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\authornames{M.~STEHR AND A.~R\O NN-NIELSEN} % insert the authors here for use in running head
\shorttitle{Supplementary Material: Tail asymptotics for an infinitely divisible model} % insert short title here for use in running head

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\begin{document}\recd{2 September 2019}{10 June 2020}

\title{Supplementary Material: Tail Asymptotics of\\ an Infinitely Divisible Space-Time Model with\\ Convolution Equivalent L\'evy Measure} % insert title - use \\ if it requires more than one line.

\authorone[Aarhus University]{Mads Stehr} % Affiliation is just the name of your university or institution
\authortwo[Copenhagen Business School]{Anders R\O nn-Nielsen}


\addressone{Centre for Stochastic Geometry and Advanced Bioimaging (CSGB), 
	Department of Mathematics, Aarhus University,
	Ny Munkegade 118, 8000 Aarhus C, Denmark} % Your postal address goes here.
\addresstwo{Center for Statistics,
	Department of Finance, Copenhagen Business School,
	Solbjerg Pl. 3, 2000 Frederiksberg, Denmark} % Your postal address goes here.

\normalsize

%\today

%%%%%%%%
	\section{Proofs of \cref{sec:maintheorem}}
	\label{sec:sup_proofs}
%%%%%%%%


\begin{proof}[Proof of \cref{lem:convolutionequivalencephi}]
For sufficiently large $x$ we find that
\begin{align*}
	\PP(Z\phi(U,S)> x) &
	= \frac{1}{\nu(A)} F \Bigl( \Bigl\{ (u,s,z)\in \RR \times \R \times \R_+\,:\, z\phi(u,s)> x  \Bigr\}\Bigr) \\&
	= \frac{1}{\nu(A)} \int_{\KS} L\Bigl(\frac{x}{c}\Bigr) \exp\Bigl(-\beta \frac{x}{c} \Bigr) m(\du,\ds) \\& \quad 
	+ \frac{1}{\nu(A)}\int_{(\KS)^c} L\Bigl(\frac{x}{\phi(u,s)}\Bigr) \exp\Bigl(-\beta \frac{x}{\phi(u,s)} \Bigr) m(\du,\ds) ,
\end{align*}
where the first term equals $L(x/c) \exp(-\beta x/c)$ times the desired limit. 
The result follows when the latter integral is shown to be of order $o(L(x/c)\exp(-\beta x/c))$, as $x \to \infty$. Let $h(u,s;x)$ denote the integrand. For all $(u,s)\not\in \KS$ we have $\phi(u,s)< c$. Combined with \cref{eq:Lconv}, this implies the existence of $\gamma >0$ and $C>0$ such that
\[
	\frac{h(u,s;x)}{L(x/c)\exp(-\beta x/c)}
	\le  C \exp(-\gamma x)
\]
for sufficiently large $x$. Thus, the integrand $h(u,s;x)$ is $o(L(x/c)\exp(-\beta x/c))$ at infinity. By dominated convergence, the integral is of order $o(L(x/c)\exp(-\beta x/c))$ if we can find an integrable function $g:\RRR\to\R$ such that
\[
	\frac{h(u,s;x)}{L(x/c)\exp(-\beta x/c)} \le g(u,s)
\]
for all $(u,s)\in\RRR$.
Returning to \cref{eq:Lfracbound} we see that for all $0<\gamma<\beta/c$ there is $C>0$ and $x_0$ such that
\begin{equation}\label{eq:phi_boundedintegrand}
	\frac{h(u,s;x)}{L(x/c)\exp(-\beta x/c)}
	\le C \exp\Bigl(-x_0(\beta-\gamma c)\Bigl(\frac{1}{\phi(u,s)} -\frac{1}{c}\Bigr)\Bigr)
\end{equation}
for all $x\ge x_0$. 
Independent of $(u,s)$ we can find a finite constant $\tilde C$ such that the right hand side of \cref{eq:phi_boundedintegrand} is bounded by
$\tilde C \phi(u,s)$, which is integrable by assumption.
This shows the desired order of convergence.

From \cite[Lemma~2.4(i)]{Pakes2004} the distribution of $Z\phi(U,S)$ is convolution equivalent with index $\beta/c$. The integrability result follows from \cite[Corollary~2.1(ii)]{Pakes2004}.
\end{proof}

\begin{cor}
If $V^1,V^2,\dots$ are i.i.d. fields with distribution $\nu_1$, then
\[
	\EE \Bigl[ \exp\Bigl( \beta \sup_{u\in B} \sup_{s\in [0,T]} \lambdausfct{(\V^1 + \cdots + \V^n)_\vt} \Bigr)\Bigr] < \infty
\]
for all $n\in\NN$.
\end{cor}

\begin{proof}
Because each $V^i$ can be represented by $(Z^i f(|v-U^i|,t-S^i)_\vtKS$, the result follows from \cref{eq:bound1} and \cref{eq:phiintegrability}.
\end{proof}

\begin{proof}[Proof of \cref{thm:Vconvolution}]
We will show the claim by induction over $n$: We note that the case $n=1$ follows easily from \cref{thm:singlejump}. Now assume that the result holds true for some $n\in\NN$ and let for convenience $V^{*n}=V^1 + \dots + V^n$. Also, let $y^*=\sup_{\vtKS} \y$. Using \cref{eq:bound2} and the representation $V^i = Z^i f(|v-U^i|,t-S^i)$, we find
\begin{align}\nonumber
	\MoveEqLeft[1] 
	\PP(\Psi(\V^{*n} + \V^{n+1} + \y) > x) \\ 
	\begin{split}\label{eq:inductionexpansion}
	& \le \PP \Bigl( \sum_{i=1}^n Z^i \phi(U^i,S^i)>\frac{x-y^*}{2} \,, Z^{n+1} \phi(U^{n+1},S^{n+1})>\frac{x-y^*}{2}\,, \\ & \hspace{10mm}
	 \Psi(\V^{*n} + \V^{n+1} + \y) > x \Bigr) \\ & \quad	
	 + \PP \Bigl(\sum_{i=1}^n Z^i \phi(U^i,S^i)\le\frac{x-y^*}{2} \,,\Psi(\V^{*n} + \V^{n+1} + \y) > x \Bigr)
	\\ 
	& \quad
	+ \PP \Bigl(Z^{n+1} \phi(U^{n+1},S^{n+1})\le\frac{x-y^*}{2} \,,\Psi(\V^{*n} + \V^{n+1} + \y) > x \Bigr)
	\end{split}	.
\end{align}
The first term in \cref{eq:inductionexpansion} is bounded from above by
\[
	\PP \Big( \sum_{i=1}^n Z^i \phi(U^i,S^i)>\frac{x-y^*}{2} \Big)
	\PP\Big(Z^{n+1} \phi(U^{n+1},S^{n+1})>\frac{x-y^*}{2} \Big) .
\]
In \cref{lem:convolutionequivalencephi} we showed that the distribution of $Z^i \phi(U^i,S^i)$ is convolution equivalent with index $\beta/c$, and hence, from \cite[Corollary~2.11]{Cline1987} and \cref{eq:convolutionequivalencephi}, both factors are asymptotically equivalent to $\rho_1((x/(2c),\infty))$ as $x\to\infty$. Following the proof of \cite[Lemma~2]{Cline1986} we see that the product is $o((\rho_1*\rho_1)((x/c,\infty)))$, and as such the first term in \cref{eq:inductionexpansion} is $o(\rho_1((x/c,\infty)))$ due to the convolution equivalence of $\rho_1$. By \cref{thm:singlejump} it is of order $o(\PP(\Psi(\V^1) > x))$ as $x\to\infty$.

By independence, the two remaining terms in \cref{eq:inductionexpansion} divided by $\PP(\Psi(\V^1) > x)$ are
\begin{equation}\label{eq:induction_integral}
\begin{aligned}
	&	
	\int_{C_x} \frac{\PP ( \Psi(\sum_{i=1}^n z^i f(|v-u^i|,t-s^i) + \V^{n+1} + \y) > x)}
	{\PP( \Psi(\V^1) > x)} \\& \qquad 
	\times F_1^{\otimes n}(\dd(u^1,s^1,z^1;\dots;u^n,s^n,z^n)) \\& 
	+
	\int_{\tilde C_x} \frac{\PP (\Psi(\V^{*n} + z^1 f(|v-u^1|,t-s^1) + \y) > x)}
	{\PP(\Psi(\V^1) > x)} F_1(\dd(u^1,s^1,z^1))  ,
\end{aligned}
\end{equation}
where $F_1^{\otimes n}$ is the $n$-fold product measure of $F_1$ and
\begin{align*}
	& C_{x} = \Bigl\{(u^1,s^1,z^1;\dots;u^n,s^n,z^n)\,:\, \sum_{i=1}^n z^i \phi(u^i,s^i) \le \frac{x-y^*}{2} \Bigr\} , \\
	& \tilde C_{x} = \Bigl\{(u^1,s^1,z^1)\,:\, z^1 \phi(u^1,s^1) \le \frac{x-y^*}{2} \Bigr\} .
\end{align*}
Above we used the representation $V^i = Z^i f(|v-U^i|,t-S^i)$ again. By \cref{thm:singlejump} and the induction assumption, the integrands of \cref{eq:induction_integral} have the following limits as $x\to\infty$,
\begin{align*}
	\MoveEqLeft[0]	 
	f_1(u^1,s^1,z^1;\dots;u^n,s^n,z^n)  \\&
	= \frac{\int_B\int_0^T \exp\left(\beta \lambdausfct{\sum_{i=1}^n z^i f(|v-u^i|,t-s^i)  + \y}\right) \ds \du}{m(\BT)} ,\\
	\MoveEqLeft[0]	 
	f_2(u^1,s^1,z^1) \\&
	=\frac{n \int_B \int_0^T \EE \left[\exp\left( \beta \lambdausfct{\V^1 + \dots + \V^{n-1} + z^1 f(|v-u^1|,t-s^1) + \y} \right) \right] \ds \du}{m(\BT)} ,
\end{align*}
respectively. When integrated with respect to the relevant measures we find
\begin{align*}
	& \int f_1(u^1,s^1,z^1;\dots;u^n,s^n,z^n) F_1^{\otimes n}(\dd(u^1,s^1,z^1;\dots;u^n,s^n,z^n)) \\ & \qquad
		+ \int f_2(u^1,s^1,z^1) F_1(\dd(u^1,s^1,z^1))\\ 
	= & \frac{n+1}{m(\BT)} \int_B \int_0^T \EE \left[\exp\left( \beta \lambdausfct{\V^1 + \dots + \V^{n} + \y} \right) \right] \ds \du  ,
\end{align*}
which is the desired expression. To show convergence of the integrals in \cref{eq:induction_integral}, using Fatou's lemma, it suffices to find integrable functions $g_1(u^1,s^1,z^1;\dots;u^n,s^n,z^n;x)$ and $g_2(u^1,s^1,z^1;x)$ that are upper bounds of the integrands such that their limits exist when $x\to\infty$ and such that
\begin{align*}
	& \int g_1(u^1,s^1,z^1;\dots;u^n,s^n,z^n;x) F_1^{\otimes n}(\dd(u^1,s^1,z^1;\dots;u^n,s^n,z^n)) \\ & \qquad
	+ \int g_2(u^1,s^1,z^1;x) F_1(\dd(u^1,s^1,z^1)) \\
	\to &
	\int \lim_{x\to\infty} g_1(u^1,s^1,z^1;\dots;u^n,s^n,z^n;x) F_1^{\otimes n}(\dd(u^1,s^1,z^1;\dots;u^n,s^n,z^n))  \\ & \qquad
	+ \int \lim_{x\to\infty} g_2(u^1,s^1,z^1;x) F_1(\dd(u^1,s^1,z^1))
\end{align*} 
as $x\to\infty$. Using \cref{eq:bound2} and properties of $\Psi$, we can choose the functions
\[
	g_1(u^1,s^1,z^1;\dots;u^n,s^n,z^n;x)
	= \boldsymbol 1_{C_x}\frac{\PP(Z^1 \phi(U^1,Z^1)> x - y^* - \sum_{i=1}^n z^i\phi(u^i,s^i))}
	{\PP(\Psi(\V^1) > x)}
\]
and
\[
	g_2(u^1,s^1,z^1;x)
	= \boldsymbol 1_{\tilde C_x}\frac{\PP(\sum_{i=1}^n Z^i \phi(U^i,Z^i)> x - y^* - z^1\phi(u^1,s^1))}
	{\PP(\Psi(\V^1) > x)} .
\]
From \cref{thm:singlejump,eq:convolutionequivalencephi} we find that
\begin{equation}\label{eq:phiVequivalence}
	\PP(Z^1 \phi(U^1,S^1)>x) \sim \frac{m(\KS)}{m(\BT)} \PP(\Psi(\V^1) > x)
\end{equation}
as $x\to\infty$. The fact that the distribution of $Z^1 \phi(U^1,S^1)$ is convolution equivalent and in particular has an exponential tail implies
\[
	g_1(u^1,s^1,z^1;\dots;u^n,s^n,z^n;x) 
	\to \frac{m(\KS)}{m(\BT)} \exp \Bigl( \frac{\beta}{c} \Bigl( y^* + \sum_{i=1}^n z^i\phi(u^i,s^i) \Bigr)\Bigr)
\]
as $x\to\infty$. Similarly, \cref{eq:phiVequivalence} and an application of \cite[Corollary~2.11]{Cline1987} gives 
\begin{align*}
	\MoveEqLeft[1]
	g_2(u^1,s^1,z^1;x)  \\&
	\to \frac{m(\KS)}{m(\BT)} n\exp \Bigl( \frac{\beta}{c} ( y^* + z^1\phi(u^1,s^1) )\Bigr)
	\Bigl(\EE \exp\Bigl( \frac{\beta}{c} Z^1 \phi(U^1,S^1)\Bigr) \Bigr)^{n-1} 
\end{align*}
as $x\to\infty$, and we conclude that
\begin{align} \nonumber
	& \int \lim_{x\to\infty} g_1(u^1,s^1,z^1;\dots;u^n,s^n,z^n;x) F_1^{\otimes n}(\dd(u^1,s^1,z^1;\dots;u^n,s^n,z^n))  \\ \nonumber & \qquad
	+ \int \lim_{x\to\infty} g_2(u^1,s^1,z^1;x) F_1(\dd(u^1,s^1,z^1)) \\ \label{eq:boundinnerconvergence}
	= &
	\frac{m(\KS)}{m(\BT)} (n+1)\exp(\beta y^*/c)
	\Bigl(\EE \exp\Bigl( \frac{\beta}{c} Z^1 \phi(U^1,S^1)\Bigr) \Bigr)^{n}  .
\end{align} 
For notational convenience, we let $\mu$ denote the distribution of $Z^i\phi(U^i,S^i)$. Then, again by \cite[Corollary~2.11]{Cline1987} and \cref{eq:phiVequivalence}, \cref{eq:boundinnerconvergence} equals
\begin{equation}\label{eq:boundinnerconvergence2}
	\lim_{x\to\infty} \frac{m(\KS)}{m(\BT)}
	\frac{ \mu^{*(n+1)}((x-y^*, \infty))}{ \mu((x,\infty))}
	= \lim_{x\to\infty} \frac{ \mu^{*(n+1)}((x-y^*, \infty))}{\PP(\Psi(\V^1) > x)} .
\end{equation}
Furthermore, we see
\begin{equation*}
\begin{aligned}	
	\MoveEqLeft[1] 
	\PP(\Psi(\V^1) > x)  
	\Bigl( \int g_1(z^1;\dots;z^n;x) \mu^{\otimes n}(\dd (z^1;\dots;z^n)) + \int g_2(z;x) \mu(\dd z) \Bigr) \\ &
	= \int_0^{(x-y^*)/2} \mu ((x-y^*-z,\infty)) \mu^{*n}(\dd z) + \int_0^{(x-y^*)/2} \mu^{*n}((x-y^*-z,\infty)) \mu(\dd z)  .
\end{aligned}
\end{equation*}
Since, in particular, the tails of $\mu$ and $\mu^{*n}$ are exponential with index $\beta/c$, we see from \cite[Lemma~2]{Cline1986} that the sum of integrals is asymptotically equivalent to $\mu^{*(n+1)}((x-y^*,\infty))$. Returning to \cref{eq:boundinnerconvergence2} concludes the proof.
\end{proof}

Before proving the theorem on the extremal behaviour of $X^1$, we need the following lemma for a dominated convergence argument.

\begin{lem}\label{lem:dominatedconvergence}
Let $V^1,V^2,\dots$ be i.i.d. fields with distribution $\nu_1$, and let $(U,S,Z)$ be distributed according to $F_1$. There exist a constant $K$ such that
\[
	\PP(\Psi(\V^1 + \dots + \V^n) > x ) \le K^n \PP(Z\phi(U,S)>x)
\]
for all $n\in\NN$ and all $x\ge0$.
\end{lem}

\begin{proof}
By \cref{lem:convolutionequivalencephi} the distribution of $Z\phi(U,S)$ is convolution equivalent, and it follows from \cite[Lemma~2.8]{Cline1987} that there is a constant $K$ such that 
\[
	\PP \Bigl(\sum_{i=1}^n Z^i\phi(U^i,S^i)>x \Bigr) 
	\le K^n \PP(Z\phi(U,S)>x) ,
\]
for i.i.d. variables $(U^1,S^1,Z^1), (U^2,S^2,Z^2), \dots$ with distribution $F_1$. The result follows directly from \cref{eq:bound2}.
\end{proof}

\begin{proof}[Proof of \cref{thm:extremeX1}]
From \cref{eq:bound1} and the representation $V^i = (Z^i f(|v-U^i|, t-S^i))_\vt$, we see that
\[
	\EE \left[\exp\bigl(\beta \lambdausfct{\X^1} \bigr) \right]
	\le \exp \left( \nu(A) \big(\EE [\exp( \tfrac{\beta}{c} Z \phi(U,S)]-1\big)\right) .
\]
The first claim now follows from \cref{eq:phiintegrability}.

For the limit result, we find by independence and \cref{lem:dominatedconvergence},
\begin{align*}
	\MoveEqLeft[1]	
	\PP(\Psi(\X^1 + \y) > x) \\ &
	= \e^{-\nu(A)} \sum_{n=1}^\infty \frac{\nu(A)^n}{n!} \PP(\Psi(\V^1 + \dots + \V^n + \y) > x) \\ &
	\le \PP(Z\phi(U,S)>x-y^*) \e^{-\nu(A)} \sum_{n=1}^\infty \frac{\nu(A)^n K^n}{n!} ,
\end{align*}
where $y^*=\sup_{\vt}\y$ and $\e^{-\nu(A)} \sum_{n=1}^\infty \frac{\nu(A)^n K^n}{n!}<\infty$. With the convention that $\V^1 + \dots + \V^{n-1} = 0$ for $n=1$, by dominated convergence, \cref{thm:singlejump,thm:Vconvolution,lem:convolutionequivalencephi} yield
\begin{align*}
	\MoveEqLeft[1]	
	\lim_{x\to\infty} \frac{\PP(\Psi(\X^1+\y) > x)}
	{L(x/c)\exp(-\beta x/c)}\\&
	= \frac{n}{\nu(A)} \e^{-\nu(A)} \sum_{n=1}^\infty \frac{\nu(A)^n}{n!}
	\int_B \int_0^T \EE \left[ \e^{ \beta \lambdausfct{\V^1 + \dots + \V^{n-1} + \y} } \right] \ds \du \\&
	= \e^{-\nu(A)} \sum_{n=0}^\infty \frac{\nu(A)^n}{n!}
	\int_B \int_0^T \EE \left[ \e^{ \beta \lambdausfct{\V^1 + \dots + \V^{n} + \y} } \right] \ds \du \\&
%	= \int_B \int_0^T \EE \left[\e^{ \beta \lambdausfct{\V^1 + \dots + \V^{N} + \y} } \right] \ds \du \\&
	= \int_B\int_0^T \EE \left[\e^{ \beta \lambdausfct{\X^1 + \y} } \right] \ds \du .
\end{align*}
This concludes the proof.
\end{proof}


\begin{proof}[Proof of \cref{lem:extremeX2}]
First we show that
\begin{equation}\label{eq:expomomentX2}
		\EE \exp(\gamma \supp \X^2) < \infty
\end{equation}
for all $\gamma > 0$. Applying arguments as in \cref{sec:preliminary}, we write $X^2$ as the independent sum 
$
	\X^2 = \Y^1 + \Y^2
$.
Here $Y^1$ is a compound Poisson sum 
\[
	\Y^1 = \sum_{k=1}^M J_{v,t}^k
\]
with finite intensity $\nu(A^c\cap D)<\infty$ and jump distribution $\nu_2 = \nu_{A^c \cap D} / \nu(A^c \cap D)$, where 
$
	D = \{z\in\R^\K \,:\, \inf_\vtKQ z_{v,t} < -1\} 
$.
Furthermore, $Y^2$ is infinitely divisible with Lévy measure $\nu_{A^c \cap D^c}$, the restriction of $\nu$ to the set 
$
	A^c \cap D^c = \{z\in\R^\K \,:\, \sup_\vtKQ |z_{v,t}| \le 1 \} 
$.
By arguments as before, both fields have t-càdlàg extensions to $\KS$.
For each $k$, $J_{v,t}^k   \le 0$ for all $\vtKS$ almost surely, and in particular 
$
	\EE \exp(\gamma \supp \Y^1) < \infty
$ 
for all $\gamma >0$.
As $(\Y^2)_\vtKS$ is t-càdlàg on the compact set $\KS$, we find that $\PP(\supp |\Y^2|<\infty)=1$. Since also 
$
	\nu_{A^c\cap D^c} ( \{z\in\R^\K \,:\, \sup_\vtKQ |z_{v,t}| > 1 \})=0 ,
$
we obtain from \cite[Lemma~2.1]{Bravermann1995} that $\EE \exp(\gamma \supp|\Y^2|)<\infty$ for all $\gamma> 0$, which yields the claim \cref{eq:expomomentX2}. 

Appealing to properties of $\lambda_{u,s}$ we find that
\[
	\lambdausfct{\X} 
	\le \lambda_{u,s} \bigl(\X^1 + \supp \X^2\bigr)
	= \lambdausfct{\X^1} + \frac{\supp \X^2}{c} .
\]
The assertion now follows from \cref{eq:expomomentX2} and the first claim of \cref{thm:extremeX1}.
\end{proof}



\begin{proof}[Proof of \cref{thm:main1}]
Let $\pi$ be the distribution of $(\X^2)_\vtKS$. Conditioning on $(\X^2)_\vt=(\y)_\vt$ we find by independence that 
\[
	\frac{\PP(\Psi(\X) > x)}{\PP(\Psi(\X^1) > x)} 
	= \int \frac{\PP(\Psi(\X^1 + \y) > x)}{\PP(\Psi(\X^1) > x)} \pi(\dd y)= \int f(y;x) \pi(\dd y)
\]
with $f(y;x)=\PP(\Psi(\X^1 + \y) > x)/ \PP(\Psi(\X^1) > x)$, which, according to \cref{thm:extremeX1}, satisfies
\[
	f(y;x) \to f(y) = 
	\frac{\int_B\int_0^T \EE \left[\exp \left( \beta \lambdausfct{\X^1 + \y} \right) \right] \ds \du}
	{\int_B\int_0^T \EE \left[\exp \left( \beta \lambdausfct{\X^1} \right) \right] \ds \du}
\]
as $x\to\infty$. By another application of \cref{thm:extremeX1} and since
\[
	\int f(y)\pi(\dd y) = \frac{\int_B\int_0^T \EE \left[\exp \left( \beta \lambdausfct{\X} \right) \right] \ds \du} 
	{\int_B\int_0^T \EE \left[\exp \left( \beta \lambdausfct{\X^1} \right) \right] \ds \du} ,
\]
the proof is completed if we can find non-negative and integrable functions $g(y;x)$ and $g(y)=\lim_{x\to\infty} g(y;x)$ such that $f(y;x)\le g(y;x)$ and such that
\begin{equation*}%\label{eq:fatou_step2}
	\int g(y;x) \pi( \dd y) \to \int g(y) \pi(\dd y)
\end{equation*}
as $x\to\infty$. With $y^*=\supp \y$ we use the function 
\[
	g(y;x)=\PP(\Psi(\X^1) + y^* > x)/ \PP(\Psi(\X^1) > x)
\] 
which, according to properties of $\lambda_{u,s}$ and \cref{thm:extremeX1}, satisfies
\[
	g(y;x)  \to g(y) = \exp(\beta y^*/c)
\]
as $x\to\infty$. From \cite[Lemma~2.4(i)]{Pakes2004} and \cref{thm:extremeX1} the distribution of $\Psi(\X^1)$ is convolution equivalent with index $\beta/c$. Now let $G$ and $H$ denote the distributions of $\Psi(\X^1)$ and $\supp \X^2$, respectively. If $\overline H(x) = o(\overline G(x))$, $x\to\infty$, it follows from the integrability statement \cref{eq:expomomentX2} and \cite[Lemma~2.1]{Pakes2004} that
\begin{align*}
	\int g(y;x) \pi (\dd y)  &
	= \frac{\PP(\Psi(\X^1) + \supp \X^2 > x)}{\PP(\Psi(\X^1) > x)} \\ &
	\to  \EE \exp\Bigl(\frac{\beta}{c} \supp \X^2 \Bigr)
	= \int g(y) \pi (\dd y)
\end{align*}
as $x\to\infty$. From \cref{eq:expomomentX2} we find that $\lim_{x\to\infty}\e^{\gamma x}\PP(\supp \X^2>x)=0$ for all $\gamma >0$. Combined with the convolution equivalence of the distribution of 
$
	\Psi(\X^1)
$,
this yields $\overline H(x) = o(\overline G(x))$ and the claim follows.
\end{proof}


%%%%%%%%
	\section{Proofs of \cref{sec:app_cont}}
	\label{sec:sup_cont}
%%%%%%%%

\begin{proof}[Proof of \cref{lem:omega2relation}]
Let $\omega\in \Omega_1'$ and $(s_n)\subset \tilde S$ such that $s_n\downarrow t\in [0,S]$. For all $k\in\NN$ there exists $N\in\NN$ such that
\begin{equation}\label{eq:limitargument}
	\un{\bZ_{s_n}(\omega)-\bZ_{s_N}(\omega)} \le \tfrac1k
	\quad \text{for all}\quad
	n\ge N .
\end{equation}
This is seen by contradiction as follows: Assume that for any $N\in\NN$ there exists $n\ge N$ such that
\[
	\un{\bZ_{s_n}(\omega)-\bZ_{s_N}(\omega)} > \tfrac{1}{k} .
\]
Now fix $p\in\NN$. By this there exist $n_0<n_1<n_2<\cdots<n_p$ such that
\[
	\un{\bZ_{s_{n_j}}(\omega)-\bZ_{s_{n_{j-1}}}(\omega)} > \tfrac1k
	\quad \text{for}\quad
	j=1,\dots,p
\]
and we conclude that $\bZ(\omega)$ has $\tfrac1k$-oscillation $p$ times in $\tilde S$ for any $p$. Hence $\omega \in A_k^c$, which is a contradiction. From \cref{eq:limitargument} and the fact that the metric space $(\mathcal{C}(K,\R),\un{\cdot})$ is complete, we know that $\lim_{n\to\infty} \bZ_{s_n}(\omega)$ exists with respect to $\un{\cdot}$ as a continuous function on $K$. To show uniqueness of the limit, let $(t_n)\subset \tilde S$ be another sequence such that $t_n\downarrow t$. Then $\lim_{n\to\infty} \bZ_{s_n}(\omega) = \lim_{n\to\infty} \bZ_{t_n}(\omega)$: Let $(r_n)$ be the union of $(s_n)$ and $(t_n)$ ordered such that $r_n\downarrow t$. Then similarly for any $\epsilon > 0$ there is $N'$ such that
\[
	\un{\bZ_{r_n}(\omega)-\bZ_{r_{N'}}(\omega)} < \tfrac{\epsilon}{2}
	\quad\text{for}\quad
	n\ge N' .
\]
Also there is $N\in\NN$ such that $(s_n)_{n\ge N},(t_n)_{n\ge N} \subseteq (r_n)_{n\ge N'}$, and hence
\[
	\un{\bZ_{s_n}(\omega)-\bZ_{t_n}(\omega)} \le \un{\bZ_{s_n}(\omega)-\bZ_{r_{N'}}(\omega)} + \un{\bZ_{t_n}(\omega)-\bZ_{r_{N'}}(\omega)} < \epsilon
\]
for all $ n\ge N$. Thus, the limit $\lim_{s\in\mathbb{Q},s\downarrow t} \bZ_s(\omega)$ exists uniquely with respect to $\un{\cdot}$. Similarly for $\lim_{s\in\mathbb{Q},s\uparrow t} \bZ_s(\omega)$.
\end{proof}

We let
\[
	B(p,\epsilon,D) = \{\omega \in \Omega \mid \bZ(\omega) \text{ has }\epsilon\text{-oscillation $p$ times in } D\} ,
\]
with $D\subseteq \mathbb{Q}\cap [0,\infty)$, and 
\[
	\alpha_{\epsilon}(r) = \sup \{\PP(\un{\bZ_t}\ge \epsilon) \mid t\in[0,r]\cap \mathbb{Q}\} .
\]
Note that a direct consequence of the stochastic continuity from \cref{lem:Levyfield} is that $\alpha_\epsilon(r) \to 0$ as $r\to 0$ for all $\epsilon >0$.

\begin{lem}\label{lem:Bbound}
Let $p$ be a positive integer, $D=\{t_1,\ldots,t_n\}\subseteq \mathbb{Q}\cap [0,\infty)$ and $u,r\in \mathbb{Q}$ such that $0\le u \le t_1 < \cdots < t_n \le r$. Then $\PP(B(p,4\epsilon,D)) \le \left(2\alpha_{\epsilon}(r-u)\right)^p$.
\end{lem}

\begin{proof}
We will show the statement by induction in $p$. For this, define
\begin{align*}
	C_k & = \{\un{\bZ_{t_j} - \bZ_u}\le 2\epsilon,\ j=1,\dots,k-1,\ \un{\bZ_{t_k}-\bZ_u} > 2\epsilon\}\ ,\\
	D_k & = \{\un{\bZ_{t_k}-\bZ_r}> \epsilon\} 
\end{align*}
and note that $C_1,\dots,C_n$ are disjoint and 
\begin{align*}
	B(1,4\epsilon,D) &
	\subseteq \bigcup_{k=1}^n \{\un{\bZ_{t_k}-\bZ_u} > 2\epsilon\} 
	=\bigcup_{k=1}^n C_k \\ &
	= \bigcup_{k=1}^n (C_k\cap D_k^c) \cup (C_k \cap D_k)  
	\subseteq \{\un{\bZ_r-\bZ_u} \ge \epsilon\} \cup \bigcup_{k=1}^n (C_k\cap D_k) .
\end{align*}
By the Lévy properties in \cref{lem:Levyfield} we have $\PP(\un{\bZ_r-\bZ_u} \ge \epsilon) \le \alpha_{\epsilon}(r-u)$ and furthermore that $\PP(C_k\cap D_k)=\PP(C_k)\PP(D_k) \le \PP(C_k)\alpha_{\epsilon}(r-u)$. 
The fact that $C_1,\dots,C_n$ are disjoint then implies
\[
	\PP(B(1,4\epsilon,D)) \le \PP(\un{\bZ_r-\bZ_u} \ge \epsilon) + \sum_{k=1}^n \PP(C_k\cap D_k) \le 2 \alpha_{\epsilon}(r-u) ,
\]
which is the desired expression for $p=1$. We now assume the result to be true for arbitrary $p\in\NN$. We define the sets
\begin{align*}
  F_k = \{\omega :\ 
  & \bZ(\omega) \text{ has }4\epsilon\text{-oscillation $p$ times in } \{t_1,\dots,t_k\}, \\
  & \text{but does not have }4\epsilon\text{-oscillation $p$ times in } \{t_1,\dots,t_{k-1}\} 
  \} \ , \\
  G_k = \{\omega :\ & \bZ(\omega) \text{ has }4\epsilon\text{-oscillation one time in } \{t_{k},\dots,t_n\}\} .
\end{align*}
Then $F_1,\dots,F_n$ are disjoint, and $\PP(F_k\cap G_k)=\PP(F_k)\PP(G_k)$ for all $k=1,\dots,n$ due to the Lévy properties. Also $B(p,4\epsilon,D)=\cup_{k=1}^n F_k$, and furthermore 
\[
	B(p+1,4\epsilon,D) = \bigcup_{k=1}^n (F_k\cap G_k)
\]
with the inclusion $\subseteq$ seen as follows: If $\omega\in B(p+1,4\epsilon,D)$ then $\bZ(\omega)$ has $4\epsilon$-oscillation $p+1$ times in some $\{t_{n_0},\dots,t_{n_{p+1}}\} \subseteq D$ with $n_0<n_1<\dots<n_{p+1}$. Hence there is $k\le n_p$ such that $\omega \in F_k$. Also $\un{\bZ_{t_{n_{p+1}}}(\omega)-\bZ_{t_{n_p}}(\omega)}> 4\epsilon$ and as such also $\omega \in G_k$. From the induction assumption, the case $p=1$ and the fact that $F_1,\ldots,F_n$ are disjoint we find that
\begin{align*}
	\PP(B(p+1,4\epsilon,D)) & 
	= \sum_{k=1}^n\PP(G_k)\PP(F_k) 	\le 2\alpha_{\epsilon}(r-u) \PP\Bigl(\bigcup_{k=1}^n F_k\Bigr) \\&
	= 2\alpha_{\epsilon}(r-u) \PP(B(p,4\epsilon,M)) 	\le \left(2\alpha_{\epsilon}(r-u)\right)^{p+1} .
\end{align*}
\end{proof}

\begin{proof}[Proof of \cref{lem:omega1prime}]
To show that $\PP(\Omega_1')=1$ it suffices to prove $\PP(A_k^c)=0$ for any fixed $k\in\NN$. Since $\alpha_{\epsilon}(r) \to 0$ as $r \downarrow 0$ for any $\epsilon > 0$, we can choose $\ell\in\NN$ such that 
$2 \alpha_{1/(4k)}(S/\ell)<1$. 
Then by continuity of $\PP$ we get
\begin{align*}
	\PP(A_k^c) &
	\le \PP( \bZ \text{ has }\tfrac{1}{k}\text{-oscillation infinitely often in } \tilde S) \\&
	\le \sum_{j=1}^\ell\PP( \bZ \text{ has }\tfrac{1}{k}\text{-oscillation infinitely often in } [\tfrac{j-1}{\ell}S,\tfrac{j}{\ell}S]\cap \mathbb{Q}) \\&
	= \sum_{j=1}^\ell \lim_{p\to\infty} \PP(B(p,\tfrac{1}{k},[\tfrac{j-1}{\ell}S,\tfrac{j}{\ell}S]\cap \mathbb{Q})) .
\end{align*}
Now fix $j=1,\dots,\ell$, and enumerate the elements of $[\tfrac{j-1}{\ell}S,\tfrac{j}{\ell}S]\cap \mathbb{Q}$ by $(t_m)_{m\in\NN}$. From \cref{lem:Bbound} we know that
\[
	\PP (B(p,\tfrac{1}{k},\{t_1,\dots,t_n\})) \le (2\alpha_{1/(4k)}(\tfrac{S}{\ell}))^{p}
\]
for any $n\in\NN$. By continuity of $\PP$ we see that
\begin{align*}
	\PP (B(p,\tfrac{1}{k},[\tfrac{j-1}{\ell}S,\tfrac{j}{\ell}S]\cap \mathbb{Q})) 
	= \lim_{n\to\infty} \PP (B(p,\tfrac{1}{k},\{t_1,\dots,t_n\}))
	\le (2\alpha_{1/(4k)}(\tfrac{S}{\ell}))^{p}
\end{align*}
which tends to 0 as $p\to\infty$ since $\ell$ is chosen such that 
$2 \alpha_{1/(4k)}(S/\ell)<1$. As this holds for all $j=1,\dots,\ell$ we conclude that $\PP(A_k^c)=0$.
\end{proof}


\acks
This work was supported by the Centre for Stochastic Geometry and Advanced Bioimaging, funded by a grant from the Villum Foundation.
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% Reference list
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% References should be in the following form (or the BibTeX file
% apt.bst should be used):
%
% For a journal:
% Surname, Initial (year). Title of paper. {\em Journal title}
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\begin{thebibliography}{99}
\footnotesize
\bibliographystyle{apt}

\bibitem{Bravermann1995}
{\sc Braverman, M. and Samorodnitsky, G.} (1995).
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\newblock {\em Stochastic Process. Appl.\/} {\bf 56,} 207--231.

\bibitem{Cline1986}
{\sc Cline, D. B.~H.} (1986).
\newblock Convolution tails, product tails and domains of attraction.
\newblock {\em Probab. Theory Related Fields\/} {\bf 72,} 529--557.

\bibitem{Cline1987}
{\sc Cline, D. B.~H.} (1987).
\newblock Convolutions of distributions with exponential and subexponential
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\newblock {\em J. Aust. Math. Soc.\/} {\bf 43,} 347--365.

\bibitem{Pakes2004}
{\sc Pakes, A.~G.} (2004).
\newblock Convolution equivalence and infinite divisibility.
\newblock {\em J. Appl. Probab.\/} {\bf 41,} 407--424.

\end{thebibliography}

\end{document}
