_{u,s}$ defined by

\begin{equation*}\lambda_{u,s}((y_{v,t})_{(v,t)\in B^{\prime}\times [0,T]}) = \lambda_u((y_{v,s})_{v\in B^{\prime}}),\end{equation*}

we claim that Assumption 3.2 is satisfied. For notational convenience, we write $C=-\lambda_u(y_{v,s})$.

For all sufficiently large x, we can choose $t_x\in [s,T]$, $v_x\in B$, and $R_x \in SO(d)$ such that

\begin{equation*}\sup_{v\in D^{R_x}(v_x)} \dfrac{x - y_{v,t_x}}{f(\lvert{v-u}\rvert,t_x-s)}= \inf_{t\in[0,T]}\inf_{v_0\in B} \inf_{R\in SO(d)} \sup_{v\in D^R(v_0)} \dfrac{x - y_{v,t}}{f(\lvert{v-u}\rvert,t-s)} .\end{equation*}

With $y^* = \sup_{(v,t)\in B^{\prime} \times [0,T]} y_{v,t}$ and $y_* = \inf_{(v,t)\in B^{\prime} \times [0,T]} y_{v,t}$, we then find

\begin{align*}\dfrac{x - y^*}{\inf_{v\in D^{R_x}(v_x)} f(\lvert{v-u}\rvert,t_x-s)} &\le \sup_{v\in D^{R_x}(v_x)} \dfrac{x - y_{v,t_x}}{f(\lvert{v-u}\rvert,t_x-s)} \\* &\le \sup_{v\in D^{R_x}(u)} \dfrac{x - y_{v,s}}{f(\lvert{v-u}\rvert,0)}\\*&\le \dfrac{x - y_*}{f(r/2,0)} .\end{align*}

Going to the limit $x\to\infty$ and using that $\inf_{v\in D^R(v_0)} f(\lvert{v-u}\rvert,t-s)\leq f(r/2,0)$ shows that

\begin{equation*}\inf_{v\in D^{R_x}(v_x)} f(\lvert{v-u}\rvert,t_x-s) \to f(r/2,0)\quad \text{as $x\to\infty$.}\end{equation*}

Since in fact $\inf_{v\in D^R(v_0)} f(\lvert{v-u}\rvert,t-s)<f(r/2,0)$ for all $(v_0,t)\neq (u,s)$ and $R\in SO(d)$, the convergence implies that also $v_x \to u$ and $t_x \to s$. We will show the desired convergence

\begin{equation*}\sup_{v\in D^{R_x}(v_x)} \dfrac{x - y_{v,t_x}}{f(\lvert{v-u}\rvert,t_x-s)} - \dfrac{x}{f(r/2,0)}\to C\quad(x\to\infty)\end{equation*}

by contradiction. Since

\begin{equation*}\sup_{v\in D^{R_x}(v_x)} \dfrac{x - y_{v,t_x}}{f(\lvert{v-u}\rvert,t_x-s)}\le\inf_{v_0\in B} \inf_{R\in SO(d)} \sup_{v\in D^R(v_0)} \dfrac{x - y_{v,s}}{f(\lvert{v-u}\rvert,0)} ,\end{equation*}

we assume the existence of $\varepsilon >0$ and a sequence $(x_n)$, $x_n\to\infty$, such that

(4.4) \begin{equation}\sup_{v\in D^{R_n}(v_n)} \dfrac{x_n - y_{v,t_n}}{f(\lvert{v-u}\rvert,t_n-s)} - \dfrac{x_n}{f(r/2,0)}\le C - \varepsilon\end{equation}

for all n, where $t_n = t_{x_n}$, $v_n=v_{x_n}$ and $R_n = R_{x_n}$. By t-càdlàg properties of the y-field, we can find $n_0$ such that

\begin{equation*}\sup_{v\in B^{\prime}} \biggl| \dfrac{y_{v,t_n} - y_{v,s}}{f(\lvert{v-u}\rvert,0)}\biggr|\le \dfrac{\varepsilon}{2}\end{equation*}

for all $n\ge n_0$. Consequently, and using that f is decreasing and (4.4),

\begin{align*} \sup_{v\in D^{R_n}(v_n)} \dfrac{x_n - y_{v,s}}{f(\lvert{v-u}\rvert,0)} - \dfrac{x_n}{f(r/2,0)} & \leq \sup_{v\in D^{R_n}(v_n)} \dfrac{x_n - y_{v,t_n}}{f(\lvert{v-u}\rvert,0)} - \dfrac{x_n}{f(r/2,0)} +\dfrac{\varepsilon}{2}\\* & \leq \sup_{v\in D^{R_n}(v_n)} \dfrac{x_n - y_{v,t_n}}{f(\lvert{v-u}\rvert,t_n-s)} - \dfrac{x_n}{f(r/2,0)}+\dfrac{\varepsilon}{2}\\* &\leq C-\dfrac{\varepsilon}{2},\end{align*}

which contradicts the limit relation (4.3).

Theorem 4.1. Let the field X be given by (2.10), where the Lévy basis M satisfies Assumption 2.1 and the kernel function f satisfies Assumptions 2.3 and 4.1. Let $D\subseteq C_{r/2}(0)$ have radius $r/2>0$ and let $\Psi$ be defined by

\begin{equation*}\Psi(y_{v,t}) = \sup_{t\in [0,T]}\sup_{v_0\in B} \sup_{R\in SO(d)}\inf_{v\in D^R(v_0)} y_{v,t} .\end{equation*}

Furthermore, let $\lambda_{u,s}$ be the functional given in Lemma 4.1 and write $c=f(r/2,0)$. Then

\begin{equation*}\lim_{x\to\infty} \dfrac{\mathbb{P}( \Psi(X_{v,t})>x)}{\rho((x/c,\infty))}= \int_B\int_0^T{\mathbb E} [ \exp(\beta \lambda_{u,s} (X_{v,t}) )]\,\mathrm{d}s \,\mathrm{d}u.\end{equation*}

Proof. The result follows from Theorem 3.4 and Lemma 4.1 once we show the existence of a function $\phi$ satisfying Assumption 3.3. Now define $\phi$ as

\begin{equation*}\phi(u,s) = f(r/2,0) \, \textbf{1}_{B^{\prime} \times [0,T]}(u,s)+ \sup_{t\in [0,T]} \sup_{v\in B \oplus C_{r/2}} f(\lvert{v-u}\rvert, t-s) \, \textbf{1}_{(B^{\prime} \times [0,T])^c}(u,s) ,\end{equation*}

which is integrable by (2.12) and satisfies (3.6) by (4.1). Appealing to Lemma 4.1 and [Reference Rønn-Nielsen and Jensen20, Lemma 3.2],

(4.5) \begin{equation}\lambda_{u,s}(y_{v,t}) = \dfrac{1}{2 f(r/2,0)}\sup_{\alpha\in\mathbb{S}^{d-1}}(y_{u+\alpha r/2,s} + y_{u-\alpha r/2,s})\end{equation}

for all ${(u,s)\in B\times [0,T]}$, if $D=\{-\alpha r/2, \alpha r/2\}$ for some $\alpha \in \mathbb{S}^{d-1}$. Adapting the proof of [Reference Rønn-Nielsen and Jensen20, Lemma 3.3] to this time-dependent setting, it is seen using (4.5) that (3.7) and (3.8) follow when it is shown that

(4.6) \begin{equation}\dfrac{1}{2} \big( y^i_{u+\alpha r/2,s} + y^i_{u-\alpha r/2,s} \big)\le z^i \phi\big(u^i,s^i\big)\quad \text{for all }{(u,s)\in B\times [0,T]} ,\ \alpha \in \mathbb{S}^{d-1},\end{equation}

with $y_{v,t}^i$ defined as just before Assumption 3.3. Since f is decreasing in both coordinates,

\begin{equation*}\dfrac{1}{2} \big( y^i_{u+\alpha r/2,s} + y^i_{u-\alpha r/2,s} \big)\le \dfrac{z^i}{2} \big( f\big( \lvert u+\alpha r/2 - u^i\rvert,0\big) + f\big( \lvert u-\alpha r/2 - u^i\rvert,0\big) \big).\end{equation*}

Using the upper bound g assumed by (4.2), arguments as in [Reference Rønn-Nielsen and Jensen20, Lemma 3.3] show that (4.6) is satisfied when $(u^i,s^i)\in B^{\prime} \times [0,T]$. When $(u^i,s^i)\in (B^{\prime} \times [0,T])^c$ it is immediately seen that

\begin{equation*}\dfrac{1}{2} \big( y^i_{u+\alpha r/2,s} + y^i_{u-\alpha r/2,s} \big)\le z^i\sup_{t\in [0,T]} \sup_{v\in B \oplus C_{r/2}} f\big(\lvert v-u^i\rvert, t-s^i\big)= z^i \phi\big(u^i,s^i\big) .\end{equation*}

This concludes the proof.

In the setting of Example 3.2 the following is assumed.

Assumption 4.2. For the set $D\subseteq C_r(0) \subseteq {\mathbb{R}^d}$ the kernel function f satisfies

(4.7) \begin{equation}\int_{D(v_0)} \int_{I(t_0)} f(\lvert{v-u}\rvert, t-s) \,\mathrm{d} t \,\mathrm{d} v < \int_{D} \int_{0}^\ell f(\lvert{v}\rvert,t) \,\mathrm{d} t \,\mathrm{d} v\end{equation}

for all $(v_0,t_0) \neq (u,s) \in {{\mathbb{R}^d} \times {\mathbb{R}}}$.

Lemma 4.2. Let $y=(y_{v,t})_{(v,t)\in B^{\prime}\times T^{\prime}}$ be a t-càdlàg field. For all ${(u,s)\in B\times [0,T]}$ it holds that

(4.8) \begin{align} & \inf_{t_0\in[0,T]} \inf_{v_0\in B} \inf_{R\in SO(d)} \dfrac{x-{{K}^{-1}}\int_{D^R(v_0)} \int_{I(t_0)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v} {{{K}^{-1}}\int_{D^R(v_0)} \int_{I(t_0)} f(\lvert{v-u}\rvert, t-s) \, \mathrm{d} t \,\mathrm{d} v} \notag \\[5pt] &\quad\, - \dfrac{x}{c} + \sup_{R\in SO(d)} \dfrac{1}{c} \dfrac{1}{K}\int_{D^R(u)}\int_{I(s)} y_{v,t} \,\mathrm{d} t \,\mathrm{d} v\to 0\end{align}

as $x\to\infty$, where

\begin{equation*}c=\dfrac{1}{K} \int_D \int_0^\ell f(\lvert{v}\rvert,t) \,\mathrm{d} t \,\mathrm{d} v.\end{equation*}

That is, with $\Psi$ and $\psi_{x,u,s,f}$ as in Example 3.2, and with

\begin{equation*}\lambda_{u,s}(y_{v,t})=\sup_{R \in SO(d)} \dfrac{1}{c} \dfrac{1}{K}\int_{D^R(u)}\int_{I(s)} y_{v,t} \,\mathrm{d} t \,\mathrm{d} v,\end{equation*}

Assumption 3.2 is satisfied.

Proof. For all sufficiently large $x>0$, choose $t_x\in[s,T]$, $v_x\in B$, and $R_x\in SO(d)$ with

\begin{equation*}\inf_{t_0,v_0,R} \dfrac{x-{{K}^{-1}}\int_{D^R(v_0)} \int_{I(t_0)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v}{{{K}^{-1}}\int_{D^R(v_0)} \int_{I(t_0)} f(\lvert{v-u}\rvert, t-s) \, \mathrm{d} t \,\mathrm{d} v}=\dfrac{x-{{K}^{-1}}\int_{D^{R_x}(v_x)} \int_{I(t_x)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v}{{{K}^{-1}}\int_{D^{R_x}(v_x)} \int_{I(t_x)} f(\lvert{v-u}\rvert, t-s) \, \mathrm{d} t \,\mathrm{d} v} .\end{equation*}

By definition of $t_x$ and $v_x$ we find that

\begin{align*}\dfrac{x-y^*}{{{K}^{-1}}\int_{D^{R_x}(v_x)} \int_{I(t_x)} f(\lvert{v-u}\rvert, t-s) \, \mathrm{d} t \,\mathrm{d} v} &\le\dfrac{x-{{K}^{-1}}\int_{D^{R_x}(v_x)} \int_{I(t_x)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v}{{{K}^{-1}}\int_{D^{R_x}(v_x)} \int_{I(t_x)} f(\lvert{v-u}\rvert, t-s) \, \mathrm{d} t \,\mathrm{d} v} \\[4pt] &\le\inf_{R\in SO(d)}\dfrac{x-{{K}^{-1}}\int_{D^{R}(u)} \int_{I(s)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v}{{{K}^{-1}}\int_{D(u)} \int_{I(s)} f(\lvert{v-u}\rvert, t-s) \, \mathrm{d} t \,\mathrm{d} v} \\[4pt] &= \dfrac{x-\sup_{R\in SO(d)}{{K}^{-1}}\int_{D^{R}(u)} \int_{I(s)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v}{c},\end{align*}

where $y^* = \sup y_{v,t}$. Rearranging and noting that

\begin{equation*}\dfrac{1}{K}\int_{D^R(v_0)} \int_{I(t_0)} f(\lvert{v-u}\rvert, t-s) \, \mathrm{d} t \,\mathrm{d} v < c\end{equation*}

for all $(v_0,t_0)\neq (u,s)$ and any $R\in SO(d)$, we conclude that

\begin{equation*}\dfrac{1}{K}\int_{D^{R_x}(v_x)} \int_{I(t_x)} f(\lvert{v-u}\rvert, t-s) \, \mathrm{d} t \,\mathrm{d} v \to c\end{equation*}

and consequently $v_x \to u$ and $t_x \to s$ as $x\to\infty$. Since the field $(y_{v,t})$ is t-càdlàg, we furthermore find that, as $x\to\infty$,

\begin{equation*}\sup_{R\in SO(d)} \int_{D^{R}(v_x)} \int_{I(t_x)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v\to\sup_{R\in SO(d)} \int_{D^{R}(u)} \int_{I(s)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v .\end{equation*}

Recalling that

\begin{align*}\dfrac{1}{K}\int_{D^{R_x}(v_x)} \int_{I(t_x)} f(\lvert{v-u}\rvert, t-s) \, \mathrm{d} t \,\mathrm{d} v \le c \quad\text{for all \textit{x},}\end{align*}

and turning to the inequalities above, we conclude (4.8) by

\begin{align*}0 &\le\dfrac{x-\sup_{R}{{K}^{-1}}\int_{D^{R}(u)} \int_{I(s)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v}{c}- \dfrac{x-{{K}^{-1}}\int_{D^{R_x}(v_x)} \int_{I(t_x)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v}{{{K}^{-1}}\int_{D^{R_x}(v_x)} \int_{I(t_x)} f(\lvert{v-u}\rvert, t-s) \, \mathrm{d} t \,\mathrm{d} v} \\[4pt] &\le\dfrac{{{K}^{-1}}\int_{D^{R_x}(v_x)} \int_{I(t_x)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v-\sup_{R}{{K}^{-1}}\int_{D^{R}(u)} \int_{I(s)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v}{c} \\[4pt] &\le\dfrac{\sup_R {{K}^{-1}}\int_{D^{R}(v_x)} \int_{I(t_x)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v-\sup_{R}{{K}^{-1}}\int_{D^{R}(u)} \int_{I(s)} y_{v,t} \, \mathrm{d} t \,\mathrm{d} v} {c} \\* & \to 0\end{align*}

as $x\to\infty$.

Theorem 4.2. Let the field X be given by (2.10), where the Lévy basis M satisfies Assumption 2.1 and the kernel function f satisfies Assumptions 2.3 and 4.2. Let $D\subseteq C_r(0)\subseteq {\mathbb{R}^d}$ for $r\ge 0$ be given, and let $\Psi$ be defined by

\begin{equation*}\Psi(y_{v,t}) = \sup_{t_0\in [0,T]} \sup_{v_0\in B} \sup_{R\in SO(d)} \dfrac{1}{K}\int_{D^R(v_0)} \int_{I(t_0)} {y_{v,t}} \, \mathrm{d} t \,\mathrm{d} v ,\end{equation*}

where $K=\int_D \int_0^\ell 1\,\mathrm{d} t \,\mathrm{d} v$. Furthermore, let

\begin{equation*}c=\dfrac{1}{K} \int_D \int_0^\ell f(\lvert{v}\rvert,t) \mathrm{d} t \,\mathrm{d} v.\end{equation*}

Then

\begin{equation*}\lim_{x\to\infty} \dfrac{\mathbb{P}( \Psi(X_{v,t})>x)}{\rho((x/c,\infty))}= m(B\times [0,T]){\mathbb E} \biggl[ \exp\biggl(\beta \sup_{R\in SO(d)} \dfrac{1}{c} \dfrac{1}{K}\int_{D^R(u)}\int_{I(s)} X_{v,t} \mathrm{d} t \,\mathrm{d} v \biggr)\biggr],\end{equation*}

where ${(u,s)\in B\times [0,T]}$ is chosen arbitrarily.

Proof of Theorem 4.2. The result follows from Theorem 3.4 and Lemma 4.2 once we show the existence of a function $\phi$ satisfying Assumption 3.3. Note that the integrand in the limit in Theorem 3.4 is constant due to the stationarity of X and $\lambda_{u,s}$. Define

\begin{equation*}\phi(u,s) = c \, \textbf{1}_{B^{\prime}\times T^{\prime}}(u,s) +\sup_{t_0\in[0,T]} \sup_{v_0\in B} \dfrac{1}{K} \int_{D(v_0)} \int_{I(t_0)} f(\lvert{v-u}\rvert, t-s) \,\mathrm{d} t \,\mathrm{d} v \, \textbf{1}_{(B^{\prime}\times T^{\prime})^c}(u,s),\end{equation*}

which is integrable by (2.12) and satisfies (3.6) by (4.7). Now let $n\in\mathbb{N}$ be fixed, and let $(y_{v,t}^i)_{(v,t)\in B^{\prime}\times T^{\prime}}$ for $i=1,\ldots,n$ be t-càdlàg fields. Then

\begin{align*}\Psi\biggl( \sum_{i=1}^n y_{v,t}^i \biggr) &= \sup_{t_0\in [0,T]} \sup_{v_0\in B} \sup_{R\in SO(d)} \dfrac{1}{K}\int_{D^R(v_0)} \int_{I(t_0)} { \sum_{i=1}^n y_{v,t}^i} \, \mathrm{d} t \,\mathrm{d} v \\* &\le \sum_{i=1}^n \sup_{t_0\in [0,T]} \sup_{v_0\in B} \sup_{R\in SO(d)} \dfrac{1}{K}\int_{D^R(v_0)} \int_{I(t_0)} {y_{v,t}^i} \, \mathrm{d} t \,\mathrm{d} v\\* & = \sum_{i=1}^n \Psi(y_{v,t}^i) .\end{align*}

Furthermore, if $y_{v,t}^i = z^i f(\lvert{v-u^i}\rvert,t-s^i)$, it is easily seen that $\Psi(y_{v,t}^i) \le z^i \phi(u^i,s^i) ,$ and hence (3.7) is satisfied. Since

\begin{equation*}\sup_{s\in [0,T]} \sup_{u\in B} \lambda_{u,s}(y_{v,t}^i)= \dfrac{1}{c} \Psi(y_{v,t}^i),\end{equation*}

(3.8) is also satisfied, which concludes the proof.

As mentioned in Example 3.2, the case of $\mathbb{P}( \sup_{t\in [0,T]} \sup_{v\in B}X_{v,t}>x)$ follows from Theorem 4.2 by letting $\ell=0$ and $D=\{0\}$. In this case the constant $c=f(0,0)=1$ and (4.7) translates into $f(\lvert{v_0 - u}\rvert,t_0-s)< f(0,0)$ for all $(v_0,t_0)\neq (u,s)$, or equivalently $f(x,y)<f(0,0)$ for all $(x,y)\neq (0,0)$.

Theorem 4.3. Let the field X be given by (2.10), where the Lévy basis M satisfies Assumption 2.1 and the kernel function f satisfies Assumption 2.3 and $f(x,y)< f(0,0)$ for all $(x,y)\neq (0,0)$. Then

\begin{equation*}\lim_{x\to\infty} \dfrac{\mathbb{P}( \sup_{t\in [0,T]} \sup_{v\in B}X_{v,t}>x)}{\rho((x,\infty))}= m(B\times [0,T]){\mathbb E} [ \exp(\beta X_{u,s} )],\end{equation*}

where ${(u,s)\in B\times [0,T]}$ is chosen arbitrarily.

5. Continuity properties

The main purpose of this section is to show that the field defined in (2.10) has a version with t-càdlàg sample paths. This result will be obtained in Theorem 5.3 below. However, the proof involves showing two other results on continuity properties of related random fields of independent value. Therefore these results are formulated as separate theorems; see Theorems 5.1 and 5.2 below. Only the main results, Theorems 5.1–5.3, are stated fully with all assumptions included in the statement. The rest are to be understood in relation to the context. As stated in Section 2, Assumption 2.1 on the Lévy basis is partly used to guarantee that the field is t-càdlàg. However, if the aim is solely to obtain the t-càdlàg property, we can relax the assumption. In this section we therefore consider Assumption 5.1 below. It will be referred to with the dimension of the Lévy basis being d and $d+1$. Thus both the assumption and the subsequent Theorem 5.1 will be formulated with $m\in\mathbb{N}$ indicating the dimension.

Assumption 5.1. The Lévy basis M on ${\mathbb{R}}^{m}$ is stationary and isotropic satisfying (2.1). Moreover, the Lévy measure, denoted $\rho$, satisfies

(5.1) \begin{equation}\int_{\lvert{y}\rvert>1} \lvert{y}\rvert^k \rho ( \mathrm{d} y) < \infty\quad \textit{for all $k\in \mathbb{N}$.}\end{equation}

For the first result in this section, consider a compact set $K\subseteq {\mathbb{R}}^{m}$ and define the random field $Y=(Y_v)_{v\in K}$ by

(5.2) \begin{equation}Y_v = \int_{{\mathbb{R}}^m} h(\lvert{v-u}\rvert) M(\mathrm{d}u),\end{equation}

where M is a Lévy basis on ${\mathbb{R}}^{m}$ satisfying Assumption 5.1. It is shown in [Reference Rønn-Nielsen and Jensen19, Theorem A.1] that such a field has a continuous version when $h\colon[0,\infty) \to {\mathbb{R}}$ satisfies certain properties including being differentiable. Under much less restrictive assumptions on the kernel function h, we show that this is still the case. We only assume that h is bounded and integrable,

(5.3) \begin{equation}\int_{{\mathbb{R}}^m} h(\lvert{u}\rvert) \,\mathrm{d}u < \infty ,\end{equation}

and that h is Lipschitz-continuous. That is, there exist $C_L>0$ such that

(5.4) \begin{equation} \lvert{h(x)-h(y)}\rvert \le C_L \lvert{x-y}\rvert\end{equation}

for all $x,y\ge 0$. Having Assumption 5.1 satisfied for the basis M and (5.3) and (5.4) satisfied for the bounded kernel function ensures in particular that the integral (5.2) exists; see [Reference Rajput and Rosinski18, Theorem 2.7].

To show continuity, we appeal to a result in [Reference Adler1], in which finite moments and cumulants of the spot variable M^′ of the basis M are needed. As already mentioned, (5.1) is equivalent to saying that M^′ has finite moments and thus cumulants of any order; see [Reference Peccati and Taqqu17, Corollary 3.2.2] for the relation between moments and cumulants.

Proof. For $r\in{\mathbb{R}}$ and $n\in\mathbb{N}$ we shall consider moments of the form ${\mathbb E} [ (Y_{v+r}-Y_v)^n]$. Note that only indices in K are relevant, so in particular, $0\le \lvert{r}\rvert \le\mathrm{diam}(K)$. By (5.4) and the triangle inequality, there is a finite C such that

\begin{equation*}\lvert h(\lvert{v + r- u}\rvert)-h(\lvert{v - u}\rvert) \rvert \le C \lvert{r}\rvert .\end{equation*}

Now let $\kappa_n[{}\cdot{}]$ denote the nth cumulant of a random variable; for a brief overview of the relation between cumulants and moments we refer to [Reference Peccati and Taqqu17, Chapter 3] and in particular [Reference Peccati and Taqqu17, Corollary 3.2.2]. The cumulants $\kappa_n$ of the difference $Y_{v+r} - Y_v$ satisfy $\kappa_1[Y_{v+r} - Y_v]=0$ and, for $n>1$,

\begin{align*}\bigl|{ \kappa_n[Y_{v+r} - Y_v] }\bigr| &\le \lvert {\kappa_n[M^{\prime}]}\rvert \int_{{\mathbb{R}}^m} \bigl| { h(\lvert{v + r- u}\rvert)-h(\lvert{v - u}\rvert)}\bigr|^n \,\mathrm{d}u \\[4pt] &\le \lvert {\kappa_n[M^{\prime}]}\rvert C^{n-1} |r|^{n-1} \int_{{\mathbb{R}}^m} \bigl| { h(\lvert{v + r- u}\rvert)-h(\lvert{v - u}\rvert)}\bigr| \,\mathrm{d}u\\[4pt] &\le C_n |r|^{n-1} ,\end{align*}

where $C_n\ge 0$ is a finite constant, chosen independently of r and $v\in K$ by

\begin{equation*}C_n= \lvert {\kappa_n[M^{\prime}]}\rvert C^{n-1} 2\int_{{\mathbb{R}}^m} h(\lvert{u}\rvert) \,\mathrm{d}u < \infty\end{equation*}

(see e.g. [Reference Rønn-Nielsen and Jensen21, Appendix A] for the cumulant formulas). Consequently, for all $n\in\mathbb{N}$ there exist finite constants $C^{\prime}_n$ and natural numbers $n^{\prime} \ge n/2$ such that

\begin{equation*}{\mathbb E} [( Y_{v+r}-Y_v)^n ] \le C^{\prime}_n {|r|^{n^{\prime}}}\end{equation*}

with the equality $n^{\prime}=n/2$ whenever n is even (see [Reference Peccati and Taqqu17, Corollary 3.2.2]). Using the fact that $\lvert{r}\rvert\le\mathrm{diam}(K)$, we find finite $C'\ge 0$ and $\eta > 4(m+1)$ such that

\begin{equation*}{\mathbb E} \lvert{Y_{v+r}-Y_v}|^{4(m+1)}\le C^{\prime}_{4(m+1)} |r|^{2m} |r|^2\le \dfrac{C^{\prime} |r|^{2m}}{| \log |r| |^{1+\eta}}\end{equation*}

for all $v\in K$. From a corollary to [Reference Adler1, Theorem 3.2.5] we conclude that $(Y_v)_{v\in K}$ has a continuous version on K.

Next we consider a field indexed by ${{\mathbb{R}^d} \times {\mathbb{R}}}$ allowing for discontinuities in time, and we show that it has a t-càdlàg version. For compact sets $K\subseteq{\mathbb{R}^d}$ and [0, S], $S>0$, we let the random field $Z=(Z_{v,t})_{(v,t)\in K\times [0,S]}$ be given by

(5.5) \begin{equation}Z_{v,t}=\int_{\mathbb{R}^d} \int_{[0,t]} g(\lvert{v-u}\rvert) M(\mathrm{d}s,\mathrm{d}u),\end{equation}

where M is a Lévy basis satisfying Assumption 5.1 with $m=d+1$, and the integration kernel $g\colon[0,\infty) \to {\mathbb{R}}$ is assumed to be bounded, integrable, and Lipschitz-continuous, i.e. it satisfies (5.3) and (5.4) with $m=d$.

Choose $0=t_0<\dots<t_n$ in [0, S] and $v\in K$. Arguing as in Section 2, the cumulant function for $(Z_{v,t_1}, Z_{v,t_2}-Z_{v,t_1},\ldots,Z_{v,t_n}-Z_{v,t_{n-1}})$ can be found to be

\begin{align*}&C(\boldsymbol{\lambda} \dagger (Z_{v,t_1}, Z_{v,t_2}-Z_{v,t_1},\ldots,Z_{v,t_n}-Z_{v,t_{n-1}})) \\[3pt] &\quad = \sum_{j=1}^n (t_j - t_{j-1})\biggl({\mathrm{i}} \lambda_j a \int_{\mathbb{R}^d} g(\lvert{v-u}\rvert) \,\mathrm{d}u- \dfrac12 \theta \lambda_j^2 \int_{\mathbb{R}^d} g(\lvert{v-u}\rvert)^2 \,\mathrm{d}u \\[3pt] & \quad\quad\, + \int_{{\mathbb{R}^d}} \int_{\mathbb{R}} \mathrm{e}^{{\mathrm{i}} g(\lvert{v-u}\rvert) \lambda_j z} - 1 - {\mathrm{i}} g(\lvert{v-u}\rvert)\lambda_j z 1_{[-1,1]}(z) \rho(\mathrm{d} z) \,\mathrm{d}u\biggr)\end{align*}

where $\boldsymbol{\lambda} = (\lambda_1,\ldots,\lambda_n)\in {\mathbb{R}}^n$. By a change of measure, we see for fixed $v\in\mathbb{K}$ that $(Z_{v,t})_{t\in [0,S]}$ is a one-dimensional Lévy process in law. In the following we shall extend this to a result concerning the process of random fields indexed by time.

In this section we will often consider the field as being a collection of real-valued functions defined on space K or $\tilde K = K \cap \mathbb{Q}^d$, with the functions indexed by time in [0, S] or $\tilde S = [0,S] \cap \mathbb{Q}$. As such we introduce the notation $\textbf{Z}_t = (Z_{v,t})_{v\in K}$, with the entire field denoted by $\textbf{Z} = (\textbf{Z}_t)_{t\in [0,S]}$ when considered as a collection of random functions. We use the same notation when space and time are indexed by $\tilde K$ and $\tilde S$, respectively, although when it is unclear which is meant and it is necessary to distinguish the cases, we explicitly state it.

Let $t\in[0,S]$ be fixed and choose $v_1,\ldots,v_n\in K$. Then $(Z_{v_1,t},\ldots,Z_{v_n,t})$ has cumulant function given by

(5.6) \begin{align} & C(\boldsymbol{\lambda} \dagger (Z_{v_1,t},\ldots,Z_{v_n,t})) \notag \\* &\quad = t{\mathrm{i}} a \int_{\mathbb{R}^d} \sum_{j=1}^n\lambda_j g(\lvert{v_j-u}\rvert) \,\mathrm{d}u \notag \\* & \quad \quad\, - t\dfrac12 \theta \int_{\mathbb{R}^d} \biggl( \sum_{j=1}^n\lambda_j g(\lvert{v_j-u}\rvert) \biggr)^2 \,\mathrm{d}u \notag \\* & \quad \quad\, + t \int_{{\mathbb{R}^d}} \int_{\mathbb{R}} \mathrm{e}^{{\mathrm{i}}\sum_{j=1}^n\lambda_j g(\lvert{v_j-u}\rvert)z} - 1 - {\mathrm{i}}\sum_{j=1}^n\lambda_j g(\lvert{v_j-u}\rvert) z 1_{[-1,1]}(z) \rho(\mathrm{d} z) \,\mathrm{d}u .\end{align}

Replacing $(a,\theta,\rho)$ with $(ta,t\theta,t\rho)$, we see from (5.6) that $\textbf{Z}_t$ is the type of field defined in (5.2). Thus, by Theorem 5.1, $(Z_{v,t})_{v\in \tilde K}$ is almost surely uniformly continuous. This holds jointly for all rational time points $t\in\tilde S$, and therefore a version of $(\textbf{Z}_t)_{t\in \tilde S}$ can be chosen with $\textbf{Z}_t$ being continuous for all $t\in \tilde S$, i.e. it has values in the space of real-valued functions on the compact set K. It will be useful in the following that this space equipped with the uniform norm, here denoted $(\mathcal{C}(K,{\mathbb{R}}),\|\cdot\|_{\infty})$, is a separable Banach space; see [Reference Kechris14, Theorem 4.19]. The following lemma concerns this specific version of $(\textbf{Z})_{t\in \tilde S}$ taking its values in $(\mathcal{C}(K,{\mathbb{R}}),\|\cdot\|_{\infty})$.

Proof. As $\textbf{Z}_t$ is a version of the field studied in (5.6) for each $t\in \tilde S$, the cumulant function for $(Z_{v_1,t},\ldots,Z_{v_n,t})$ will also be as in (5.6). With similar considerations but heavier notation, it can be realized that for $v_1,\ldots,v_n \in K$ and $0=t_0<t_1 < \dots < t_m \in \tilde S$, and defining $Z_{t_j}^n = (Z_{v_1,t_j},\ldots,Z_{v_n,t_j})$ for $j=1,\ldots,m$, it holds that

\begin{align*}&C\big(\boldsymbol{\lambda} \dagger \big(Z^n_{t_1}, Z^n_{t_2}-Z^n_{t_1},\ldots,Z^n_{t_m}-Z^n_{t_{m-1}}\big)\big) = \sum_{j=1}^mC\big(\lambda_j \dagger Z^n_{t_j} - Z^n_{t_{j-1}}\big)= \sum_{j=1}^mC\big(\lambda_j \dagger Z^n_{t_j- t_{j-1}}\big) ,\end{align*}

where $\boldsymbol{\lambda} = (\lambda_1,\ldots,\lambda_n)$ and each $\lambda_j \in {\mathbb{R}}^n$, and the natural convention $Z_0^n=(0,\ldots,0)$ is applied. This shows that $(\textbf{Z}_t)_{t\in\tilde S}$ has stationary and independent increments.

To show stochastic continuity it suffices to show that

\begin{equation*}\lim_{n\to\infty} \mathbb{P}(\|\textbf{Z}_{t_n}\|_{\infty}\ge \varepsilon) = 0\end{equation*}

for any rational sequence $(t_n)$ satisfying $t_n\downarrow 0$. As $(\mathcal{C}(K,{\mathbb{R}}),\|\cdot\|_{\infty})$ is a separable Banach space, this is equivalent to showing that $\textbf{Z}_{t_n}$ converges to $\delta_{\textbf{0}}$ in law in the uniform norm, where $\delta_{\textbf{0}}$ is the degenerate probability measure concentrated at $\textbf{0}$. For $t\in\tilde S$, let $\nu_t$ denote the distribution of $\textbf{Z}_{t}$ and let $\hat \nu_{t_n}$ be its characteristic function defined on the dual space of $\mathcal{C}(K,{\mathbb{R}})$; see [Reference Linde15, Section 1.7]. Since $(\mathcal{C}(K,{\mathbb{R}}),\|\cdot\|_{\infty})$ is separable, $\nu_t$ is a Radon measure [Reference Linde15, Proposition 1.1.3] and the results in [Reference Linde15, Chapters 2 and 5] apply. Due to the infinite divisibility, $\nu_1 = \nu_{t_n}*\nu_{1-t_n}$ for any $n\in \mathbb{N}$ (assuming $t_n\le 1$), and we conclude that $\{\nu_{t_n}\}$ is relatively shift compact [Reference Linde15, Theorem 2.3.1]. Following the proofs of [Reference Linde15, Propositions 5.1.4 and 5.1.5] we obtain that $\lim_{n\to\infty} \hat \nu_{t_n} \to 1$ uniformly on bounded sets of the dual space. Combining [Reference Linde15, Propositions 2.3.9 and 1.8.2] shows that $\textbf{Z}_{t_n}$ converges in law to $\delta_{\textbf{0}}$ as claimed.

The next theorem states that the field Z defined in (5.5) indeed has a t-càdlàg version.

The desired t-càdlàg version will be an extension of the field $(\textbf{Z}_t)_{t\in \tilde S}$ studied in Lemma 5.1. Thus $(\textbf{Z}_t)_{t\in\tilde S}$ will still be a version chosen such that each $\textbf{Z}_t$ is a continuous random field. The result relies on a sequence of lemmas that are shown using an adaptation of the ideas of [Reference Sato24, Theorems 11.1 and 11.5] and [Reference Sato24, Lemmas 11.2–11.4] for Lévy processes on ${\mathbb{R}}$. Lemmas 5.2 and 5.3 are shown using similar techniques for the Lévy process $(\textbf{Z}_t)_{t\in\tilde S}$, and therefore we omit the proofs here and refer to the supplementary material [Reference Stehr and Rønn-Nielsen27] for completeness.

For the statement and proof of these lemmas, the following notation will be useful. We say that $\textbf{Z}(\omega)$ has $\varepsilon$-oscillation n times in a set $M\subseteq\mathbb{Q} \cap [0,\infty)$ if there exist $t_0<t_1<\dots < t_n \in M$ such that

\begin{equation*}\|\textbf{Z}_{t_j}(\omega) - \textbf{Z}_{t_{j-1}}(\omega)\|_{\infty} = \sup_{v\in K} |Z_{v,t_j}(\omega) - Z_{v,t_{j-1}}(\omega)| > \varepsilon\end{equation*}

for all $j=1,\ldots,n$. We say that $\textbf{Z}(\omega)$ has $\varepsilon$-oscillation infinitely often in M if it has $\varepsilon$-oscillation n times in M for any $n\in\mathbb{N}$. Consider $\Omega_1$ given by

\begin{align*}\Omega_1 = \Bigl\{\omega\in\Omega \mid& \lim_{s\in\mathbb{Q},s\downarrow t} \textbf{Z}_s(\omega) \text{ exists with respect to $\|\cdot\|_{\infty}$ for all $t\in [0,S]$ and} \\*& \lim_{s\in\mathbb{Q},s\uparrow t} \textbf{Z}_s(\omega) \text{ exists with respect to $\|\cdot\|_{\infty}$ for all $t\in [0,S]$} \Bigr\} .\end{align*}

Furthermore define the sets

\begin{equation*}A_k = \{\omega \in \Omega \mid \textbf{Z}(\omega) \text{ does not have }{{1}/{k}}\text{-oscillation infinitely often in } \tilde S\} ,\end{equation*}

and from these define $\Omega^{\prime}_1=\cap_{k\in\mathbb{N}} A_k$. Each $A_k$ is measurable as each $\textbf{Z}_t$ is continuous on K for $t\in \tilde S$, such that $\|\cdot\|_{\infty} = \sup_{v\in K}|\cdot| = \sup_{v\in \tilde K}|\cdot| $.

Having established that $\lim_{s\in\mathbb{Q},s\downarrow t} \textbf{Z}_s$ and $\lim_{s\in\mathbb{Q},s\uparrow t} \textbf{Z}_s$ exist almost surely, we now prove the main result, Theorem 5.2, on the existence of a t-càdlàg version of Z.

Proof of Theorem 5.2. We have $\mathbb{P}(\Omega^{\prime}_1)=1$ by Lemma 5.3. For all $t\in [0,S]$, define $\textbf{Z}^{\prime}_t(\omega) =\textbf{1}_{\Omega_1^{\prime}}(\omega)(\lim_{s\in\mathbb{Q},s\downarrow t} \textbf{Z}_s(\omega))$, where the limit is with respect to $\|\cdot\|_{\infty}$, and exists according to Lemma 5.2. The càdlàg assertion is trivially true for $\omega\not\in\Omega_1^{\prime}$. Now consider $\omega\in\Omega_1^{\prime}$ but suppress $\omega$ to ease notation. By definition of $\textbf{Z}^{\prime}_t$, for all $\epsilon >0$ there exists N such that

(5.7) \begin{equation}\|\mathrm{Z}_t^{\prime}-\mathrm{Z}_s\|_\infty<\epsilon \quad \text{for all } s\in (t,t+ 1/N)\cap \mathbb{Q} .\end{equation}

Let $(t_n)$ be any sequence satisfying $t_n\downarrow t$. Fix $\varepsilon > 0$, and let $N\in\mathbb{N}$ satisfy (5.7) with the bound ${{\varepsilon}/{2}}$. There exists $n_0\in\mathbb{N}$ such that $|t_n-t|<{{1}/{N}}$ for all $n\ge n_0$. Now fix such n. By another application of (5.7) there exists $N_n$ such that $t_n+{{1}/{N_n}}\le t+{{1}/{N}}$ and $\|\textbf{Z}_{t_n}^{\prime}-\textbf{Z}_s\|_{\infty}<{{\varepsilon}/{2}}$ for all $s\in (t_n,t_n+{{1}/{N_n}})\cap \mathbb{Q}$. For any of those s we find in particular that

\begin{equation*}\|\textbf{Z}_{t_n}^{\prime}-\textbf{Z}_t^{\prime}\|_{\infty} \le \|\textbf{Z}_{t_n}^{\prime}-\textbf{Z}_s\|_{\infty} + \|\textbf{Z}_{t}^{\prime}-\textbf{Z}_s\|_{\infty} < \varepsilon .\end{equation*}

As this is true for all $n\ge n_0$, we conclude that $\textbf{Z}^{\prime}=(\textbf{Z}^{\prime}_t)_{t\in [0,S]}$ is right-continuous with respect to $\|\cdot\|_{\infty}$. Similar arguments show that $\textbf{Z}^{\prime}$ has limits from the left and that the limits are unique. The mapping $v \mapsto Z^{\prime}_{v,t}$ is continuous because the space $(\mathcal{C}(K,{\mathbb{R}}),\|\cdot\|_{\infty})$ is complete, and $\textbf{Z}^{\prime}_t$ is defined as the limit of such functions.

We now argue that $\textbf{Z}^{\prime}$ is indeed a version of $\textbf{Z}$. If $(t_n)\subset \tilde S$ with $t_n\downarrow t$ then $Z_{v,t_n} \overset{\mathbb{P}}{\to} Z_{v,t}$ for all $v\in K$ as $(Z_{v,t})_{t\in [0,S]}$ is a Lévy process in law and thus especially stochastically continuous. Since $\mathbb{P}(\Omega^{\prime}_1)=1$ we have $Z_{v,t_n} \to Z^{\prime}_{v,t}$ almost surely, and by uniqueness of limits we conclude that $\mathbb{P}(Z^{\prime}_{v,t}=Z_{v,t}) = 1$ for all ${(v,t)\in K\times [0,S]}$.

It remains to show that Z^′ is t-càdlàg. For given ${(v,t)\in K\times [0,S]}$ we can write

\begin{equation*}|Z^{\prime}_{v,t} - Z^{\prime}_{u,s}|\le \lvert{Z^{\prime}_{v,t} - Z^{\prime}_{u,t}}\rvert + \|\textbf{Z}^{\prime}_t-\textbf{Z}^{\prime}_s\|_{\infty}\end{equation*}

for any choice of ${(u,s)} \in K\times [0,S]$, and thus we conclude that $\lim_{(u,s)\to(v,t^+)} Z^{\prime}_{u,s} = Z^{\prime}_{v,t}$, from the continuity of $\textbf{Z}^{\prime}_t$ and the uniform càdlàg property of $(\textbf{Z}^{\prime}_t)_{t\in [0,S]}$. Similar arguments give that the limit $\lim_{(u,s)\to(v,t^-)} Z^{\prime}_{u,s}$ exists in ${\mathbb{R}}$ and that it is unique.

Theorem 5.3 below is stated under Assumption 2.1. However, in order to establish t-càdlàg sample paths, the milder Assumption 5.1 would have been sufficient.

Proof. We decompose the field $(X_{v,t})$ as

\begin{align*}X_{v,t} & = \int_{{\mathbb{R}^d} \times[0,t]} f(\lvert{v-u}\rvert, t-s) - f(|v-u|,0) M(\mathrm{d}s,\mathrm{d}u) \\[5pt] & \quad\, + \int_{{\mathbb{R}^d} \times({-}\infty,0)} f(\lvert{v-u}\rvert, t-s) M(\mathrm{d}s,\mathrm{d}u)+ \int_{{\mathbb{R}^d} \times[0,t]} f(|v-u|,0) M(\mathrm{d}s,\mathrm{d}u) .\end{align*}

By Theorem 5.2, choosing $g(\cdot) = f(\cdot,0)$, the third term has a t-càdlàg version. Due to continuity of the integrands, the first and second terms have continuous versions by arguments similar to those in the proof of Theorem 5.1: defining the continuous function $\phi\colon[0,\infty)\times{\mathbb{R}} \to {\mathbb{R}}$ by

\begin{equation*}\phi(x,y)=\textbf{1}_{[0,\infty)}(y)(f(x,y)-f(x,0)) ,\end{equation*}

the first term above reads

\begin{equation*}Y_{v,t}^{\prime} = \int_{{\mathbb{R}^d} \times [0,\infty)} \phi(\lvert{v-u}\rvert, t-s) M(\mathrm{d}u,\mathrm{d}s) = Y_{v,t} + y_{v,t},\end{equation*}

where $y_{v,t} = {\mathbb E} Y_{v,t}^{\prime}$ and $Y_{v,t} = Y_{v,t}^{\prime} - y_{v,t}$. The field $(Y_{v,t})$ is continuous by previous arguments, replacing assumptions (5.3) and (5.4) with the conditions

(5.8) \begin{equation}\int_{\mathbb{R}^d} \int_0^\infty \bigl| { \phi(|{u}|,T+\ell - s) }\bigr| \,\mathrm{d}s \,\mathrm{d}u < \infty\end{equation}

and the Lipschitz continuity of $\phi$,

\begin{equation*}\bigl| { \phi(|u_1|,t_1-s) - \phi(|u_2|,t_2-s) }\bigr|\le C \lvert {(u_1-u_2, t_1 - t_2)}\rvert\end{equation*}

for all $u_1,u_2\in {\mathbb{R}^d}$ and $t_1,t_2\in T^{\prime}$. These conditions are easily seen to be satisfied under Assumption 2.2. As

\begin{equation*}y_{v,t} = y_{0,t}= {\mathbb E} [M^{\prime}] \int_{\mathbb{R}^d} \int_0^\infty \phi(\lvert{u}\rvert, t-s) \,\mathrm{d}s \,\mathrm{d}u < \infty ,\end{equation*}

the deterministic field $(y_{v,t})$ is continuous by a dominated convergence argument using (5.8). The continuity of the second term follows similarly.