Nerman’s martingales at complex parameters.

###  6.1 Centered characteristics

In this section we study the fluctuations of $\mathcal{Z}^{\chi}_{t}$ as
$t\to\infty$ for centered characteristics, that is, for characteristics $\chi$
satisfying $\mathds{E}[\chi(t)]=0$ for all $t\in\mathds{R}$. Theorem 6.3 below
plays a key role in the proof of our main result Theorem 2.14. Before we state
it, we give a preparatory lemma.

######  Lemma 6.1.

Suppose that (A1) through (A2.9) hold. Let $\theta\geq 0$ and
$f:[0,\infty)\to[0,\infty)$ be a continuous function with $f(x)=O(x^{\theta})$
as $x\to\infty$ such that $x^{-\theta}f(x)$ is uniformly continuous on
$[1,\infty)$ and the limit

$\lim_{t\to\infty,\,t\in\mathds{G}}\frac{1}{t^{\theta+1}}\int_{[0,t]}f(x)\,\ell(\mathrm{d}\mathit{x})=\vcentcolon
c\in(0,\infty)$ (6.1)

exists. Then, for $\varphi(t)\vcentcolon=e^{\alpha
t}f(t)\mathds{1}_{[0,\infty)}(t)$, we have $\sup_{t\geq 1}e^{-\alpha
t}t^{-\theta-1}\mathds{E}[\mathcal{Z}^{\varphi}_{t}]<\infty$ and

$\displaystyle\frac{e^{-\alpha
t}}{t^{\theta+1}}\mathcal{Z}^{\varphi}_{t}\to\frac{cW}{\beta}\quad\text{as
}t\to\infty,\;t\in\mathds{G}\quad\text{a.\,s.}$ (6.2)

###### Proof.

For any $t\geq 0$, we have $e^{-\alpha t}N((t-1,t])=e^{-\alpha
t}\mathcal{Z}_{t}^{\mathds{1}_{[0,1)}}$. Taking expectations and using (4.6)
gives

$\displaystyle e^{-\alpha t}\mathds{E}[N((t-1,t])]$ $\displaystyle=e^{-\alpha
t}\mathds{E}[\mathcal{Z}_{t}^{\mathds{1}_{[0,1)}}]=\int\mathds{1}_{[0,1)}(t-x)e^{-\alpha(t-x)}\,\mathsf{U}(\mathrm{d}\mathit{x}),$

which converges to a finite constant as $t\to\infty$, $t\in\mathds{G}$ by the
key renewal theorem and (A1). Hence,

$\displaystyle\frac{e^{-\alpha
t}}{t^{\theta+1}}\mathds{E}[\mathcal{Z}^{\varphi}_{t}]$ $\displaystyle\leq
C_{\theta}\bigg{(}\frac{\mathds{E}[N(\\{0\\})]}{t}+\frac{e^{-\alpha
t}}{t^{\theta+1}}\sum_{n=0}^{\lfloor
t\rfloor}e^{\alpha(t-n)}(t-n)^{\theta}\mathds{E}[N((n,n+1])]\bigg{)}$

is bounded for $t\geq 1$. It remains to show (6.2). To this end, first assume
that $|t^{-\theta}f(t)|\leq C_{\theta}<\infty$ for all $t>0$. In particular,
$f(0)=0$ if $\theta>0$. First notice that for any fixed $r>0$ and $t\geq r$,

$\displaystyle\frac{e^{-\alpha
t}}{t^{\theta+1}}\sum_{\begin{subarray}{c}u\in\mathcal{I}:\\\ S(u)\leq
r\end{subarray}}e^{\alpha(t-S(u))}f(t\\!-\\!S(u))\leq\frac{C_{\theta}}{t}\cdot
N([0,r])\to 0$

almost surely as $t\to\infty$. Hence, almost surely, the limiting behavior of
$e^{-\alpha t}t^{-\theta-1}\mathcal{Z}^{\varphi}_{t}$ as $t\to\infty$,
$t\in\mathds{G}$, is the same as that of

$\displaystyle e^{-\alpha
t}t^{-\theta-1}\sum_{\begin{subarray}{c}u\in\mathcal{I}:\\\ r<S(u)\leq
t\end{subarray}}e^{\alpha(t\\!-\\!S(u))}f(t-S(u)).$ (6.3)

Now first consider the lattice case and notice that by [33, Corollary 3.1(b)],
for given $\varepsilon>0$, with probability $1$ we may choose (a random)
$r\in\mathds{N}$ so large that

$(1-\varepsilon)e^{\alpha k}\frac{W}{\beta}\leq
N(\\{k\\})\leq(1+\varepsilon)e^{\alpha k}\frac{W}{\beta}$

for all $k\in\mathds{N}$, $k\geq r$. Then, for $t\in\mathds{N}$ with $t>r$,

$\displaystyle e^{-\alpha
t}t^{-\theta-1}\\!\\!\\!\sum_{\begin{subarray}{c}u\in\mathcal{I}:\\\ r\leq
S(u)\leq t\end{subarray}}\\!\\!\\!e^{\alpha(t\\!-\\!S(u))}f(t-S(u))$
$\displaystyle=\frac{1}{t^{\theta+1}}\sum_{k=r}^{t}f(t-k)e^{-\alpha
k}N(\\{k\\})$ $\displaystyle\leq\frac{(1+\varepsilon)W}{\beta
t^{\theta+1}}\sum_{k=0}^{t-r}f(k)\to(1+\varepsilon)\frac{cW}{\beta}\quad\text{as
}t\to\infty$

by (6.1). The corresponding lower bound can be obtained analogously. Now (6.2)
follows by letting $\varepsilon\to 0$.

Next, we turn to the non-lattice case and fix small $\varepsilon,\delta>0$. By
[33, Corollary 3.1(a)], with probability $1$, we may choose (a random)
$r\in\delta\mathds{N}$, $r\geq 1$ so large that

$(1-\varepsilon)e^{\alpha
t}\frac{e^{\alpha\delta}-1}{\alpha}\frac{W}{\beta}\leq
N((t,t+\delta])\leq(1+\varepsilon)e^{\alpha
t}\frac{e^{\alpha\delta}-1}{\alpha}\frac{W}{\beta}$

for all $t\geq r-\delta$. For $t\geq r$, define
$I_{k}^{\delta}\vcentcolon=[k\delta,(k+1)\delta)$ for
$k=0,\ldots,t_{\delta}-1$ where
$t_{\delta}\vcentcolon=\lfloor\frac{t-r}{\delta}\rfloor$, and
$I_{t_{\delta}}^{\delta}\vcentcolon=[t_{\delta}\delta,t-r)$. Notice that
$t-t_{\delta}\delta\geq r$. Hence, almost surely,

$\displaystyle\limsup_{t\to\infty}\frac{e^{-\alpha t}}{t^{\theta+1}}\\!\\!\\!$
$\displaystyle\sum_{\begin{subarray}{c}u\in\mathcal{I}:\\\ r<S(u)\leq
t\end{subarray}}\\!\\!\\!e^{\alpha(t\\!-\\!S(u))}f(t\\!-\\!S(u))$
$\displaystyle\leq\limsup_{t\to\infty}\frac{e^{-\alpha
t}}{t^{\theta+1}}\sum_{k=0}^{t_{\delta}}\\!\\!\\!\sum_{\begin{subarray}{c}u\in\mathcal{I}:\\\
t-S(u)\in I_{k}^{\delta}\end{subarray}}\\!\\!\\!e^{\alpha\sup
I_{k}^{\delta}}f(t\\!-\\!S(u))$
$\displaystyle\leq\limsup_{t\to\infty}\frac{e^{-\alpha
t}}{t^{\theta+1}}\sum_{k=0}^{t_{\delta}}e^{\alpha\sup
I_{k}^{\delta}}N(t-I_{k}^{\delta})\sup_{x\in I_{k}^{\delta}}f(x)$
$\displaystyle\leq(1+\varepsilon)\frac{e^{\alpha\delta}-1}{\alpha\delta}\frac{W}{\beta}\limsup_{t\to\infty}\frac{1}{t^{\theta+1}}\sum_{k=0}^{t_{\delta}}\delta\sup_{x\in
I_{k}^{\delta}}f(x).$ (6.4)

Write $w(\delta)\vcentcolon=\sup_{x,y\geq
1,\,|x-y|\leq\delta}|x^{-\theta}f(x)-y^{-\theta}f(y)|$ for the modulus of
continuity of $x^{-\theta}f(x)$ on $[1,\infty)$. By uniform continuity,
$w(\delta)\to 0$ as $\delta\to 0$. We now estimate the $\limsup$ in (6.4):

$\displaystyle\limsup_{t\to\infty}\frac{1}{t^{\theta+1}}\sum_{k=0}^{t_{\delta}}\delta\sup_{x\in
I_{k}^{\delta}}f(x)$
$\displaystyle~{}\leq\limsup_{t\to\infty}\bigg{(}\frac{1}{t^{\theta+1}}\\!\\!\int\limits_{0}^{t-r+\delta}\\!f(x)\,\mathrm{d}\mathit{x}+\frac{1}{t^{\theta+1}}\\!\\!\int\limits_{0}^{t-r+\delta}\\!((x+\delta)^{\theta}(x^{-\theta}f(x)+w(\delta))-x^{\theta}x^{-\theta}f(x))\,\mathrm{d}\mathit{x}\bigg{)}$
$\displaystyle=c+\limsup_{t\to\infty}\bigg{(}\frac{C_{\theta}}{t^{\theta+1}}\\!\\!\int\limits_{0}^{t-r+\delta}\\!\big{(}(x+\delta)^{\theta}-x^{\theta}\big{)}\,\mathrm{d}\mathit{x}+\frac{w(\delta)}{t^{\theta+1}}\\!\\!\int\limits_{0}^{t-r+\delta}\\!(x+\delta)^{\theta}\mathrm{d}\mathit{x}\bigg{)}$
$\displaystyle\leq c+\frac{w(\delta)}{\theta+1}.$

Using this in (6.4) gives

$\displaystyle\limsup_{t\to\infty}\frac{e^{-\alpha t}}{t^{\theta+1}}\\!\\!\\!$
$\displaystyle\sum_{\begin{subarray}{c}u\in\mathcal{I}:\\\ r\leq S(u)\leq
t\end{subarray}}\\!\\!\\!e^{\alpha(t\\!-\\!S(u))}f(t\\!-\\!S(u))$
$\displaystyle\leq(1+\varepsilon)\frac{e^{\alpha\delta}-1}{\alpha\delta}\frac{W}{\beta}\Big{(}c+\frac{w(\delta)}{\theta+1}\Big{)}$

almost surely. Letting $\varepsilon,\delta\to 0$ yields the upper bound of
(6.2). The lower bound can be obtained analogously.

For the general case, we split $f=f_{1}+f_{2}$ with

$f_{2}(x)=\begin{cases}f(1)x^{\theta}&\text{for }0\leq x\leq 1,\\\
f(x)&\text{for }x\geq 1\end{cases}$

and $f_{1}\vcentcolon=f-f_{2}$. Then $f_{1},f_{2}$ are continuous and the
previous part of the proof applies to $f_{2}$. Further, the limit in (6.1) is
the same if $f$ is replaced by $f_{2}$. Define
$\varphi_{i}(t)\vcentcolon=e^{\alpha t}f_{i}(t)\mathds{1}_{[0,\infty)}(t)$ for
$i=1,2$ so that $\varphi=\varphi_{1}+\varphi_{2}$. We conclude

$\displaystyle\frac{e^{-\alpha
t}}{t^{\theta+1}}\mathcal{Z}^{\varphi_{2}}_{t}\to\frac{cW}{\beta}\quad\text{as
}t\to\infty,\;t\in\mathds{G}\quad\text{a.\,s.}$

On the other hand, as $\varphi_{1}$ is bounded and supported on $[0,1)$, we
have $e^{-\alpha t}\mathcal{Z}^{\varphi_{1}}_{t}$ converges a. s. to an a. s.
finite limit by [36, Theorem 5.4] and [15, Theorem 3.2], which finishes the
proof. ∎

######  Remark 6.2.

Notice that in the proof of Lemma 6.1, we actually do not use the full power
of assumptions (A2.8) and (A2.9). Indeed, we only need the assumptions
regarding $\xi$ that allow us to apply [33, Corollary 3.1]. What is more, we
could replace the application of [33, Corollary 3.1(a)] in the non-lattice
case by an application of [36, Theorem 5.4] and the application of [33,
Corollary 3.1(b)] in the lattice case by an application of [15, Theorem 3.2]
to get the assertion of the lemma under the even weaker assumptions of [36,
Theorem 5.4] and [15, Theorem 3.2], respectively.

The following theorem gives the central limit theorem in the case of a
centered characteristic $\chi$. Recall that $\mathds{G}=\mathds{Z}$ in the
lattice case and $\mathds{G}=\mathds{R}$ in the non-lattice case and that
$\mathcal{F}=\sigma(\pi_{u}:u\in\mathcal{I})$ where $\pi_{u}$ is the
projection onto the life space of individual $u$ (in particular,
$(\xi_{u},\zeta_{u},\chi_{u})$ is $\sigma(\pi_{u})$-measurable).

######  Theorem 6.3.

Suppose that (A1) through (A2.9) hold. Let $\chi$ be a real-valued, centered
characteristic, and let $\mathcal{N}$ be a standard normal random variable
independent of $\mathcal{F}$.

  1. (i)

Suppose that (A5) holds for the characteristic $\chi$. Then

$e^{-\frac{\alpha}{2}t}\mathcal{Z}^{\chi}_{t}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{st}$\cr\kern-0.21529pt\cr$\to$\cr}}}\bigg{(}\frac{W}{\beta}\int\limits_{\mathds{G}}\mathds{E}[\chi^{2}](x)e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x})\bigg{)}^{\\!\\!1/2}\mathcal{N}\quad\text{as
}t\to\infty,\,t\in\mathds{G}.$ (6.5)

  2. (ii)

Suppose that there are $\theta\geq 0$ and a function $f$ not vanishing
identically on $\mathds{G}$ and satisfying the conditions of Lemma 6.1,
$\mathds{E}[\chi^{2}(t)]=e^{\alpha t}f(t)\mathds{1}_{[0,\infty)}(t)$ and

$\displaystyle\mathds{E}\big{[}\chi^{2}(t)\mathds{1}_{\\{\chi^{2}(t)>\varepsilon
e^{\alpha t}t^{\theta+1}\\}}\big{]}=o(t^{\theta}e^{\alpha t})\quad\text{as
}t\to\infty$ (6.6)

for every $\varepsilon>0$. Then

$\bigg{(}e^{\alpha t}\int\limits_{[0,t]}\mathds{E}[\chi^{2}(x)]e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x})\bigg{)}^{\\!\\!-1/2}{\mathcal{Z}_{t}^{\chi}}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{st}$\cr\kern-0.21529pt\cr$\to$\cr}}}\big{(}\tfrac{W}{\beta}\big{)}^{\\!1/2}\mathcal{N}\text{
as }t\to\infty,\,t\in\mathds{G}.$ (6.7)

###### Proof of Theorem 6.3.

Consider an admissible ordering $v_{1},v_{2},\ldots$ of $\mathcal{I}$ and put
$\mathcal{I}_{n}\vcentcolon=\\{v_{1},\ldots,v_{n}\\}$ and
$\mathcal{G}_{n}\vcentcolon=\sigma(\pi_{v_{j}}:j=1,\ldots,n)$. Now we set

$a_{t}\vcentcolon=\begin{cases}\frac{1}{\beta}\int_{\mathds{G}}\mathds{E}[\chi^{2}](x)e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x})\cdot e^{\alpha t}&\quad\text{in case
(i)},\\\\[6.99997pt]
\frac{1}{\beta}\int_{0}^{t}\mathds{E}[\chi^{2}](x)e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x})\cdot e^{\alpha t}&\quad\text{in case
(ii)}\end{cases}$

for all $t\in\mathds{G}$. If $\|\chi\|_{L^{2}(\mathrm{d}\mathds{P}\otimes
e^{-\alpha
t}\ell(\mathrm{d}\mathit{t}))}=\int\mathds{E}[\chi^{2}(x)]e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x})=0$ in case (i), then the assertion is trivial.
Hence, we exclude this case and may thus assume that $a_{t}>0$ for all
sufficiently large $t\in\mathds{G}$. The latter is automatic in case (ii) in
view of the assumption that $f$ does not vanish identically on $\mathds{G}$
and is uniformly continuous. For $t$ with $a_{t}>0$, we define

$M_{n}(t)\vcentcolon=a_{t}^{-1/2}\sum_{u\in\mathcal{I}_{n}}\chi_{u}(t-S(u)).$

Then $(M_{n}(t),\mathcal{G}_{n})_{n\in\mathds{N}_{0}}$ is a centered,
martingale and bounded in $L^{2}$ by Lemma 4.1 We write $M(t)$ for its limit
(almost sure and in $L^{2}$). Let $(t_{n})_{n\in\mathds{N}}$ be an increasing
sequence in $\mathds{G}$ that diverges to infinity. Then there exists an
increasing sequence $(k_{n})_{n\in\mathds{N}}$ such that
$\mathds{E}[(M(t_{n})-M_{k_{n}}(t_{n}))^{2}]\leq 2^{-n}$ for every
$n\in\mathds{N}$ and, therefore, $M(t_{n})-M_{k_{n}}(t_{n})$ converges to $0$
almost surely as $n\to\infty$. In view of Slutsky’s theorem [41, Theorem
8.6.1], in order to prove the convergence in distribution of
$M(t_{n})=M(t_{n})-M_{k_{n}}(t_{n})+M_{k_{n}}(t_{n})$ as $n\to\infty$, it
suffices to prove convergence in distribution of $M_{k_{n}}(t_{n})$ as
$n\to\infty$. For the latter, we rely on the martingale central limit theorem
[17, Corollary 3.1 on p. 58]. To apply the cited theorem, it suffices to
verify that

$\displaystyle
a_{t_{n}}^{-1}\sum_{j=1}^{k_{n}}\mathds{E}\Big{[}\chi_{v_{j}}^{2}(t_{n}-S(v_{j}))\Big{|}\mathcal{G}_{j-1}\Big{]}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathds{P}$\cr\kern-0.21529pt\cr$\to$\cr}}}W\quad\text{as
}n\to\infty$ (6.10) $\displaystyle
a_{t_{n}}^{-1}\sum_{j=1}^{k_{n}}\mathds{E}\Big{[}\chi_{v_{j}}^{2}(t_{n}-S(v_{j}))\mathds{1}_{\\{|\chi_{v_{j}}(t_{n}-S(v_{j}))|>\varepsilon
a_{t_{n}}^{1/2}\\}}\Big{|}\mathcal{G}_{j-1}\Big{]}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathds{P}$\cr\kern-0.21529pt\cr$\to$\cr}}}0\quad\text{as
}n\to\infty$ (6.13)

for every $\varepsilon>0$. To prove (6.10) observe that

$\displaystyle
a_{t_{n}}^{-1}\mathds{E}\bigg{[}\sum_{j=k_{n}+1}^{\infty}\mathds{E}\Big{[}\chi_{v_{j}}^{2}(t_{n}-S(v_{j}))\Big{|}\mathcal{G}_{j-1}\Big{]}\bigg{]}=\mathds{E}\big{[}(M(t_{n})-M_{k_{n}}(t_{n}))^{2}\big{]}\leq
2^{-n}$

and hence (6.10) is equivalent to

$\displaystyle
a_{t_{n}}^{-1}\sum_{j=1}^{\infty}\mathds{E}\Big{[}\chi_{v_{j}}^{2}(t_{n}-S(v_{j}))\Big{|}\mathcal{G}_{j-1}\Big{]}=a_{t_{n}}^{-1}\sum_{u\in\mathcal{I}}\mathds{E}[\chi^{2}](t_{n}-S(u))\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathds{P}$\cr\kern-0.21529pt\cr$\to$\cr}}}W.$
(6.16)

In case (i), (6.16) is equivalent to

$\displaystyle e^{-\alpha
t_{n}}\sum_{u\in\mathcal{I}}\mathds{E}[\chi^{2}](t_{n}-S(u))=e^{-\alpha
t_{n}}\mathcal{Z}_{t_{n}}^{\mathds{E}[\chi^{2}]}~{}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathds{P}$\cr\kern-0.21529pt\cr$\to$\cr}}}\frac{W}{\beta}\int
e^{-\alpha x}\mathds{E}[\chi^{2}](x)\,\ell(\mathrm{d}\mathit{x}),$

which follows from [25, Theorem 6.1] in the non-lattice case. The lattice case
is analogous. Lemma 6.1 gives (6.16) in case (ii).

Now we show (6.13). Let
$v_{2}(t,s)\vcentcolon=\mathds{E}[\chi^{2}(t)\mathds{1}_{\\{|\chi(t)|>s\\}}]$
for $t\in\mathds{R}$ and $s\geq 0$. In case (i), for any $\varepsilon>0$,

$\displaystyle\limsup_{n\to\infty}e^{-\alpha t_{n}}$
$\displaystyle\sum_{j=1}^{k_{n}}\mathds{E}\big{[}\chi_{v_{j}}^{2}(t_{n}-S(v_{j}))\mathds{1}_{\\{|\chi_{v_{j}}(t_{n}-S(v_{j}))|>\varepsilon
e^{\alpha t_{n}/2}\\}}\,\big{|}\,\mathcal{G}_{j-1}\big{]}$
$\displaystyle\leq\limsup_{n\to\infty}e^{-\alpha
t_{n}}\sum_{j=1}^{\infty}v_{2}(t_{n}-S(v_{j}),\varepsilon e^{\alpha t_{n}/2})$
$\displaystyle\leq\liminf_{s\to\infty}\limsup_{n\to\infty}e^{-\alpha
t_{n}}\mathcal{Z}_{t_{n}}^{v_{2}(\cdot,s)}$
$\displaystyle=\liminf_{s\to\infty}\frac{W}{\beta}\int v_{2}(x,s)e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x})=0\qquad\text{a.\,s.}$

by [25, Theorem 6.1] in the non-lattice case and the dominated convergence
theorem. The lattice case is analogous.

We turn to case (ii) and fix $\varepsilon>0$. We infer from (6.6) that for any
$\varepsilon,\delta>0$ there is a $T\geq 0$ such that, for all $t\geq T$,

$v_{2}(t,\varepsilon e^{\alpha
t/2}t^{\frac{\theta+1}{2}})=\mathds{E}\Big{[}\chi(t)^{2}\mathds{1}_{\\{|\chi(t)|>\varepsilon{e^{\alpha
t/2}t^{\frac{\theta+1}{2}}}\\}}\Big{]}\leq\delta e^{\alpha t}t^{\theta}.$

Therefore, with
$\|\mathds{E}[\chi^{2}]\|_{[0,T]}\vcentcolon=\sup_{x\in[0,T]}\mathds{E}[\chi^{2}](x)$,

$\displaystyle\limsup_{n\to\infty}\frac{e^{-\alpha
t_{n}}}{t_{n}^{\theta+1}}\sum_{j=1}^{k_{n}}\mathds{E}\Big{[}\chi_{v_{j}}^{2}(t_{n}\\!-\\!S(v_{j}))\mathds{1}_{\\{|\chi_{v_{j}}(t_{n}-S(v_{i}))|>\varepsilon{e^{\alpha
t_{n}/2}t_{n}^{\frac{\theta+1}{2}}}\\}}\,\Big{|}\,\mathcal{G}_{j-1}\Big{]}$
$\displaystyle~{}\leq\limsup_{n\to\infty}\frac{e^{-\alpha
t_{n}}}{t_{n}^{\theta+1}}\sum_{u\in\mathcal{I}}v_{2}(t_{n}\\!-\\!S(u),\varepsilon{e^{\alpha
t_{n}/2}t_{n}^{\frac{\theta+1}{2}}})$
$\displaystyle~{}=\limsup_{n\to\infty}\frac{e^{-\alpha
t_{n}}}{t_{n}^{\theta+1}}\bigg{(}\\!\\!\\!\\!\sum_{\begin{subarray}{c}u\in\mathcal{I}:\\\
S(u)\leq
t_{n}\\!-\\!T\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!v_{2}(t_{n}\\!-\\!S(u),\varepsilon{e^{\alpha
t_{n}/2}t_{n}^{\frac{\theta+1}{2}}})+\|\mathds{E}[\chi^{2}]\|_{[0,T]}\cdot
N((t_{n}-T,t_{n}])\bigg{)}$
$\displaystyle~{}\leq\limsup_{n\to\infty}\frac{e^{-\alpha
t_{n}}}{t_{n}^{\theta+1}}\bigg{(}\delta\\!\\!\\!\\!\sum_{\begin{subarray}{c}u\in\mathcal{I}:\\\
S(u)\leq
t_{n}\\!-\\!T\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!e^{\alpha(t_{n}\\!-\\!S(u))}(t_{n}-S(u))^{\theta}+\|\mathds{E}[\chi^{2}]\|_{[0,T]}\cdot
N((t_{n}-T,t_{n}])\bigg{)}$ $\displaystyle~{}\leq\frac{\delta
W}{\beta(\theta+1)}\qquad\text{a.\,s.}$

by Lemma 6.1 with $f(t)=t^{\theta}$, $t\geq 0$ and the fact that $e^{-\alpha
t_{n}}N((t_{n}-T,t_{n}])$ converges a. s. by [36, Theorem 5.4] in the non-
lattice case and by [15, Theorem 3.2] in the lattice case. Since $\delta>0$
was arbitrary, we conclude that the limit is zero and, therefore, (6.13) holds
true in both cases.

It remains to justify that the convergence is stable and that limiting random
variable $\mathcal{N}$ is independent of $\mathcal{F}$. Although, this is not
stated explicitly in [17, Theorem 3.2], it follows from the proof of the
preceding Lemma 3.1 of [17], cf. Eq. (3.15) there, that is, for any
$E\in\mathcal{F}$, we have

$\displaystyle\mathds{E}\Big{[}e^{\mathrm{i}\theta
M_{k_{n}}(t_{n})}\mathds{1}_{E}\Big{]}\to\mathds{E}\Big{[}e^{-W\frac{\theta^{2}}{2}}\mathds{1}_{E}\Big{]}$

for every $\theta\in\mathds{R}$. The latter is equivalent, by a standard
approximation argument, to say that for any $\mathcal{F}$-measurable random
variable $Y$

$\displaystyle\mathds{E}\Big{[}e^{\mathrm{i}\theta
M_{k_{n}}(t_{n})}e^{\mathrm{i}\eta
Y}\Big{]}\to\mathds{E}\Big{[}e^{-W\frac{\theta^{2}}{2}}e^{\mathrm{i}\eta
Y}\Big{]}=\mathds{E}\Big{[}e^{\mathrm{i}\theta\sqrt{W}\mathcal{N}}e^{\mathrm{i}\eta
Y}\Big{]}$

for a standard normal variable $\mathcal{N}$ independent of $(W,Y)$. This also
implies the stable convergence by [1, Proposition 1].

∎

###  6.2 Deterministic characteristics

Let $f$ be a deterministic characteristic, i.e., a càdlàg function
$f:\mathds{R}\to\mathds{R}$. We investigate the behavior of
$\mathcal{Z}^{f}_{t}$ as $t\to\infty$ by means of an auxiliary centered random
characteristic $\chi_{f}$ defined by

$\chi_{f}(t)\vcentcolon=f*\xi*V(t)-f*\mu*V(t)=m^{f}*\xi(t)-m^{f}*\mu(t),$
(6.17)

where
$V(\cdot)=\sum_{n=0}^{\infty}\mu^{*n}(\cdot)=\mathds{E}[\sum_{u\in\mathcal{I}}\delta_{S(u)}(\cdot)]$
and $*$ denotes Lebesgue-Stieltjes convolution. For instance, for every
$t\in\mathds{R}$,

$f*V(t)=\int
f(t-x)\,V(\mathrm{d}\mathit{x})=\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}}f(t-S(u))\bigg{]}=m_{t}^{f}$

if the integrals are well-defined. However, the latter is not guaranteed a
priori. The following lemma provides a sufficient condition along with an
important connection between $\mathcal{Z}^{f}_{t}$ and
$\mathcal{Z}^{\chi_{f}}_{t}$.

######  Lemma 6.4.

Assume that (A1) holds. Let $f:\mathds{R}\to\mathds{R}$ be a deterministic
càdlàg function such that $t\mapsto|f(t)|e^{-\alpha t}$ is directly Riemann
integrable.

  1. (a)

The characteristic $\chi_{f}$ given by (6.17) is well-defined and has almost
surely càdlàg paths.

  2. (b)

If (A2.9) holds and the function $t\mapsto
m_{t}^{f}e^{-\frac{\alpha}{2}t}(1+t^{2})$ is bounded, then the characteristic
$\chi_{f}$ satisfies (A5) and (A6).

  3. (c)

If (A2.9) holds and the function $t\mapsto
m_{t}^{f}e^{-\frac{\alpha}{2}t}(1+t^{2})$ is bounded, then for any
$t\in\mathds{R}$, $\mathds{E}[\chi_{f}(t)]=0$ and

$\mathcal{Z}^{f}_{t}-m^{f}_{t}=\mathcal{Z}^{\chi_{f}}_{t}\qquad\text{a.\,s.\quad
for all }t\in\mathds{R}.$ (6.18)

  4. (d)

If $f$ is supported on $[0,\infty)$, i.e., if $f(t)=0$ for all $t<0$, then
(6.18) also holds.

###### Proof.

(a) In order to see that $\chi_{f}$ is well-defined, it suffices to check that
$|f|*\mu*V(t)$ is finite for all $t\in\mathds{R}$. Indeed, as
$\mu*V\leq\delta_{0}+\mu*V=V$, putting $g(t)\vcentcolon=|f(t)|e^{-\alpha t}$
we obtain

$\displaystyle|f|*\mu*V(t)\leq|f|*V(t)$
$\displaystyle=\sum_{n=0}^{\infty}\mathds{E}\bigg{[}\sum_{|u|=n}|f|(t-S(u))\bigg{]}$
$\displaystyle=e^{\alpha
t}\sum_{n=0}^{\infty}\mathds{E}\bigg{[}\sum_{|u|=n}e^{-\alpha
S(u)}g(t-S(u))\bigg{]}=e^{\alpha t}g*\mathsf{U}(t)<\infty,$ (6.19)

where we have used the many-to-one formula (4.2) in the next-to-last step and
the direct Riemann integrability of $g$ in combination with (4.4) in the last.
To prove that $\chi_{f}$ has càdlàg paths almost surely, it suffices to show
that $f^{*}*\mu*V$ is finite. This is justified by the fact that $f$ has
càdlàg paths together with the dominated convergence theorem. Since
$f^{*}*\mu*V\leq f^{*}*V$, we have to check that the latter is finite.
Further, $e^{-\alpha t}f^{*}(t)\leq e^{\alpha}g^{*}(t)$. Therefore, by a
calculation analogous to (6.19), it is enough to show that $g^{*}*\mathsf{U}$
is finite, which, in view of (4.4), is true if $g^{*}$ is directly Riemann
integrable. This however follows from the converse part of Proposition 2.5(a)
since $g$ is directly Riemann integrable.

(b) By Proposition 2.5(c), it suffices to show that

$\int\mathds{E}\big{[}(\chi_{f}^{*})^{2}\big{]}(x)e^{-\alpha
x}\,\mathrm{d}\mathit{x}<\infty,$ (6.20)

To this end, note that, since $\chi_{f}=m^{f}*\xi-m^{f}*\mu$, we have, for any
$x\in\mathds{R}$,

$\displaystyle\mathds{E}\big{[}|\chi^{*}_{f}(x)|^{2}\big{]}\leq
2\mathds{E}\Big{[}\sup_{|t-x|\leq
1}|m^{f}\\!*\xi(t)|^{2}\Big{]}+2\\!\\!\\!\sup_{|t-x|\leq
1}|m^{f}\\!*\mu(t)|^{2}\leq 4\mathds{E}\Big{[}\sup_{|t-x|\leq
1}|m^{f}\\!*\xi(t)|^{2}\Big{]},$

where we have used Jensen’s inequality. For $|t-x|\leq 1$, we obtain

$\displaystyle 1+(x-X_{j})^{2}$ $\displaystyle=1+(t-X_{j}+x-t)^{2}\leq
1+2(t-X_{j})^{2}+2(x-t)^{2}\leq 3+2(t-X_{j})^{2}$ $\displaystyle\leq
3(1+(t-X_{j})^{2})$

for $j=1,\ldots,N$, and, therefore, with
$C\vcentcolon=3e^{\frac{\alpha}{2}}\sup_{t\in\mathds{R}}e^{-\frac{\alpha}{2}t}(1+t^{2})|m_{t}^{f}|$,

$\displaystyle|m^{f}\\!*\xi(t)|^{2}=\bigg{|}\sum_{j=1}^{N}m_{t-X_{j}}^{f}\bigg{|}^{2}\leq
C^{2}\sum_{1\leq i,j\leq
N}\frac{e^{\frac{\alpha}{2}(x-X_{i})}}{1+(x-X_{i})^{2}}\frac{e^{\frac{\alpha}{2}(x-X_{j})}}{1+(x-X_{j})^{2}}.$

Thus, since

$\displaystyle\int\frac{1}{1+(x-X_{i})^{2}}\frac{1}{1+(x-X_{j})^{2}}\,\mathrm{d}\mathit{x}\leq\int\frac{\mathrm{d}\mathit{x}}{1+(x-X_{i})^{2}}=\pi,$

we conclude

$\displaystyle\int\mathds{E}\big{[}|\chi^{*}_{f}(x)|^{2}\big{]}e^{-\alpha
x}\,\mathrm{d}\mathit{x}\leq 4\pi C^{2}\mathds{E}\bigg{[}\sum_{1\leq i,j\leq
N}e^{-\frac{\alpha}{2}X_{i}}e^{-\frac{\alpha}{2}X_{j}}\bigg{]}<\infty$

from assumption (A2.9).

(c) By part (a), $|f|*\mu*V(t)$ is finite for all $t\in\mathds{R}$ and, hence,
$\mathds{E}[\chi_{f}](t)=\mathds{E}[f*\xi*V(t)]-f*\mu*V(t)=f*\mu*V(t)-f*\mu*V(t)=0$
for all $t\in\mathds{R}$. Further, since $f$ satisfies (A4) by assumption and
trivially also (A5), Proposition 2.2 implies that $\mathcal{Z}^{f}_{t}$
converges in $L^{1}$ for every $t\in\mathds{R}$. The characteristic $\chi_{f}$
on the other hand satisfies (A5) by part (b) and trivially also (A4) because
it is centered. Thus, Proposition 2.2 yields that also the series defining
$\mathcal{Z}^{\chi_{f}}_{t}$ converges unconditionally in $L^{1}$ for all
$t\in\mathds{R}$.

In particular, for any $n\in\mathds{N}$, the infinite series

$\sum_{0\leq|u|\leq n}\chi_{f,u}(t-S(u))$

also converges unconditionally in $L^{1}$ and so is well-defined and converges
to $\mathcal{Z}^{\chi_{f}}_{t}$ as $n\to\infty$ in $L^{1}$. Moreover, due to
the fact that $f$ is deterministic, $\chi_{f}$ is $\xi$-measurable. For
$u\in\mathcal{I}$, we have $\chi_{f,u}(t)=f*\xi_{u}*V(t)-f*\mu*V(t)$. Using
this and $V=\delta_{0}+\mu*V$, we infer

$\displaystyle\sum_{0\leq|u|\leq n}\chi_{f,u}(t-S(u))$
$\displaystyle~{}=\sum_{0\leq|u|\leq
n}\big{(}f*\xi_{u}*V(t-S(u))-f*\mu*V(t-S(u))\big{)}$
$\displaystyle~{}=\sum_{1\leq|u|\leq n+1}\\!f*V(t-S(u))-\sum_{0\leq|u|\leq
n}f*\mu*V(t-S(u))$ $\displaystyle~{}=\sum_{1\leq|u|\leq
n}\\!f(t-S(u))-f*\mu*V(t)+\sum_{|u|=n+1}f*V(t-S(u))$
$\displaystyle=\sum_{0\leq|u|\leq
n}\\!f(t-S(u))-f*V(t)+\sum_{|u|=n+1}f*V(t-S(u)).$

In the last line, $f*V(t)=m_{t}^{f}$. The manipulations in the above chain of
equalities are justified by the fact that

$\mathds{E}\bigg{[}\sum_{0\leq|u|\leq n+1}|f*V(t-S(u))|\bigg{]}<\infty.$

The finiteness of the above expectation follows from

$\displaystyle\mathds{E}\bigg{[}\sum_{|u|=k}\\!|f*V(t-S(u))|\bigg{]}$
$\displaystyle\leq\mathds{E}\bigg{[}\sum_{|u|=k}\\!|f|*V(t-S(u))\bigg{]}=\mathds{E}\bigg{[}\sum_{|u|\geq
k}\\!|f|(t-S(u))\bigg{]}$
$\displaystyle\leq\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}}\\!|f|(t-S(u))\bigg{]}=m^{|f|}_{t}<\infty.$

The dominated convergence theorem yields

$\sum_{|u|=n+1}f*V(t-S(u))\to 0\quad\text{and}\sum_{0\leq|u|\leq
n}\\!f(t-S(u))\to\mathcal{Z}^{f}_{t}$

as $n\to\infty$ in $L^{1}$.

(d) The last calculation carries over if $f(t)=0$ for all $t<0$. Indeed, the
latter condition implies $\chi_{f}(t)=0$ for all $t<0$ and hence with
probability one, all sums have only finitely many non-vanishing terms (since
only finitely many individuals are born before any fixed time almost surely by
[23, Theorem 6.2.3]). ∎

Lemma 6.4 has the following corollary.

######  Corollary 6.5.

If the assumptions of Lemma 6.4(b) are satisfied, then, for every
$t\in\mathds{R}$,

$\operatorname{\mathrm{Var}}[\mathcal{Z}^{f}_{t}]=m^{\chi_{f}^{2}}_{t}<\infty.$

###### Proof.

By Lemma 6.4, we have
$\mathcal{Z}^{f}_{t}=\mathcal{Z}^{\chi_{f}}_{t}+m^{f}_{t}$ where
$\mathcal{Z}^{\chi_{f}}_{t}$ is centered and

$e^{-\alpha t}\operatorname{\mathrm{Var}}[\chi_{f}](t)=e^{-\alpha
t}\mathds{E}[\chi_{f}^{2}](t)$

is directly Riemann integrable. Using
$\operatorname{\mathrm{Var}}[\mathcal{Z}^{f}_{t}]=\operatorname{\mathrm{Var}}[\mathcal{Z}^{\chi_{f}}_{t}+m^{f}_{t}]=\operatorname{\mathrm{Var}}[\mathcal{Z}^{\chi_{f}}_{t}]=\mathds{E}[(\mathcal{Z}^{\chi_{f}}_{t})^{2}]$
we infer with the help of (4.10) that

$\mathds{E}[(\mathcal{Z}^{\chi_{f}}_{t})^{2}]=\mathds{E}[\mathcal{Z}^{\chi_{f}^{2}}_{t}]\quad\text{for
all }t\in\mathds{R}.$

∎

###  6.3 Slowly growing mean process with signed characteristics

We now treat the case where $m_{t}^{\varphi}$ grows relatively slowly as
$|t|\to\infty$. Later, we shall reduce the general case to this one.

######  Theorem 6.6.

Suppose that (A1) through (A2.9) hold, the random characteristic $\varphi$
satisfies (A4), (A5) and that the function $t\mapsto
e^{-\frac{\alpha}{2}t}(1+t^{2})m_{t}^{\varphi}$ is bounded. If
$\mathds{G}=\mathds{R}$, assume in addition that (A6) holds. Then, with
$\mathcal{N}$ a standard normal random variable independent of $\mathcal{F}$,

$e^{-\frac{\alpha}{2}t}\mathcal{Z}^{\varphi}_{t}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sigma_{\varphi}\sqrt{\tfrac{W}{\beta}}\mathcal{N}\quad\text{
(stably)},$

with

$\sigma^{2}_{\varphi}\vcentcolon=\int\operatorname{\mathrm{Var}}[\varphi(x)+\xi*m^{\varphi}_{x}]e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x}).$

Moreover, $\sigma^{2}_{\varphi}=0$ if and only if $\mathcal{Z}^{\varphi}_{t}$
is equal to its expectation a. s. for every $t\in\mathds{G}$.

###### Proof.

Clearly, $m^{\mathds{E}[\varphi]}=m^{\varphi}$ where $\mathds{E}[\varphi]$
denotes the function $t\mapsto\mathds{E}[\varphi(t)]$. In view of Lemma 6.4,
we can write

$\displaystyle\mathcal{Z}_{t}^{\varphi}$
$\displaystyle=\mathcal{Z}^{\varphi-\mathds{E}[\varphi]}_{t}+\mathcal{Z}^{\mathds{E}[\varphi]}_{t}=\mathcal{Z}^{\varphi-\mathds{E}[\varphi]}_{t}+\mathcal{Z}^{\chi_{\mathds{E}[\varphi]}}_{t}+m^{\varphi}_{t}=\mathcal{Z}^{\varphi-\mathds{E}[\varphi]+\chi_{\mathds{E}[\varphi]}}_{t}+m^{\varphi}_{t}$
$\displaystyle=\mathcal{Z}_{t}^{\varphi+m^{\varphi}*\xi-\mathds{E}[\varphi+m^{\varphi}*\xi]}+m^{\varphi}_{t}\quad\text{a.\,s.}$
(6.21)

By assumption, $e^{-\alpha t/2}m^{\varphi}_{t}\to 0$ as $t\to\infty$. Hence,
it suffices to show that
$e^{-\frac{\alpha}{2}t}\mathcal{Z}^{\varphi-\mathds{E}[\varphi]+\chi_{\mathds{E}[\varphi]}}_{t}$
converges in distribution to the claimed distribution. Since the
characteristic

$t\mapsto\varphi(t)-\mathds{E}[\varphi(t)]+\chi_{\mathds{E}[\varphi]}(t)$
(6.22)

is centered, it is reasonable to apply Theorem 6.3(i). To this end, we need to
check that (A5) holds for the characteristic in (6.22), i.e., that the
function

$\displaystyle t$ $\displaystyle\mapsto e^{-\alpha
t}\operatorname{\mathrm{Var}}[\varphi(t)-\mathds{E}[\varphi(t)]+\chi_{\mathds{E}[\varphi]}(t)]$
$\displaystyle=e^{-\alpha
t}\operatorname{\mathrm{Var}}[\varphi(t)+\chi_{\mathds{E}[\varphi]}(t)]\text{
is directly Riemann integrable.}$ (6.23)

From Lemma 6.4 we conclude that $\chi_{\mathds{E}[\varphi]}(t)$ satisfies (A5)
and (A6). This is also true for $\varphi$. Hence, (6.23) holds by Remark 2.4.

Finally, if $\sigma_{\varphi}=0$, then $\varphi(x)+m^{\varphi}*\xi(x)$ is
equal to its expectation a. s. for $\ell$-almost every $x\in\mathds{G}$, i.e.,
$\varphi(x)+m^{\varphi}*\xi(x)-(\mathds{E}[\varphi](x)+m^{\varphi}*\mu(x))=0$
a. s. for $\ell$-almost every $x\in\mathds{G}$. On the other hand, Lemma
6.4(a) implies that the left-hand side of the above equation has càdlàg paths
a. s. and thus vanishes for every $x\in\mathds{G}$ a. s. Consequently, by
(6.21),

$\displaystyle\mathcal{Z}_{t}^{\varphi}=\mathcal{Z}_{t}^{\varphi+m^{\varphi}*\xi-\mathds{E}[\varphi+m^{\varphi}*\xi]}+m^{\varphi}_{t}=m^{\varphi}_{t}\quad\text{a.\,s.\
for all }t\in\mathds{G},$

i.e., $\mathcal{Z}_{t}^{\varphi}$ is a. s. deterministic for every
$t\in\mathds{G}$.

∎

###  6.4 Proof of Theorem 2.14

In the proof of Theorem 2.14, we use the following fact.

######  Lemma 6.7.

Let $\eta_{1},\ldots,\eta_{m}$ be different real numbers and let
$(Y_{j,l})_{1\leq j\leq m,0\leq l\leq n}$ be a collection of centered, square-
integrable random variables. Then for

$\chi(t)\vcentcolon=e^{\frac{\alpha}{2}t}\sum_{l=0}^{n}\sum_{j=1}^{m}t^{l}e^{\mathrm{i}\eta_{j}t}Y_{j,l}$

it holds that

$\frac{1}{t^{2n+1}}\int_{[0,t]}\operatorname{\mathrm{Var}}\big{[}\chi(x)\big{]}e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x})\to\frac{1}{2n+1}\sum_{j=1}^{m}\operatorname{\mathrm{Var}}[Y_{j,n}]\quad\text{as
}t\to\infty,\,t\in\mathds{G}$ (6.24)

and, for any $\varepsilon>0$,

$\displaystyle\mathds{E}\big{[}|\chi(t)|^{2}\mathds{1}_{\\{|\chi(t)|^{2}>\varepsilon
t^{2n+1}e^{\alpha t}\\}}\big{]}=o(t^{2n}e^{\alpha t})\quad\text{as
}t\to\infty,\,t\in\mathds{G}.$ (6.25)

In other words, $\chi(t)$ fulfills the assumption of Theorem 6.3(ii) with
$\theta=2n$.

###### Proof.

Expanding the variance gives

$\displaystyle\operatorname{\mathrm{Var}}\bigg{[}\sum_{l=0}^{n}\sum_{j=1}^{m}x^{l}e^{\mathrm{i}\eta_{j}x}Y_{j,l}\bigg{]}$
$\displaystyle=x^{2n}\sum_{j=1}^{m}\operatorname{\mathrm{Var}}[Y_{j,n}]$
$\displaystyle\hphantom{=}~{}+x^{2n}\sum_{j\not=k}e^{\mathrm{i}(\eta_{j}-\eta_{k})x}\operatorname{\mathrm{Cov}}[Y_{j,n},Y_{k,n}]+O(x^{2n-1})$
(6.26)

as $x\to\infty$. Further notice that, for $\eta\in\mathds{R}$,

$\displaystyle\frac{1}{t^{{2n}+1}}\int_{[0,t]}x^{2n}e^{\mathrm{i}\eta
x}\,\ell(\mathrm{d}\mathit{x})\to\begin{cases}\frac{1}{2n+1},&\text{if
}\eta=0,\\\ 0,&\text{if }\eta\not=0.\end{cases}$ (6.27)

This follows from the fundamental theorem of calculus in the non-lattice case
and integration by parts if $\eta\not=0$, whereas in the lattice case, it
follows from Faulhaber’s formula if $\eta=0$ and from summation by parts if
$\eta\not=0$. Relation (6.24) now follows from (6.4) and (6.27).

In order to prove that (6.25) holds we set
$\chi_{j}(t)\vcentcolon=e^{\frac{\alpha}{2}t}\sum_{l=0}^{n}t^{l}e^{\mathrm{i}\eta_{j}t}Y_{j,l}$.
Since for any complex numbers $c_{1},\dots,c_{m}$ and $y>0$

$|c_{1}+\cdots+c_{m}|^{2}\mathds{1}_{\\{|c_{1}+\cdots+c_{m}|>y\\}}\leq
m^{2}\big{(}|c_{1}|^{2}\mathds{1}_{\\{|c_{1}|>y/m\\}}+\cdots+|c_{m}|^{2}\mathds{1}_{\\{|c_{m}|>y/m\\}}\big{)},$

it suffices to prove (6.25) for $\chi_{j}$ instead of $\chi$. By Markov’s
inequality, we have

$\mathds{P}(|\chi_{j}(t)|^{2}>\varepsilon t^{2n+1}e^{\alpha
t})\leq\frac{(n+1)^{2}}{\varepsilon
t^{2n+1}}\sum_{l=0}^{n}t^{2l}\mathds{E}[|Y_{j,l}|^{2}]\to 0\quad\text{as
}t\to\infty$

as the sum is of the order $t^{2n}$ as $t\to\infty$. Consequently,

$\displaystyle\mathds{E}\big{[}|\chi_{j}(t)|^{2}\mathds{1}_{\\{|\chi_{j}(t)|^{2}>\varepsilon
t^{2n+1}e^{\alpha t}\\}}\big{]}$ $\displaystyle~{}\leq 2t^{2n}e^{\alpha
t}\mathds{E}\big{[}|Y_{j,n}|^{2}\mathds{1}_{\\{|\chi_{j}(t)|^{2}>\varepsilon
t^{2n+1}e^{\alpha t}\\}}\big{]}+O(t^{2n-1}e^{\alpha t})=o(t^{2n}e^{\alpha t})$

as $t\to\infty$. This proves (6.25). ∎

We now turn to the proof of Theorem 2.14.

###### Proof of Theorem 2.14.

Suppose that $\varphi$ is a random characteristic satisfying

$\displaystyle
m^{\varphi}_{t}=\mathds{1}_{[0,\infty)}(t)\sum_{\lambda\in\Lambda_{\geq}}\sum_{l=0}^{k(\lambda)-1}a_{\lambda,l}t^{l}e^{\lambda
t}+r(t),\quad t\in\mathds{G}$ (2.20)

for some constants $a_{\lambda,l}\in\mathds{R}$ and a function $r$ such that
$|r(t)|\leq Ce^{\frac{\alpha}{2}t}/(1+t^{2})$ for a finite constant $C>0$. For
any $\lambda\in\Lambda_{\geq}$, we put

$\vec{a}_{\lambda}\vcentcolon=\sum_{l=1}^{k(\lambda)}a_{\lambda,l-1}\mathsf{e}_{l}$

and consider the following characteristic

$\displaystyle\psi_{\Lambda}(t)=\sum_{\lambda\in\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}(\phi_{\lambda}(t)+\chi_{\lambda}(t))\mathsf{e}_{1}$

for $\phi_{\lambda}$ and $\chi_{\lambda}$ defined in (5.5) and (5.6),
respectively. Then, by Lemma 5.5, for $t\geq 0$,

$\displaystyle\mathcal{Z}_{t}^{\psi_{\Lambda}}$
$\displaystyle=\sum_{\lambda\in\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}\mathcal{Z}_{t}^{\phi_{\lambda}+\chi_{\lambda}}\mathsf{e}_{1}=\sum_{\lambda\in\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}\exp(\lambda,t,k(\lambda))W(\lambda,k(\lambda))\mathsf{e}_{1}=H_{\Lambda}(t),$
(6.28)

where the definition of $H_{\Lambda}$ should be recalled from (2.21). Further,
by Lemma 5.2

$\displaystyle m_{t}^{\psi_{\Lambda}}$
$\displaystyle=\sum_{\lambda\in\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}\mathds{E}[\mathcal{Z}_{t}^{\phi_{\lambda}+\chi_{\lambda}}]\mathsf{e}_{1}$
$\displaystyle=\sum_{\lambda\in\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}\mathds{E}[\mathcal{Z}_{t}^{\phi_{\lambda}}]\mathsf{e}_{1}=\mathds{1}_{[0,\infty)}(t)\sum_{\lambda\in\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}\exp(\lambda,t,k(\lambda))\mathsf{e}_{1}.$
(6.29)

Similarly, putting

$\displaystyle\psi_{\partial\Lambda}(t)$
$\displaystyle=\sum_{\lambda\in\partial\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}\mathds{E}[\phi_{\lambda}(t)]\mathsf{e}_{1},$

again by Lemma 5.2, we obtain, for any $t\in\mathds{R}$,

$\displaystyle m^{\psi_{\partial\Lambda}}_{t}$
$\displaystyle=\mathds{E}[\mathcal{Z}_{t}^{\psi_{\partial\Lambda}}]=\sum_{\lambda\in\partial\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}\mathds{E}[\mathcal{Z}_{t}^{\phi_{\lambda}}]\mathsf{e}_{1}$
$\displaystyle=\mathds{1}_{[0,\infty)}(t)\sum_{\lambda\in\partial\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}\exp{(\lambda,t,k(\lambda))}\mathsf{e}_{1}=\mathds{1}_{[0,\infty)}(t)H_{\partial\Lambda}(t),$
(6.30)

where the definition of $H_{\partial\Lambda}$ should be recalled from (2.22).
In view of Remark 2.15, we have
$\overline{\vec{a}_{\lambda}}=\vec{a}_{\overline{\lambda}}$,
$\overline{\phi_{\lambda}}=\phi_{\overline{\lambda}}$ and
$\overline{\chi_{\lambda}}=\chi_{\overline{\lambda}}$. We thus conclude that
both characteristics $\psi_{\Lambda}$ and $\psi_{\partial\Lambda}$ are in fact
real-valued.

Now write

$\displaystyle\mathcal{Z}_{t}^{\varphi}=\mathcal{Z}_{t}^{\psi_{\Lambda}}+\mathcal{Z}_{t}^{\psi_{\partial\Lambda}}+\mathcal{Z}_{t}^{\varrho}=H_{\Lambda}(t)+\mathcal{Z}_{t}^{\psi_{\partial\Lambda}}+\mathcal{Z}_{t}^{\varrho},$
(6.31)

where $\varrho\vcentcolon=\varphi-\psi_{\Lambda}-\psi_{\partial\Lambda}$.
Next, since $\psi_{\partial\Lambda}$ is deterministic, we may consider the
associated centered characteristic $\chi_{\psi_{\partial\Lambda}}$ defined in
(6.17), namely,

$\displaystyle\chi_{\psi_{\partial\Lambda}}(t)$
$\displaystyle=\xi*m^{\psi_{\partial\Lambda}}_{t}-\mu*m^{\psi_{\partial\Lambda}}_{t}$
$\displaystyle=\sum_{\lambda\in\partial\Lambda}\sum_{j=1}^{N}\vec{a}_{\lambda}^{\mathsf{T}}\exp{(\lambda,t-X_{j},k(\lambda))}\mathsf{e}_{1}\mathds{1}_{[0,\infty)}(t-X_{j})-\mu*m^{\psi_{\partial\Lambda}}_{t}.$

If we set now

$\displaystyle\psi_{\lambda}(t)$
$\displaystyle\vcentcolon=\mathds{1}_{[0,\infty)}(t)\sum_{j=1}^{N}\vec{a}_{\lambda}^{\mathsf{T}}\exp(\lambda,t-X_{j},k(\lambda))\mathsf{e}_{1}$
$\displaystyle\hphantom{\vcentcolon=}~{}-\mathds{1}_{[0,\infty)}(t)\mathds{E}\bigg{[}\sum_{j=1}^{N}\vec{a}_{\lambda}^{\mathsf{T}}\exp(\lambda,t-X_{j},k(\lambda))\mathsf{e}_{1}\bigg{]},$
$\displaystyle\chi(t)$
$\displaystyle\vcentcolon=\sum_{\lambda\in\partial\Lambda}\psi_{\lambda}(t)$

and

$\displaystyle\phi_{\partial\Lambda}(t)$
$\displaystyle\vcentcolon=\sum_{\lambda\in\partial\Lambda}\sum_{j=1}^{N}\vec{a}_{\lambda}^{\mathsf{T}}\exp(\lambda,t-X_{j},k(\lambda))\mathsf{e}_{1}\mathds{1}_{[0,X_{j})}(t)$
$\displaystyle\hphantom{\vcentcolon=}~{}-\sum_{\lambda\in\partial\Lambda}\mathds{E}\bigg{[}\sum_{j=1}^{N}\vec{a}_{\lambda}^{\mathsf{T}}\exp(\lambda,t-X_{j},k(\lambda))\mathsf{e}_{1}\mathds{1}_{[0,X_{j})}(t)\bigg{]}$
$\displaystyle=\sum_{\lambda\in\partial\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}\phi_{\lambda}(t)\mathsf{e}_{1}-\sum_{\lambda\in\partial\Lambda}\mathds{E}\big{[}\vec{a}_{\lambda}^{\mathsf{T}}\phi_{\lambda}(t)\mathsf{e}_{1}\big{]}.$

we get the following decomposition

$\displaystyle\chi_{\psi_{\partial\Lambda}}(t)=\chi(t)-\phi_{\partial\Lambda}(t).$

The fact that all the expectations above are finite and thus the
characteristics are well-defined follows from (A2.8) and (A7). Note also that,
for every $\lambda\in\partial\Lambda$,

$\displaystyle\sum_{i=1}^{N}\vec{a}_{\lambda}^{\mathsf{T}}\exp(\lambda,t-X_{i},k(\lambda))\mathsf{e}_{1}$
$\displaystyle=\sum_{i=1}^{N}e^{\lambda(t-X_{i})}\sum_{l=0}^{k(\lambda)-1}a_{\lambda,l}(t-X_{i})^{l}$
$\displaystyle=\sum_{i=1}^{N}e^{\lambda(t-X_{i})}\sum_{l=0}^{k(\lambda)-1}a_{\lambda,l}\sum_{j=0}^{l}\binom{l}{j}t^{j}(-X_{i})^{l-j}$
$\displaystyle=\sum_{j=0}^{k(\lambda)-1}\sum_{l=j}^{k(\lambda)-1}\sum_{i=1}^{N}e^{-\lambda
X_{i}}a_{\lambda,l}\binom{l}{j}(-X_{i})^{l-j}t^{j}e^{\lambda t}$
$\displaystyle=\sum_{j=0}^{k(\lambda)-1}R_{\lambda,j}t^{j}e^{\lambda t}$

by (2.23). With this at hand, we infer

$\displaystyle\psi_{\lambda}(t)=\mathds{1}_{[0,\infty)}(t)\sum_{l=0}^{k(\lambda)-1}(R_{\lambda,l}-\mathds{E}[R_{\lambda,l}])t^{l}e^{\lambda
t},\quad t\in\mathds{R}.$ (6.32)

Using Lemma 6.4(d) with $f=\psi_{\partial\Lambda}$ (note here that such $f$
fulfills the assumptions as for any $\lambda\in\partial\Lambda$ the
characteristic $\phi_{\lambda}$ vanishes on $(-\infty,0)$ and satisfies (A4)
by Lemma 5.1), we get that
$\mathcal{Z}_{t}^{\psi_{\partial\Lambda}}-m_{t}^{\psi_{\partial\Lambda}}=Z_{t}^{\chi_{\psi_{\partial\Lambda}}}$
and, hence,
$\mathcal{Z}_{t}^{\psi_{\partial\Lambda}}=Z_{t}^{\chi_{\psi_{\partial\Lambda}}}+m_{t}^{\psi_{\partial\Lambda}}=Z_{t}^{\chi_{\psi_{\partial\Lambda}}}+H_{\partial\Lambda}(t)$
for $t\geq 0$. Therefore, from (6.31) we obtain the following decomposition,

$\displaystyle\mathcal{Z}_{t}^{\varphi}=H_{\Lambda}(t)+H_{\partial\Lambda}(t)+\mathcal{Z}_{t}^{\varrho-\phi_{\partial\Lambda}}+\mathcal{Z}_{t}^{\chi},\quad
t\geq 0.$ (6.33)

It suffices to prove the limit theorem for
$\mathcal{Z}_{t}^{\varrho-\phi_{\partial\Lambda}}$ and
$\mathcal{Z}_{t}^{\chi}$. To this end, we invoke Theorem 6.6 for the first
process and Theorem 6.3(ii) for the second as both characteristics are real-
valued as a consequence of Remark 2.15. We begin with
$\mathcal{Z}_{t}^{\varrho-\phi_{\partial\Lambda}}$ and first notice that, in
view of Lemma 5.1 and Remark 2.4, the characteristic
$\varrho-\phi_{\partial\Lambda}$ has càdlàg paths and satisfies (A5) and (A6).
Moreover, using the fact that $\phi_{\partial\Lambda}$ is centered in
combination with (6.29), (6.30) and (2.20) we infer

$\displaystyle m^{\varrho-\phi_{\partial\Lambda}}_{t}$
$\displaystyle=m^{\varrho}_{t}=m^{\varphi-\psi_{\Lambda}-\psi_{\partial\Lambda}}_{t}=m_{t}^{\varphi}-m_{t}^{\psi_{\Lambda}}-m_{t}^{\psi_{\partial\Lambda}}$
(6.34)
$\displaystyle=m_{t}^{\varphi}-\mathds{1}_{[0,\infty)}(t)\big{(}\mathds{E}\big{[}H_{\Lambda}(t)\big{]}+H_{\partial\Lambda}(t)\big{)}=r(t).$

We may thus apply Theorem 6.6 to conclude that

$e^{-\frac{\alpha}{2}t}\mathcal{Z}^{\varrho-\phi_{\partial\Lambda}}_{t}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sigma\sqrt{\tfrac{W}{\beta}}\mathcal{N},$
(6.35)

where

$\displaystyle\sigma^{2}$ $\displaystyle\vcentcolon=\int v(x)e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x}),$ (6.36)

and
$v(t)\vcentcolon=\operatorname{\mathrm{Var}}\big{[}\varrho(t)-\phi_{\partial\Lambda}(t)+r*\xi(t)\big{]}$.
The function $v$ can be further simplified in the following way

$\displaystyle v(t)$
$\displaystyle=\operatorname{\mathrm{Var}}\big{[}\varphi(t)-\psi_{\Lambda}(t)-\psi_{\partial\Lambda}(t)-\phi_{\partial\Lambda}(t)+r*\xi(t)\big{]}$
$\displaystyle=\operatorname{\mathrm{Var}}\bigg{[}\varphi(t)-\sum_{\lambda\in\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}(\phi_{\lambda}(t)+\chi_{\lambda}(t))\mathsf{e}_{1}-\sum_{\lambda\in\partial\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}\phi_{\lambda}(t)\mathsf{e}_{1}+r*\xi(t)\bigg{]}$
$\displaystyle=\operatorname{\mathrm{Var}}\bigg{[}\varphi(t)+\sum_{j=1}^{N}\bigg{(}-\sum_{\lambda\in\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}\mathds{1}_{(-\infty,X_{j})}(t)\exp({\lambda,t-X_{j}},k(\lambda))\mathsf{e}_{1}$
$\displaystyle\hphantom{=\operatorname{\mathrm{Var}}\bigg{[}}-\sum_{\lambda\in\partial\Lambda}\vec{a}_{\lambda}\mathds{1}_{[0,X_{j})}(t)\exp({\lambda,t-X_{j}},k(\lambda))\mathsf{e}_{1}+r(t-X_{j})\bigg{)}\bigg{]}$
$\displaystyle=\operatorname{\mathrm{Var}}\bigg{[}\varphi(t)+\sum_{j=1}^{N}\bigg{(}-\sum_{\lambda\in\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}\exp({\lambda,t-X_{j}},k(\lambda))\mathsf{e}_{1}$
$\displaystyle\hphantom{=\operatorname{\mathrm{Var}}\bigg{[}}-\mathds{1}_{[0,\infty)}(t)\sum_{\lambda\in\partial\Lambda}\vec{a}_{\lambda}^{\mathsf{T}}\exp({\lambda,t-X_{j}},k(\lambda))\mathsf{e}_{1}+m^{\varphi}(t-X_{j})\bigg{)}\bigg{]}.$

This proves the theorem under the assumption that $\rho_{l}=0$ for all $l\geq
0$. Indeed, in this case $\mathcal{Z}_{t}^{\chi}=0$ and
$\sum_{j=1}^{N}\vec{a}_{\lambda}^{\mathsf{T}}\exp({\lambda,t-X_{j}},k(\lambda))\mathsf{e}_{1}$
is a. s. constant for any $\lambda\in\partial\Lambda$, and (2.27) follows.

Theorem 6.6 implies (2.26) if $\sigma^{2}>0$. If, on the other hand,
$\sigma^{2}=0$, then $\mathcal{Z}_{t}^{\varrho-\phi_{\partial\Lambda}}$ is
equals to its expectation a. s. for every $t\in\mathds{G}$, which is $r(t)$ by
(6.34). This proves (i).

It remains to prove the theorem in the case where $\rho_{l}>0$ for some $l\geq
0$. First notice that by (6.35), the already established central limit theorem
for $\mathcal{Z}_{t}^{\varrho-\phi_{\partial\Lambda}}$, we have
$\mathcal{Z}_{t}^{\varrho-\phi_{\partial\Lambda}}=o(t^{\frac{1}{2}}e^{\frac{\alpha}{2}t})$
as $t\to\infty$ in probability. Let $n\in\mathds{N}_{0}$ be maximal with
$\rho_{n}>0$. We show that the characteristic $\chi$ satisfies the assumptions
of Theorem 6.3(ii) with $\theta=2n$. Observe that, for any
$\lambda\in\partial\Lambda$, $l\leq k(\lambda)-1$ and some constant
$C_{\lambda,\vec{a}_{\lambda}}$ depending on $\lambda,l$ and
$\vec{a}_{\lambda}$,

$\displaystyle|R_{\lambda,l}|\leq
C_{\lambda,\vec{a}_{\lambda}}\sum_{j=1}^{N}(1+X_{j}^{k(\lambda)-1})e^{-\frac{\alpha}{2}X_{j}}.$

In view of assumption (A2.9) the random variable $R_{\lambda,l}$ is square
integrable. Setting $R_{\lambda,l}\vcentcolon=0$ for $l\geq k(\lambda)$, we
write

$\displaystyle\chi(t)=\mathds{1}_{[0,\infty)}(t)e^{\frac{\alpha}{2}t}\cdot\sum_{\lambda\in\partial\Lambda}\sum_{l=0}^{n}(R_{\lambda,l}-\mathds{E}[R_{\lambda,l}])t^{l}e^{\mathrm{i}\,\mathrm{Im}(\lambda)t}.$

An application on Lemma 6.7 gives

$\displaystyle\frac{1}{t^{2n+1}}\int_{0}^{t}\mathds{E}[\chi^{2}(x)]e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x})\to\frac{\sum_{\lambda\in\partial\Lambda}\operatorname{\mathrm{Var}}[R_{\lambda,n}]}{2n+1}=\frac{\rho^{2}_{n}}{2n+1}$

and

$\displaystyle\mathds{E}\big{[}|\chi(t)|^{2}\mathds{1}_{\\{|\chi(t)|^{2}>\varepsilon
t^{2n+1}e^{\alpha t}\\}}\big{]}=o(t^{2n}e^{\alpha t})\quad\text{as
}t\to\infty,\;t\in\mathds{G}.$

Finally, by Theorem 6.3(ii),

$\displaystyle\Big{(}\frac{\rho^{2}_{n}t^{2n+1}}{2n+1}e^{\alpha
t}\Big{)}^{\\!-\frac{1}{2}}\mathcal{Z}_{t}^{\chi}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sqrt{\tfrac{W}{\beta}}\mathcal{N},$

which finishes the proof. ∎

######  Remark 6.8.

The proofs of Theorems 2.14 and 6.6 reveal that for any characteristic
$\varphi$ satisfying the assumptions (A4) through (A6), there exists a
decomposition $\varphi=\varphi_{1}+\varphi_{2}+\varphi_{3}$, where each term
also satisfies (A4) through (A6). Furthermore, for the corresponding Crump-
Mode-Jagers processes, it is established that
$\mathcal{Z}^{\varphi_{1}}_{t}=H(t)$ for $t\geq 0$,
$\mathcal{Z}^{\varphi_{2}}_{t}$ is centered, and
$\mathcal{Z}^{\varphi_{3}}_{t}$ is a deterministic function equal to $r(t)$.
In particular, for the characteristic
$\tilde{\varphi}\vcentcolon=\varphi_{1}+\varphi_{3}$, one obtains that for
$t\geq 0$, $\mathcal{Z}^{\tilde{\varphi}}_{t}=H(t)+r(t)$ almost surely,
indicating the lack of Gaussian fluctuations as $t$ goes to infinity.

###### Proof of Corollary 2.19.

First observe that linear combinations as well as the translations
$\varphi(\cdot)\mapsto\varphi(\cdot-s)$ preserve the conditions (A4), (A5) and
(A6). Moreover, for the characteristic $\psi(t)\vcentcolon=\varphi(t-s)$ the
mean function $m^{\psi}_{t}$ has expansion (2.20) with coefficients given by
vectors
$\big{(}{\exp(\lambda,-s,k(\lambda))}^{\mathsf{T}}\vec{a}_{\lambda}\big{)}$.
According to the Cramér–Wold device the convergence in distribution of

$t^{-\frac{d}{2}}e^{-\frac{\alpha}{2}t}\big{(}\mathcal{Z}^{\varphi}_{t-s_{1}}-H(t-s_{1}),\dots,\mathcal{Z}^{\varphi}_{t-s_{n}}-H(t-s_{n})\big{)}$

is equivalent to the convergence in distribution of

$t^{-\frac{d}{2}}e^{-\frac{\alpha}{2}t}\sum_{j=1}^{n}c_{j}\big{(}\mathcal{Z}^{\varphi}_{t-s_{j}}-H(t-s_{j})\big{)},$

for all choices $c_{1},\dots,c_{n}\in\mathds{R}$, and the latter convergence
follows from Theorem 2.14. The covariance can be obtained by the polarization
identity applied to the variance and the fact that
$m^{\psi}_{t}=m^{\varphi}_{t-s},\ h^{\psi}(t)=h^{\varphi}(t-s)$. ∎

##  7 Asymptotic expansion of the mean

In this section we are concerned with the asymptotic expansion of the mean
$m_{t}^{\varphi}=\mathds{E}[\mathcal{Z}^{\varphi}_{t}]$ of a supercritical
general branching process $(\mathcal{Z}_{t}^{\varphi})_{t\geq 0}$ as
$t\to\infty$. Throughout the section, we assume that (A1) and (A2.8) hold. We
fix some notation throughout the section. By $\theta$ we denote a parameter
from $(0,\frac{\alpha}{2})$ such that

$\displaystyle\mathcal{L}\mu(\theta)<\infty\text{ and }\mathcal{L}\mu(z)\neq
1\quad\text{whenever }\theta\leq\mathrm{Re}(z)<\tfrac{\alpha}{2}.$ (7.1)

Such a $\theta$ exists in all particular cases considered in this section and
may sometimes be enlarged in order to ensure the validity of additional
conditions. We also fix $\gamma>\frac{\alpha}{2}$ such that
$\gamma<\min\\{\mathrm{Re}(\lambda):\lambda\in\Lambda\\}$.

In the non-lattice case we work with the corresponding bilateral Laplace
transforms whereas in the lattice case, we use generating functions.

###  7.1 The lattice case

In the present subsection, we assume that $\mu$ is concentrated on the lattice
$\mathds{Z}$ (and not on a smaller lattice). We set

$\displaystyle\mathcal{G}\mu(z)$
$\displaystyle\vcentcolon=\sum_{k=0}^{\infty}\mu(\\{k\\})z^{k}=\int
z^{x}\,\mu(\mathrm{d}\mathit{x})$ (7.2)

for all $z\in\mathds{C}$ for which the series is absolutely convergent. In
particular, $\mathcal{L}\mu(z)=\mathcal{G}\mu(e^{-z})$. Note that, due to
assumption (A2.8), $\mathcal{G}\mu(e^{-\vartheta})<\infty$ and hence the power
series (7.2) defines a holomorphic function on $\\{|z|<e^{-\vartheta}\\}$.
Further, by slightly increasing the value of $\vartheta$ if necessary, we may
assume without loss of generality that there are only finitely many solutions
of the equation $\mathcal{G}\mu(z)=1$ in the disc $\\{|z|<e^{-\vartheta}\\}$.

######  Lemma 7.1.

Assume that (A1) and (A2.8) hold. Let $\theta\in(\vartheta,\frac{\alpha}{2})$
be such that there are no solutions to $\mathcal{G}\mu(z)=1$ in
$\\{z:e^{-\alpha/2}<|z|\leq e^{-\theta}\\}$. Then there are constants
$b_{\lambda,l}$, $\lambda\in\Lambda_{\geq},\,l=0,\ldots,k(\lambda)-1$ such
that, for any characteristic $\varphi$ with

$\sum_{n\in\mathds{Z}}|\mathds{E}[\varphi(n)]|(e^{-\theta n}+e^{-\alpha
n})<\infty,$ (7.3)

it holds that, for $t\in\mathds{Z}$,

$\displaystyle m^{\varphi}_{t}$
$\displaystyle=\begin{cases}\sum\limits_{\lambda\in\Lambda_{\geq}}\sum\limits_{l=0}^{k(\lambda)-1}b_{\lambda,l}\sum\limits_{n\in\mathds{Z}}\mathds{E}[\varphi(n)](t-n)^{l}e^{\lambda(t-n)}+O(e^{\theta
t})&\text{as }t\to\infty\\\ O(e^{\gamma t})&\text{as }t\to-\infty.\end{cases}$
(7.4)

######  Remark 7.2.

Defining
$\vec{b}_{\lambda}\vcentcolon=(b_{\lambda,l-1})_{l=1,\ldots,k(\lambda)}$, we
may write (7.4) more compactly in the form

$\displaystyle
m_{t}^{\varphi}=\sum_{\lambda\in\Lambda_{\geq}}\mathds{1}_{\\{t\geq 0\text{ or
}\lambda\in\Lambda\\}}\sum_{n\in\mathds{Z}}\mathds{E}[\varphi(n)]{\vec{b}_{\lambda}}^{\mathsf{T}}\exp(\lambda,t-n,k(\lambda))\mathsf{e}_{1}+O(e^{\theta
t}\wedge e^{\gamma t})$

as $t\to\pm\infty$, $t\in\mathds{Z}$.

###### Proof of Lemma 7.1.

For $r>0$, let $B_{r}=\\{|z|<r\\}$ and $\partial B_{r}=\\{|z|=r\\}$. Now fix
$r<e^{-\alpha}$. As $\mathcal{G}(\mu^{*l})=(\mathcal{G}\mu)^{l}$, for any
$l\in\mathds{N}$ and since $\mathcal{G}\mu$ is holomorphic on
$B_{e^{-\vartheta}}$, we infer from Cauchy’s integral formula that

$\displaystyle\mu^{*l}(\\{n\\})=\frac{1}{2\pi\mathrm{i}}\int\limits_{\partial
B_{r}}\frac{(\mathcal{G}\mu)^{l}(z)}{z^{n+1}}\,\mathrm{d}\mathit{z}.$

In particular,

$\displaystyle\mathds{E}[N(\\{n\\})]$
$\displaystyle=\sum_{l=0}^{\infty}\mu^{*l}(\\{n\\})=\sum_{l=0}^{\infty}\frac{1}{2\pi\mathrm{i}}\int\limits_{\partial
B_{r}}\\!\frac{(\mathcal{G}\mu)^{l}(z)}{z^{n+1}}\mathrm{d}\mathit{z}=\frac{1}{2\pi\mathrm{i}}\int\limits_{\partial
B_{r}}\\!\frac{\mathrm{d}\mathit{z}}{(1-\mathcal{G}\mu(z))z^{n+1}}$

where the last equality follows by Fubini’s theorem. For
$\lambda\in\Lambda_{\geq}$, let

$\sum_{j=-k(\lambda)}^{-1}{b_{j}(\lambda)}{(z-e^{-\lambda})^{j}}$

be the principle part of the Laurent expansion of the meromorphic function
$(1-\mathcal{G}\mu(z))^{-1}$ around $e^{-\lambda}$. Then the function

$\displaystyle
H(z)\vcentcolon=\frac{1}{1-\mathcal{G}\mu(z)}-\sum_{\lambda\in\Lambda_{\geq}}\sum_{j=-k(\lambda)}^{-1}{b_{j}(\lambda)}{(z-e^{-\lambda})^{j}}$

is holomorphic on $B_{e^{-\theta}}$. On the other hand, for any
$d\in\mathds{N}_{0}$,

$\displaystyle\frac{1}{2\pi\mathrm{i}}\int\limits_{\partial
B_{r}}\frac{(z-e^{-\lambda})^{-d}}{z^{n+1}}\,\mathrm{d}\mathit{z}=(-e^{\lambda})^{d}e^{\lambda
n}{n+d-1\choose d-1}$

by the residue theorem. Therefore,

$\displaystyle
G(z)\vcentcolon=\sum_{\lambda\in\Lambda_{\geq}}\sum_{j=-k(\lambda)}^{-1}{b_{j}(\lambda)}{(z-e^{-\lambda})^{j}}$

satisfies

$\displaystyle\frac{1}{2\pi\mathrm{i}}\int\limits_{\partial
B_{r}}\frac{G(z)}{z^{n+1}}\,\mathrm{d}\mathit{z}$
$\displaystyle=\sum_{\lambda\in\Lambda_{\geq}}p_{\lambda}(n)e^{\lambda n}$

where $p_{\lambda}$, for $\lambda\in\Lambda_{\geq}$, is a polynomial with
complex coefficients of degree $k(\lambda)-1$. From the analyticity of $H$, we
infer

$\bigg{|}\int\limits_{\partial
B_{e^{-\theta}}}\frac{H(z)}{z^{n+1}}\,\mathrm{d}\mathit{z}\bigg{|}=O(e^{\theta
n})\quad\text{as }n\to\infty,$

which in turn gives

$\displaystyle\mathds{E}[N(\\{n\\})]$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\int\limits_{\partial
B_{r}}\frac{G(z)+H(z)}{z^{n+1}}\,\mathrm{d}\mathit{z}$
$\displaystyle=\sum_{\lambda\in\Lambda_{\geq}}\vec{b}_{\lambda}^{\;\mathsf{T}}\exp(\lambda,n,k(\lambda))\mathsf{e}_{1}+O(e^{\theta
n})\quad\text{as }n\to\infty$

for some
$\vec{b}_{\lambda}=\sum_{l=1}^{k(\lambda)}b_{\lambda,l-1}\mathsf{e}_{l}\in\mathds{R}^{k(\lambda)}$.
In other words, there exists a constant $C>0$ such that, for any
$n\in\mathds{Z}$,

$\displaystyle\bigg{|}\mathds{E}[N(\\{n\\})]-\sum_{\lambda\in\Lambda_{\geq}}\mathds{1}_{\\{n\geq
0\text{ or
}\lambda\in\Lambda\\}}\vec{b}_{\lambda}^{\;\mathsf{T}}\exp(\lambda,n,k(\lambda))\mathsf{e}_{1}\bigg{|}\leq
C(e^{\theta n}\wedge e^{\gamma n}).$ (7.5)

Now we are ready to investigate the asymptotic behavior of $m_{t}^{\varphi}$
as $t\to\pm\infty$, $t\in\mathds{Z}$. Since
$m_{t}^{\varphi}=m_{t}^{\mathds{E}[\varphi]}$, we assume without loss of
generality that $\varphi=f$ is a deterministic function satisfying

$\sum_{n\in\mathds{Z}}|f(n)|(e^{-\theta n}+e^{-\alpha n})<\infty,$

Then, for $t\in\mathds{Z}$, we have
$m_{t}^{f}=\sum_{n\in\mathds{Z}}f(n)\mathds{E}[N(\\{t-n\\})]$. We write

$\displaystyle\bigg{|}m^{f}_{t}-\sum_{\lambda\in\Lambda_{\geq}}\mathds{1}_{\\{t\geq
0\text{ or
}\lambda\in\Lambda\\}}\sum_{n\in\mathds{Z}}f(n)\vec{b}_{\lambda}^{\;\mathsf{T}}\exp(\lambda,t-n,k(\lambda))\mathsf{e}_{1}\bigg{|}$
$\displaystyle\leq\sum_{n\in\mathds{Z}}|f(n)|\bigg{|}\mathds{E}[N(\\{t-n\\})]-\sum_{\lambda\in\Lambda_{\geq}}\mathds{1}_{\\{t\geq
n\text{ or
}\lambda\in\Lambda\\}}\vec{b}_{\lambda}^{\;\mathsf{T}}\exp(\lambda,t-n,k(\lambda))\mathsf{e}_{1}\bigg{|}$
$\displaystyle\hphantom{\leq}+\sum_{n\in\mathds{Z}}|f(n)|\bigg{|}\sum_{\lambda\in\partial\Lambda}|\mathds{1}_{\\{t\geq
0\\}}-\mathds{1}_{\\{t\geq
n\\}}|\vec{b}_{\lambda}^{\;\mathsf{T}}\exp(\lambda,t-n,k(\lambda))\mathsf{e}_{1}\bigg{|}.$
(7.6)

We use (7.5) to estimate the first sum on the right-hand side of (7.6) by

$\displaystyle C\sum_{n\in\mathds{Z}}|f(n)|(e^{\theta(t-n)}\wedge
e^{\gamma(t-n)})$ $\displaystyle\leq
C\bigg{(}\sum_{n\in\mathds{Z}}|f(n)|e^{\theta(t-n)}\bigg{)}\wedge\bigg{(}\sum_{n\in\mathds{Z}}|f(n)|e^{\gamma(t-n)}\bigg{)}$
$\displaystyle\leq(e^{\theta t}\wedge e^{\gamma
t})\,C\sum_{n\in\mathds{Z}}|f(n)|(e^{-\theta n}+e^{-\alpha n}).$

On the other hand, we use (4.17) to conclude that for any
$0<\epsilon<\alpha/2-\theta$ and $\lambda\in\partial\Lambda$ there is a
constant $C_{\epsilon}\geq 0$ such that $\|\exp(\lambda,n,k(\lambda))\|\leq
C_{\epsilon}e^{\frac{\alpha}{2}n+\epsilon|n|}$. Hence the second sum on the
right-hand side of (7.6) is bounded by

$\displaystyle
C_{\epsilon}\sum_{\lambda\in\partial\Lambda}|\vec{b}_{\lambda}|\sum_{n\in\mathds{Z}}|f(n)||\mathds{1}_{\\{t\geq
0\\}}-\mathds{1}_{\\{t\geq n\\}}|e^{\frac{\alpha}{2}(t-n)+\epsilon|t-n|}.$

The latter sum can be estimated as follows

$\displaystyle\sum_{n\in\mathds{Z}}|f(n)||\mathds{1}_{\\{t\geq
0\\}}-\mathds{1}_{\\{t\geq n\\}}|e^{\frac{\alpha}{2}(t-n)+\epsilon|t-n|}$
$\displaystyle=\mathds{1}_{\mathds{N}_{0}}(t)\sum_{n>t}|f(n)|e^{(\frac{\alpha}{2}-\epsilon)(t-n)}+\mathds{1}_{\mathds{Z}\setminus\mathds{N}_{0}}(t)\sum_{n\leq
t}|f(n)|e^{(\frac{\alpha}{2}+\epsilon)(t-n)}$
$\displaystyle\leq\mathds{1}_{\mathds{N}_{0}}(t)\sum_{n\in\mathds{Z}}|f(n)|e^{\theta(t-n)}+\mathds{1}_{\mathds{Z}\setminus\mathds{N}_{0}}(t)\sum_{n\leq
t}|f(n)|e^{\alpha(t-n)}=O(e^{\theta t}\wedge e^{\alpha t}),$

as $t\to\pm\infty$. ∎

###  7.2 The non-lattice case

We again work under the conditions (A1) and (A2.8) as in Section 7.1, but now
we assume that $\mu$ is non-lattice.

Similar to the lattice case, first we study the behavior of
$\mathds{E}[N(t)]$. This was already done in [28, Theorem 3.1] in the special
case where $\mathcal{L}\mu(z)-1$ has only simple roots. However, the proof
given in the cited source can be adapted to the more general setting here. In
order to make this paper self-contained and for the reader’s convenience, we
include the proof.

######  Lemma 7.3.

Suppose that, besides (A1) and (A2.8), the following condition holds:

$\limsup_{\eta\to\infty}|\mathcal{L}\mu(\tfrac{\alpha}{2}-\delta+\mathrm{i}\eta)|<1$
(7.7)

for some $\delta\in(0,\tfrac{\alpha}{2}-\vartheta]$. Then $\Lambda_{\geq}$ is
finite. In fact, the function $\mathcal{L}\mu$ takes the value $1$ only at
finitely many points in the strip
$\tfrac{\alpha}{2}-\delta<\mathrm{Re}(z)<\alpha$. Then, for any root
$\lambda\in\Lambda_{\geq}$ of multiplicity $k(\lambda)\in\mathds{N}$, there
exist constants $c_{\lambda,l}$, $l=0,\ldots,k(\lambda)-1$ such that, for any
$\theta\in(\tfrac{\alpha}{2}-\delta,\frac{\alpha}{2})$ satisfying (7.1) it
holds that

$\displaystyle\mathds{E}[N(t)]=\sum_{\lambda\in\Lambda_{\geq}}e^{\lambda
t}\sum_{l=0}^{k(\lambda)-1}c_{\lambda,l}{t^{l}}+O(e^{\theta t})\quad\text{as
}t\to\infty.$ (7.8)

######  Remark 7.4.

Note that (7.8) can be rewritten in the form

$\mathds{E}[N(t)]=\sum_{\lambda\in\Lambda_{\geq}}\vec{c}_{\lambda}^{\
\mathsf{T}}\exp(\lambda,t,k(\lambda))\mathsf{e}_{1}+O(e^{\theta t})$

as $t\to\infty$ with
$\vec{c}_{\lambda}\vcentcolon=\sum_{l=1}^{k(\lambda)}c_{\lambda,l-1}\mathsf{e}_{l}$.

######  Remark 7.5.

Suppose that (A1) and (A2.8) hold and that the intensity measure $\mu$ has a
density with respect to the Lebesgue measure. Then one can check using the
Riemann-Lebesgue lemma that (7.7) holds for any
$\delta\in(0,\tfrac{\alpha}{2}-\vartheta]$. Hence, in this case, Lemma 7.3
applies.

###### Proof of Lemma 7.3.

First, condition (7.7) implies that

$\limsup_{\eta\to\infty}|\mathcal{L}\mu(\theta+\mathrm{i}\eta)|<1$

for all $\theta\geq\tfrac{\alpha}{2}-\delta$ and that there are only finitely
many roots of the equation $\mathcal{L}\mu(z)=1$ in the strip
$\tfrac{\alpha}{2}-\delta\leq\mathrm{Re}(z)<\alpha$, see Lemmas 2.1 and 2.3 in
[28] (note that although the setup in [28] is slightly different, the proofs
carry over without changes).

Now let $f=\mathds{1}_{[0,\infty)}$ and recall that
$N(t)=\mathcal{Z}_{t}^{f}$, hence $V(t)\vcentcolon=\mathds{E}[N(t)]=m_{t}^{f}$
for $t\in\mathds{R}$. In analogy to the derivation of [28, Eq. (3.11)], we use
the recursive structure of $\mathcal{Z}_{t}^{f}$ to obtain a renewal equation
for $V(t)$ as follows. We start with

$\mathcal{Z}_{t}^{f}=f(t)+\sum_{j=1}^{N}\mathcal{Z}_{j,t-X_{j}}^{f}$

where $\mathcal{Z}^{f}_{1,t},\mathcal{Z}^{f}_{2,t}\ldots$ are independent
copies of $\mathcal{Z}^{f}_{t}$. Taking expectations, then conditioning with
respect to $\xi$, the reproduction point process of the ancestor, we infer

$m_{t}^{f}=f(t)+\int
m_{t-x}^{f}\,\mu(\mathrm{d}\mathit{x})=f(t)+\mu*m_{t}^{f},\quad
t\in\mathds{R}.$ (7.9)

Our subsequent proof relies on a smoothing technique. So let
$\rho\vcentcolon=\mathds{1}_{[0,1]}$. For any $\varepsilon>0$, we set

$\textstyle\rho_{\varepsilon}(t)\vcentcolon=\frac{1}{\varepsilon}\rho(\frac{t}{\varepsilon})=\frac{1}{\varepsilon}\mathds{1}_{[0,\varepsilon]}(t),\quad
t\in\mathds{R}.$

Then for $f_{\varepsilon}\vcentcolon=f*\rho_{\varepsilon}$ (Lebesgue
convolution), we have

$f_{\varepsilon}(t)\leq f(t)\leq f_{\varepsilon}(t+\varepsilon)$

for all $t\in\mathds{R}$, which in turn gives

$m^{f_{\varepsilon}}_{t}\leq m^{f}_{t}\leq
m^{f_{\varepsilon}}_{t+\varepsilon}.$ (7.10)

Also, one can check that $t\mapsto m^{f_{\varepsilon}}_{t}$ is a continuous
function. First, we find the asymptotic expansion of this function and then,
we let $\varepsilon$ tend to $0$ in a controlled way while letting
$t\to\infty$ to deduce the asymptotic behavior of $V(t)=m_{t}^{f}$ from that
of $m^{f_{\varepsilon}}_{t}$. From the renewal equation (7.9) we conclude that
for $\mathrm{Re}(z)>\alpha$ it holds

$\mathcal{L}m^{f_{\varepsilon}}(z)=\mathcal{L}(\rho_{\varepsilon}*m_{t}^{f})(z)=\mathcal{L}{f_{\varepsilon}}(z)+\mathcal{L}\mu(z)\mathcal{L}m^{f_{\varepsilon}}(z),$

hence,

$\mathcal{L}m^{f_{\varepsilon}}(z)=\frac{\mathcal{L}{f_{\varepsilon}}(z)}{1-\mathcal{L}\mu(z)}\quad\text{for
}\mathrm{Re}(z)>\alpha.$

The function

$\frac{\mathcal{L}{f_{\varepsilon}}(z)}{1-\mathcal{L}\mu(z)}=\frac{\mathcal{L}{\rho_{\varepsilon}}(z)\mathcal{L}f(z)}{1-\mathcal{L}\mu(z)}=\frac{1-e^{-\varepsilon
z}}{\varepsilon z^{2}(1-\mathcal{L}\mu(z))}$

defines a meromorphic extension of $\mathcal{L}m^{f_{\varepsilon}}$ on
$\mathrm{Re}(z)>\vartheta$. This function decays as $|\mathrm{Im}(z)|^{-2}$ as
$\mathrm{Im}(z)\to\pm\infty$ and $\mathrm{Re}(z)$ is constant, hence, it is
integrable along vertical lines. Thus, for any $\tau>\alpha$, the Laplace
inversion formula (see, for instance, [45, Theorem 7.3 on p. 66]) gives

$\displaystyle
m^{f_{\varepsilon}}_{t}=\frac{m^{f_{\varepsilon}}_{t+}+m^{f_{\varepsilon}}_{t-}}{2}=\frac{1}{2\pi\mathrm{i}}\int\limits_{\tau-\mathrm{i}\infty}^{\tau+\mathrm{i}\infty}e^{tz}\mathcal{L}m^{f_{\varepsilon}}(z)\,\mathrm{d}\mathit{z},\quad
t>0.$

To simplify notation, we assume without loss of generality that
$\vartheta=\frac{\alpha}{2}-\delta$ and that $\mathcal{L}\mu$ is holomorphic
on a neighborhood of $\mathrm{Re}(z)\geq\vartheta$. Then, for large enough
$R$, an application of the residue theorem gives

$\displaystyle\int\limits_{\tau-\mathrm{i}R}^{\tau+\mathrm{i}R}e^{tz}\mathcal{L}m^{f_{\varepsilon}}(z)\,\mathrm{d}\mathit{z}=2\pi\mathrm{i}\sum_{\lambda\in\Lambda_{\geq}}\operatorname{\mathrm{Res}}_{z=\lambda}\big{(}e^{tz}\mathcal{L}m^{f_{\varepsilon}}(z)\big{)}+\int\limits_{\vartheta-\mathrm{i}R}^{\vartheta+\mathrm{i}R}e^{tz}\mathcal{L}m^{f_{\varepsilon}}(z)\,\mathrm{d}\mathit{z}$
$\displaystyle+\int\limits_{\vartheta+\mathrm{i}R}^{\tau+\mathrm{i}R}e^{tz}\mathcal{L}m^{f_{\varepsilon}}(z)\,\mathrm{d}\mathit{z}-\int\limits_{\vartheta-\mathrm{i}R}^{\tau-\mathrm{i}R}e^{tz}\mathcal{L}m^{f_{\varepsilon}}(z)\,\mathrm{d}\mathit{z}.$

Here,

$\displaystyle\bigg{|}\int\limits_{\vartheta+\mathrm{i}R}^{\tau+\mathrm{i}R}e^{tz}\mathcal{L}m^{f_{\varepsilon}}(z)\,\mathrm{d}\mathit{z}\bigg{|}$
$\displaystyle\leq
e^{t\tau}\int\limits_{\vartheta}^{\tau}\Big{|}\frac{1-e^{-\varepsilon(x+\mathrm{i}R)}}{\varepsilon(x+\mathrm{i}R)^{2}(1-\mathcal{L}\mu(x+\mathrm{i}R))}\Big{|}\,\mathrm{d}\mathit{x}$
$\displaystyle\leq
Ce^{t\tau}\int\limits_{\vartheta}^{\tau}\Big{|}\frac{1}{\varepsilon(x+\mathrm{i}R)^{2}}\Big{|}\,\mathrm{d}\mathit{x}\underset{R\to\infty}{\longrightarrow}0$

for some constant $C$ that depends only on $\mu$. Here we used the fact that,
by Lemma 2.1 of [28], $\mathcal{L}\mu(x+\mathrm{i}R)$ for $x\geq\vartheta$ and
$R\geq R_{0}$ is uniformly bounded away from $1$ for some sufficiently large
$R_{0}>0$. The same bound holds for the second horizontal integral. Therefore,
by letting $R$ tend to infinity we conclude

$\displaystyle
m^{f_{\varepsilon}}_{t}=\sum_{\lambda\in\Lambda_{\geq}}\operatorname{\mathrm{Res}}_{z=\lambda}\big{(}e^{tz}\mathcal{L}m^{f_{\varepsilon}}(z)\big{)}+\frac{1}{2\pi\mathrm{i}}\int\limits_{\vartheta-\mathrm{i}\infty}^{\vartheta+\mathrm{i}\infty}e^{tz}\mathcal{L}m^{f_{\varepsilon}}(z)\,\mathrm{d}\mathit{z}.$
(7.11)

Next, denoting by $\\{b_{j}(\lambda)\\}_{j\in\mathds{Z}}$ the coefficients in
the Laurent expansion of the function $(1-\mathcal{L}\mu(z))^{-1}$ at
$z=\lambda\in\Lambda_{\geq}$ (hence, in particular, $b_{j}(\lambda)=0$ for
$j<-k(\lambda)$), we have

$\displaystyle\operatorname{\mathrm{Res}}_{z=\lambda}\big{(}e^{tz}\mathcal{L}m^{f_{\varepsilon}}(z)\big{)}$
$\displaystyle=\operatorname{\mathrm{Res}}_{z=\lambda}\Big{(}e^{tz}\frac{\mathcal{L}{f_{\varepsilon}}(z)}{1-\mathcal{L}\mu(z)}\Big{)}$
$\displaystyle=e^{\lambda t}\\!\\!\\!\sum_{\begin{subarray}{c}n,l\geq 0\\\
n+l<k(\lambda)\end{subarray}}\\!\\!\frac{t^{l}}{l!}\frac{(\mathcal{L}f_{\varepsilon})^{(n)}(\lambda)}{n!}b_{-1-n-l}(\lambda)$
$\displaystyle=e^{\lambda t}\\!\\!\\!\sum_{\begin{subarray}{c}n,l\geq 0\\\
n+l<k(\lambda)\end{subarray}}\\!\\!\frac{t^{l}}{l!}\frac{\int(-x)^{n}f_{\varepsilon}(x)e^{-\lambda
x}\,\mathrm{d}\mathit{x}}{n!}b_{-1-n-l}(\lambda)$ $\displaystyle=e^{\lambda
t}\\!\\!\\!\sum_{\begin{subarray}{c}n,l\geq 0\\\
n+l<k(\lambda)\end{subarray}}\\!\\!\frac{t^{l}}{l!}\frac{\int(-x)^{n}f(x)e^{-\lambda
x}\mathrm{d}\mathit{x}}{n!}b_{-1-n-l}(\lambda)+\varepsilon O(e^{\alpha t})$
$\displaystyle=\vcentcolon e^{\lambda t}\sum_{0\leq
l<k(\lambda)}c_{\lambda,l}t^{l}+\varepsilon O(e^{\alpha t}),$ (7.12)

where the implicit constant depends only on $\lambda$, not on $\varepsilon$.
It remains to estimate the second term in (7.11). For
$\varepsilon\leq\vartheta^{-1}$, using that $|1-e^{-z}|\leq|z|\wedge 2$ for
$\mathrm{Re}(z)\geq 0$, we infer

$\displaystyle\bigg{|}\int\limits_{\vartheta-\mathrm{i}\infty}^{\vartheta+\mathrm{i}\infty}e^{tz}\mathcal{L}m^{f_{\varepsilon}}(z)\,\mathrm{d}\mathit{z}\bigg{|}$
$\displaystyle\leq e^{\vartheta
t}\int\limits_{\vartheta-\mathrm{i}\infty}^{\vartheta+\mathrm{i}\infty}\bigg{|}\frac{1-e^{-\varepsilon
z}}{\varepsilon z^{2}(1-\mathcal{L}\mu(z))}\bigg{|}\,|\mathrm{d}\mathit{z}|$
$\displaystyle\leq Ce^{\vartheta
t}\int\limits_{\vartheta-\mathrm{i}\infty}^{\vartheta+\mathrm{i}\infty}\frac{|\varepsilon
z|^{-1}\wedge 1}{|z|}\,|\mathrm{d}\mathit{z}|$ $\displaystyle=Ce^{\vartheta
t}\int\limits_{\varepsilon\vartheta-\mathrm{i}\infty}^{\varepsilon\vartheta+\mathrm{i}\infty}(|z|^{-1}\wedge|z|^{-2})\,|\mathrm{d}\mathit{z}|$
$\displaystyle\leq Ce^{\vartheta t}\int\limits_{-\infty}^{\infty}x^{-1}\wedge
x^{-2}\wedge(\varepsilon\vartheta)^{-1}\,\mathrm{d}\mathit{x}$
$\displaystyle\leq C^{\prime}e^{\vartheta t}(|\log\varepsilon|+1)$ (7.13)

for some constant $C^{\prime}$ that depends neither on $t$ nor on
$\varepsilon$. Using (7.12) with $t+\varepsilon$ instead of $t$, we conclude
that

$\displaystyle\big{|}\operatorname{\mathrm{Res}}_{z=\lambda}\big{(}e^{(t+\varepsilon)z}\mathcal{L}m^{f_{\varepsilon}}(z)\big{)}-\operatorname{\mathrm{Res}}_{z=\lambda}\big{(}e^{tz}\mathcal{L}m^{f_{\varepsilon}}(z)\big{)}\big{|}=\varepsilon
O(e^{\alpha t}),$

where we used $k(\lambda)=1$ for $\lambda=\alpha$, and thereupon, by (7.11)
and (7.13),

$m^{f_{\varepsilon}}_{t+\varepsilon}-m^{f_{\varepsilon}}_{t}\leq O(\varepsilon
e^{\alpha t}+|\log\varepsilon|e^{\vartheta t}).$

Setting now $\varepsilon\vcentcolon=e^{-\alpha t}$, by (7.10), we infer

$m_{t}^{f}=\sum_{\lambda\in\Lambda_{\geq}}e^{\lambda t}\sum_{0\leq l\leq
k(\lambda)-1}c_{\lambda,l}t^{l}+O(te^{\vartheta t}),$

which completes the proof of the lemma. ∎

Now we are ready to provide the asymptotic expansion for the expectation
function of a general branching process counted with a random characteristic
$\varphi$.

######  Lemma 7.6.

Suppose that, besides (A1) and (A2.8), condition (7.7) holds. Then
$\Lambda_{\geq}$ is finite and there are constants $b_{\lambda,l}$,
$\lambda\in\Lambda_{\geq}$, $0\leq l<k(\lambda)$ such that, for any
$\vartheta<\theta<\frac{\alpha}{2}$ fulfilling (7.1) and any random
characteristic $\varphi$ satisfying (2.19), we have

$\displaystyle m^{\varphi}_{t}$
$\displaystyle=\begin{cases}\sum\limits_{\lambda\in\Lambda_{\geq}}\sum\limits_{l=0}^{k(\lambda)-1}b_{\lambda,l}\int(t-x)^{l}e^{\lambda(t-x)}\mathds{E}[\varphi(x)]\,\mathrm{d}\mathit{x}+O(e^{\theta
t})&\text{as }t\to\infty\\\ O(e^{\gamma t})&\text{as }t\to-\infty.\end{cases}$
(7.14)

######  Remark 7.7.

If we set
$\vec{b}_{\lambda}\vcentcolon=(b_{\lambda,l})_{l=0,\ldots,k(\lambda)-1}$, then
formula (7.14) can be written in the form

$m_{t}^{\varphi}=\sum_{\lambda\in\Lambda_{\geq}}\mathds{1}_{\\{t\geq 0\text{
or
}\lambda\in\Lambda\\}}\int\vec{b}_{\lambda}^{\;\mathsf{T}}\exp(\lambda,t-x,k(\lambda))\mathsf{e}_{1}\mathds{E}[\varphi(x)]\,\mathrm{d}\mathit{x}+O(e^{\theta
t}\wedge e^{\gamma t})$

as $t\to\pm\infty$.

###### Proof.

Without loss of generality we assume that the characteristic $\varphi=f$ is a
deterministic function. By Lemma 7.3 there exist constants $c_{\lambda,l}$,
$l=0,\ldots,k(\lambda)-1$, $\theta\in(\vartheta,\frac{\alpha}{2})$ and a
constant $C$ such that, for any $t\in\mathds{R}$,

$\bigg{|}\mathds{E}[N(t)]-\sum_{\lambda\in\Lambda_{\geq}}\mathds{1}_{\\{t\geq
0\text{ or }\lambda\in\Lambda\\}}\vec{c}_{\lambda}^{\
\mathsf{T}}\exp(\lambda,t,k(\lambda))\mathsf{e}_{1}\bigg{|}\leq C(e^{\theta
t}\wedge e^{\gamma t})$ (7.15)

and hence for the characteristic
$f(t)=\mathds{1}_{[x,\infty)}(t)=\mathds{1}_{[0,\infty)}(t-x)$, we find

$\bigg{|}m^{f}_{t}-\sum_{\lambda\in\Lambda_{\geq}}\mathds{1}_{\\{t-x\geq
0\text{ or }\lambda\in\Lambda\\}}\vec{c}_{\lambda}^{\
\mathsf{T}}\exp(\lambda,t-x,k(\lambda))\mathsf{e}_{1}\bigg{|}\leq
C(e^{\theta(t-x)}\wedge e^{\gamma(t-x)}).$

Suppose now that $f\geq 0$ is a càdlàg, nondecreasing function with

$\displaystyle\int f(x)(e^{-\alpha x}+e^{-\vartheta
x})\,\mathrm{d}\mathit{x}<\infty.$ (7.16)

Then $f$ is the measure-generating function of a locally finite measure $\nu$
on the Borel sets of $\mathds{R}$, namely, for any $y\in\mathds{R}$,

$f(y)=\nu((-\infty,y])=\int\mathds{1}_{[x,\infty)}(y)\,\nu(\mathrm{d}\mathit{x}).$

For any $t\in\mathds{R}$, by an application of Fubini’s theorem, we infer

$m^{f}_{t}=\int\mathds{E}[N(t-x)]\,\nu(\mathrm{d}\mathit{x}).$

By (4.16) we have
$\frac{\mathrm{d}}{\mathrm{d}\mathit{x}}\exp(\lambda,x,k)=J_{\lambda,k}\exp(\lambda,x,k)$.
We show that (7.14) holds with
$\vec{b}_{\lambda}\vcentcolon=J_{\lambda,k}^{\mathsf{T}}\vec{c}_{\lambda}$,
$\lambda\in\Lambda_{\geq}$. To this end, first notice that, as $f$ fulfills
(7.16), another application of Fubini’s theorem yields

$\displaystyle\int\exp(\lambda,-x,k)\,\nu(\mathrm{d}\mathit{x})=\int
J_{\lambda,k}\exp(\lambda,-x,k)f(x)\,\mathrm{d}\mathit{x}.$

We now write

$\displaystyle\bigg{|}m^{f}_{t}-\int\Big{(}\sum_{\lambda\in\Lambda_{\geq}}\mathds{1}_{\\{t\geq
0\text{ or
}\lambda\in\Lambda\\}}\vec{b}_{\lambda}^{\;\mathsf{T}}\exp(\lambda,t-x,k(\lambda))f(x)\mathsf{e}_{1}\Big{)}\,\mathrm{d}\mathit{x}\bigg{|}$
$\displaystyle~{}=\bigg{|}m^{f}_{t}-\int\Big{(}\sum_{\lambda\in\Lambda_{\geq}}\mathds{1}_{\\{t\geq
0\text{ or }\lambda\in\Lambda\\}}\vec{c}_{\lambda}^{\
\mathsf{T}}J_{\lambda,k(\lambda)}\exp(\lambda,t-x,k(\lambda))f(x)\mathsf{e}_{1}\Big{)}\,\mathrm{d}\mathit{x}\bigg{|}$
$\displaystyle~{}=\bigg{|}\int\Big{(}\mathds{E}[N(t-x)]-\sum_{\lambda\in\Lambda_{\geq}}\mathds{1}_{\\{t\geq
0\text{ or }\lambda\in\Lambda\\}}\vec{c}_{\lambda}^{\
\mathsf{T}}\exp(\lambda,t-x,k(\lambda))\mathsf{e}_{1}\Big{)}\,\nu(\mathrm{d}\mathit{x})\bigg{|}$
$\displaystyle~{}\leq\int\bigg{|}\mathds{E}[N(t-x)]-\sum_{\lambda\in\Lambda_{\geq}}\mathds{1}_{\\{t-x\geq
0\text{ or }\lambda\in\Lambda\\}}\vec{c}_{\lambda}^{\
\mathsf{T}}\exp(\lambda,t-x,k(\lambda))\mathsf{e}_{1}\bigg{|}\,\nu(\mathrm{d}\mathit{x})$
$\displaystyle\hphantom{~{}\leq}~{}+\sum_{\lambda\in\partial\Lambda}\int\big{|}\mathds{1}_{\\{t\geq
0\\}}-\mathds{1}_{\\{t-x\geq 0\\}}\big{|}\big{|}\vec{c}_{\lambda}^{\
\mathsf{T}}\exp(\lambda,t-x,k(\lambda))\mathsf{e}_{1}\big{|}\,\nu(\mathrm{d}\mathit{x}).$

For the first term, by (7.15), we have

$\displaystyle\int\Big{|}\mathds{E}[N(t-x)]-\sum_{\lambda\in\Lambda_{\geq}}\mathds{1}_{\\{t-x\geq
0\text{ or }\lambda\in\Lambda\\}}\vec{c}_{\lambda}^{\
\mathsf{T}}\exp(\lambda,t-x,k(\lambda))\mathsf{e}_{1}\Big{|}\,\nu(\mathrm{d}\mathit{x})$
$\displaystyle~{}\leq C\int(e^{\theta(t-x)}\wedge
e^{\gamma(t-x)})\,\nu(\mathrm{d}\mathit{x})$ $\displaystyle~{}\leq
C\bigg{(}\int e^{\theta(t-x)}\nu(\mathrm{d}\mathit{x})\wedge\int
e^{\gamma(t-x)}\,\nu(\mathrm{d}\mathit{x})\bigg{)}$ $\displaystyle~{}\leq
C(e^{\theta t}\wedge e^{\gamma t})\bigg{(}\theta\int f(x)e^{-\theta
x}\,\mathrm{d}\mathit{x}+\gamma\int f(x)e^{-\gamma
x}\,\mathrm{d}\mathit{x}\bigg{)}.$

Next, for $\lambda\in\partial\Lambda$, we estimate

$\displaystyle\int\big{|}\mathds{1}_{\\{t\geq 0\\}}-\mathds{1}_{\\{t-x\geq
0\\}}\big{|}\big{|}\vec{c}_{\lambda}^{\
\mathsf{T}}\exp(\lambda,t-x,k(\lambda))\mathsf{e}_{1}\big{|}\,\nu(\mathrm{d}\mathit{x})$
$\displaystyle~{}=\mathds{1}_{(-\infty,0)}(t)\int\limits_{(-\infty,t]}\big{|}\vec{c}_{\lambda}^{\
\mathsf{T}}\exp(\lambda,t-x,k(\lambda))\mathsf{e}_{1}\big{|}\,\nu(\mathrm{d}\mathit{x})$
$\displaystyle\hphantom{~{}=}~{}+\mathds{1}_{[0,\infty)}(t)\int\limits_{(t,\infty)}\big{|}\vec{c}_{\lambda}^{\
\mathsf{T}}\exp(\lambda,t-x,k(\lambda))\mathsf{e}_{1}\big{|}\,\nu(\mathrm{d}\mathit{x})$
$\displaystyle~{}\leq C\mathds{1}_{(-\infty,0)}(t)\int
e^{\alpha(t-x)}\,\nu(\mathrm{d}\mathit{x})+C\mathds{1}_{[0,\infty)}(t)\int
e^{\vartheta(t-x)}\,\nu(\mathrm{d}\mathit{x})$ $\displaystyle~{}\leq
C^{\prime}\big{(}e^{\alpha t}\wedge e^{\vartheta t}\big{)},$

where we have used (4.17) in the penultimate step. This completes the proof of
the theorem for non-decreasing $f\geq 0$.

Now let $f$ be an arbitrary càdlàg function satisfying the integrability
condition (2.19) (with $f$ in place of $\mathds{E}[\varphi]$). Define

$f_{\pm}(x)\vcentcolon=\sup\bigg{\\{}\sum_{j=1}^{n}(f(x_{j})-f(x_{j-1}))^{\pm}:-\infty<x_{0}<\ldots<x_{n}\leq
x,\ n\in\mathds{N}\bigg{\\}}$

for $x\in\mathds{R}$. Clearly, $f_{+},f_{-}:\mathds{R}\to\mathds{R}$ are
nondecreasing with $f_{\pm}\geq 0$. It is known that $f=f_{+}-f_{-}$. (This is
the Jordan decomposition of $f$ on $\mathds{R}$.) It is further known that
$f_{+}$ and $f_{-}$ are càdlàg since $f$ is. Further,
$\mathrm{V}\\!f(x)=f_{+}(x)+f_{-}(x)$ and hence (2.19) implies that both,
$f_{+}$ and $f_{-}$ satisfy (7.16). The previous part of the proof thereby
applies to $f_{+}$ and $f_{-}$ and, by linearity, extends to $f$. ∎

######  Remark 7.8.

In the situations of Lemmas 7.1 and 7.6, $m^{\varphi}_{t}$ has a
representation as in (2.20). Indeed, in both cases, $m_{t}^{\varphi}$ can be
written as

$\displaystyle m_{t}^{\varphi}$
$\displaystyle=\mathds{1}_{[0,\infty)}(t)\sum_{\lambda\in\Lambda_{\geq}}\int_{\mathds{G}}\vec{b}_{\lambda}^{\;\mathsf{T}}\exp(\lambda,t-x,k(\lambda))\mathsf{e}_{1}\mathds{E}[\varphi](x)\,\ell(\mathrm{d}\mathit{x})+O(e^{\theta
t}\wedge e^{\gamma t})$
$\displaystyle=\mathds{1}_{[0,\infty)}(t)\sum_{\lambda\in\Lambda_{\geq}}\vec{a}_{\lambda}^{\;\mathsf{T}}\exp(\lambda,t,k(\lambda))\mathsf{e}_{1}+O(e^{\theta
t}\wedge e^{\gamma t}),$

where
$\vec{a}_{\lambda}\vcentcolon=\int_{\mathds{G}}\exp(\lambda,-x,k(\lambda))^{\mathsf{T}}\vec{b}_{\lambda}\mathds{E}[\varphi](x)\,\ell(\mathrm{d}\mathit{x})$.
Consequently (cf. (6.28), (2.22) and (6.30)), we have

$\displaystyle H_{\Lambda}(t)$
$\displaystyle=\sum_{\lambda\in\Lambda}\vec{a}_{\lambda}^{\;\mathsf{T}}\exp(\lambda,t,k(\lambda))W(\lambda,k(\lambda))\mathsf{e}_{1}$
$\displaystyle=\sum_{\lambda\in\Lambda}\vec{b}_{\lambda}^{\;\mathsf{T}}\int_{\mathds{G}}\exp(\lambda,t-x,k(\lambda))W(\lambda,k(\lambda))\mathsf{e}_{1}\mathds{E}[\varphi](x)\ell(\mathrm{d}\mathit{x})$
$\displaystyle=\sum_{\lambda\in\Lambda}\sum_{l=0}^{k(\lambda)-1}b_{\lambda,l}\int_{\mathds{G}}e^{\lambda(t-x)}\sum_{j=0}^{l}{l\choose
j}(t-x)^{l-j}W^{(j)}(\lambda)\mathds{E}[\varphi](x)\ell(\mathrm{d}\mathit{x})$
$\displaystyle=\sum_{\lambda\in\Lambda}e^{\lambda
t}\sum_{l=0}^{k(\lambda)-1}b_{\lambda,l}\sum_{j=0}^{l}{l\choose
j}W^{(j)}(\lambda)\int_{\mathds{G}}(t-x)^{l-j}\mathds{E}[\varphi](x)e^{-\lambda
x}\ell(\mathrm{d}\mathit{x})$

and, similarly,

$\displaystyle H_{\partial\Lambda}(t)$
$\displaystyle=\sum_{\lambda\in\partial\Lambda}\vec{a}_{\lambda}^{\;\mathsf{T}}\exp(\lambda,t,k(\lambda))\mathsf{e}_{1}$
$\displaystyle=\sum_{\lambda\in\partial\Lambda}\vec{b}_{\lambda}^{\;\mathsf{T}}\int_{\mathds{G}}\exp(\lambda,t-x,k(\lambda))\mathsf{e}_{1}\mathds{E}[\varphi](x)\,\ell(\mathrm{d}\mathit{x})$
$\displaystyle=\sum_{\lambda\in\partial\Lambda}e^{\lambda
t}\sum_{l=0}^{k(\lambda)-1}b_{\lambda,l}\int_{\mathds{G}}(t-x)^{l}\mathds{E}[\varphi](x)e^{-\lambda
x}\,\ell(\mathrm{d}\mathit{x}).$

###  7.3  Proofs of Theorems 2.7, 2.8 and 2.9

###### Proof of Theorem 2.7.

Theorem 2.7 is a consequence of Theorem 2.14. To see this, we first check that
the assumptions of Theorem 2.7 imply those of Theorem 2.14. In a second step,
we show how the conclusion of Theorem 2.7 follows from that of Theorem 2.14.

So suppose the assumptions of Theorem 2.7, in particular, (A1) through (A2.9),
hold. Also (A7) holds because $\Lambda_{\geq}=\\{\alpha\\}$ and, by
assumption, there are no roots of the equation $\mathcal{L}\mu(z)=1$ in the
strip $\vartheta<\mathrm{Re}(z)<\alpha$. Let $\varphi$ be a characteristic
satisfying (A5), (A6) and (2.19). Notice that (2.19) implies (A4). To see
this, decompose $\mathds{E}[\varphi]=f_{1}-f_{2}$ for two non-negative, non-
decreasing functions $f_{1}$ and $f_{2}$ such that
$\mathrm{V}\\!\mathds{E}[\varphi](x)=f_{1}(x)+f_{2}(x)$. Then

$\displaystyle\int\Big{(}f_{i}(x)e^{-\alpha
x}\Big{)}^{\\!*}\,\mathrm{d}\mathit{x}\leq\int
f_{i}(x+1)e^{-\alpha(x-1)}\,\mathrm{d}\mathit{x}\leq
e^{2\alpha}\int\mathrm{V}\\!\mathds{E}[\varphi](x)e^{-\alpha
x}\,\mathrm{d}\mathit{x}<\infty.$

We conclude from Proposition 2.5 that both $x\mapsto f_{1}(x)e^{-\alpha x}$
and $x\mapsto f_{2}(x)e^{-\alpha x}$ are directly Riemann integrable and hence
so is their difference, i.e., (A4) is satisfied. We have to check that
$m_{t}^{\varphi}$ has a representation of the form (2.20). This follows from
Lemma 7.6 once we have shown that (7.7) holds (cf. Remark 7.8). However, the
latter follows from the existence of a Lebesgue density for $\mu$ and the
Riemann-Lebesgue lemma (cf. Remark 7.5).

Since $\Lambda_{\geq}=\\{\alpha\\}$ and
$\mathcal{L}\mu^{\prime}(\alpha)=-\beta\not=0$, i.e., $k(\alpha)=1$, Lemma 7.6
gives

$\displaystyle m^{\varphi}_{t}$ $\displaystyle=\begin{cases}e^{\alpha
t}b_{\alpha,0}\int\mathds{E}[\varphi(x)]e^{-\alpha
x}\,\mathrm{d}\mathit{x}+O(e^{\theta t})&\text{as }t\to\infty\\\ O(e^{\gamma
t})&\text{as }t\to-\infty\end{cases}$

for some constant $b_{\lambda,0}\in\mathds{R}$. From Nerman’s law of large
numbers [25, Theorem 6.1, see the proof on p. 246] or alternatively
Proposition 7.9, we know that $b_{\alpha,0}=\beta^{-1}$, so

$m^{\varphi}_{t}=\mathds{1}_{[0,\infty)}(t)\beta^{-1}\mathcal{L}(\mathds{E}[\varphi])(\alpha)e^{\alpha
t}+O(e^{\theta t}\wedge e^{\gamma t})$

as $t\to\pm\infty$, i.e., $m^{\varphi}_{t}$ indeed has a representation of the
form (2.20). Hence, Theorem 2.14 applies. Let
$a_{\alpha}\vcentcolon=\beta^{-1}\mathcal{L}(\mathds{E}[\varphi])(\alpha)$.
Then $H(t)=e^{\alpha t}a_{\alpha}W$. Notice that $n=-1$ and thus $\rho_{-1}=0$
in Theorem 2.14. With $\sigma\geq 0$ as in Theorem 2.14, we now infer that in
both cases, $\sigma=0$ and $\sigma>0$, that

$\displaystyle
e^{-\frac{\alpha}{2}t}\big{(}\mathcal{Z}_{t}^{\varphi}-a_{\alpha}e^{\alpha
t}W\big{)}=e^{-\frac{\alpha}{2}t}\big{(}\mathcal{Z}_{t}^{\varphi}-H(t)\big{)}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sigma\sqrt{\tfrac{W}{\beta}}\mathcal{N}\quad\text{as
}t\to\infty$

for a standard normal random variable $\mathcal{N}$ independent of $W$. ∎

###### Proof of Theorem 2.8.

As before we shall prove that the assumptions of Theorem 2.14 are fulfilled.
As in the proof of Theorem 2.7, we conclude that (2.19) implies (A4). Lemma
7.6 yields (A7) and that $m_{t}^{\varphi}$ has an expansion of the form (2.20)
with $r(t)=O(e^{\theta t}\wedge e^{\gamma t})$ as $t\to\infty$, see also
Remark 7.8. The assertion now follows from Theorem 2.14. Note, that
$H_{\partial\Lambda}(t)=O(t^{k-1}e^{\frac{\alpha}{2}t})=o(t^{k-\frac{1}{2}}e^{\frac{\alpha}{2}t})$
by the definition of $H_{\partial\Lambda}$ (cf. (2.22)) and hence this term
can be neglected in the limit theorem. ∎

###### Proof of Theorem 2.9.

Again, we first check that the assumptions of Theorem 2.9 imply those of
Theorem 2.14. So suppose that the assumptions of Theorem 2.9, in particular,
(A1) through (A2.9), hold. Regarding (A7), i.e., the finiteness of
$\Lambda_{\geq}$, notice that since $\mu$ is lattice, we have

$\Lambda_{\geq}=\\{\lambda\in\mathds{C}:\tfrac{\alpha}{2}\leq\mathrm{Re}(\lambda)\leq\alpha,\;-\pi<\mathrm{Im}(\lambda)\leq\pi,\;\mathcal{L}\mu(\lambda)=1\\}.$

By (A2.8), $\mathcal{L}\mu$ is holomorphic on
$\\{z\in\mathds{C}:\mathrm{Re}(z)>\vartheta\\}$ and non-constant by (A1).
Hence, $\mathcal{L}\mu(z)=1$ can hold for only finitely many $z$ in the
compact box $\tfrac{\alpha}{2}\leq\mathrm{Re}(z)\leq\alpha$,
$|\mathrm{Im}(z)|\leq\pi$, that is, $\Lambda_{\geq}$ is finite. Now suppose
that $\varphi$ is a characteristic satisfying
$\sum_{n\in\mathds{Z}}|\mathds{E}[\varphi(n)]|(e^{-\theta n}+e^{-\alpha
n})<\infty$ and (A5). Then $\varphi$ satisfies (A4) and (A6). Moreover, by
Lemma 7.1 and Remark 7.8 we conclude (2.20) hence, Theorem 2.14 applies. As
before, the assertion follows from Theorem 2.14 in combination with Remark
7.8. By the same argument as in the proof of Theorem 2.8, the term
$H_{\partial\Lambda}$ can be neglected. ∎

###  7.4 Determining the coefficients

Note that although the constants $\vec{c}_{\lambda}$ and $\vec{b}_{\lambda}$
are not given explicitly it is not hard to follow the proofs and provide
explicit expressions for them. However, even for small $k(\lambda)$, this
approach may lead to tedious calculations, not to mention that there can also
be several roots in the relevant strip. It seems that a more efficient way to
determine the constants $b^{\lambda}_{l}$ is an application of Lemma 7.1 or
7.6, respectively, to a characteristic for which we explicitly know the
asymptotic behavior of the expectation of the associated general branching
process.

######  Proposition 7.9.

Let $\lambda\in\Lambda_{\geq}$ be a root of $\mathcal{L}\mu(z)=1$ of
multiplicity $k$ (in the lattice case we also assume that
$\mathrm{Im}(\lambda)\in(-\pi,\pi]$). Then the vector $\vec{b}_{\lambda}$
appearing in Lemma 7.1 or 7.6, respectively, is given by
$M^{\lambda}\vec{b}_{\lambda}=\mathsf{e}_{k}$, where $M^{\lambda}$ is the
$k\times k$ upper triangular matrix such that for $j\geq i$

$(M^{\lambda})_{i,j}\vcentcolon=-\frac{(j-1)!(k-1)!}{(i-1)!(j-i+k)!}(\mathcal{L}\mu)^{(k+j-i)}(\lambda)$

in the non-lattice case. In contrast, in the lattice case,

$(M^{\lambda})_{i,j}\vcentcolon=\binom{j-1}{i-1}P_{k,j-i}\Big{(}\frac{\mathrm{d}}{\mathrm{d}\mathit{z}}\Big{)}\mathcal{L}\mu(z)_{|z=\lambda},$

where the polynomials $P_{k,l}$ are given by

$P_{k,l}(y)\vcentcolon=(-1)^{l}\sum_{m=1}^{k}\binom{k-1}{m-1}y^{k-m}\frac{B_{l+m}(-y)-B_{l+m}(0)}{l+m},$

and $B_{n}$ is the $n^{\mathrm{th}}$ Bernoulli polynomial. In particular, in
both cases, as $(\mathcal{L}\mu)^{(j)}(\lambda)=0$ for $j=1,\ldots,k-1$,

$\det(M^{\lambda})=\left(\frac{-\mathcal{L}\mu^{(k)}(\lambda)}{k}\right)^{k}\neq
0$

and the matrix $M^{\lambda}$ is invertible.

For the proof, we need a lemma which essentially is Jensen’s inequality for
the total variation operator $\mathrm{V}$ defined in (2.18).

######  Lemma 7.10.

Let $\phi=(\phi(t))_{t\in\mathds{R}}$ be a stochastic process with càdlàg
paths such that $\phi(t)\in L^{1}$ for every $t\in\mathds{R}$ and
$t\mapsto\mathds{E}[\phi](t)$ is again càdlàg. Then, finite or not,

$\displaystyle\mathrm{V}\\!\mathds{E}[\phi](t)\leq\mathds{E}[\mathrm{V}\\!\phi(t)]$
(7.17)

for every $t\in\mathds{R}$.

###### Proof.

First notice that $\mathrm{V}\\!\phi(t)$ is a random variable. Indeed, since
the paths of $\phi$ are cádlág, we have

$\displaystyle\mathrm{V}\\!\phi(t)=\sup\bigg{\\{}\sum_{j=1}^{n}|\phi(t_{j})-\phi(t_{j-1})|:-\infty<t_{0}<\ldots<t_{n}\leq
t,\;t_{0},\ldots,t_{n}\in\mathds{Q},\ n\in\mathds{N}\bigg{\\}},$

which is measurable as the supremum of a family of random variables indexed by
a countable set. Since $\mathds{E}[\phi]$ is also càdlàg, we infer

$\displaystyle\mathrm{V}\\!\mathds{E}[\phi](t)$
$\displaystyle=\sup\bigg{\\{}\sum_{j=1}^{n}|\mathds{E}[\phi(t_{j})-\phi(t_{j-1})]|:t_{0}<\ldots<t_{n}\leq
t,\,t_{0},\ldots,t_{n}\in\mathds{Q},\,n\in\mathds{N}\bigg{\\}}$
$\displaystyle\leq\sup\bigg{\\{}\mathds{E}\bigg{[}\sum_{j=1}^{n}|\phi(t_{j})-\phi(t_{j-1})|\bigg{]}:t_{0}<\ldots<t_{n}\leq
t,\,t_{0},\ldots,t_{n}\in\mathds{Q},\,n\in\mathds{N}\bigg{\\}}$
$\displaystyle\leq\mathds{E}\bigg{[}\sup\bigg{\\{}\sum_{j=1}^{n}|\phi(t_{j})-\phi(t_{j-1})|:t_{0}<\ldots<t_{n}\leq
t,\,t_{0},\ldots,t_{n}\in\mathds{Q},\,n\in\mathds{N}\bigg{\\}}\bigg{]}$
$\displaystyle=\mathds{E}[\mathrm{V}\\!\phi](t).$

∎

###### Proof of Proposition 7.9.

For $\lambda\in\Lambda_{\geq}$, consider the characteristic

$\displaystyle\phi(t)$ $\displaystyle=\mathsf{e}_{k}^{\
\mathsf{T}}\mathds{E}[\phi_{\lambda}(t)]\mathsf{e}_{1}=\mathds{1}_{[0,\infty)}(t)\mathds{E}\bigg{[}\sum_{j=1}^{N}\mathds{1}_{{\\{X_{j}>t\\}}}(t-X_{j})^{k-1}e^{\lambda(t-X_{j})}\bigg{]}$
$\displaystyle=\mathds{1}_{[0,\infty)}(t)\mathds{E}\bigg{[}\sum_{j=1}^{N}\mathsf{e}_{k}^{\
\mathsf{T}}\mathds{1}_{[0,X_{j})}(t)\exp(\lambda,t-X_{j},k)\mathsf{e}_{1}\bigg{]}$
$\displaystyle=\mathds{1}_{[0,\infty)}(t)\mathds{E}\bigg{[}\sum_{j=1}^{N}f_{X_{j}}(t)\bigg{]},\quad
t\in\mathds{R}$

where, for any $x\geq 0$,

$f_{x}(t)\vcentcolon=\mathsf{e}_{k}^{\
\mathsf{T}}\mathds{1}_{[0,x)}(t)\exp(\lambda,t-x,k)\mathsf{e}_{1}.$

By Lemma 5.2, the following holds for all $t\geq 0$

$\displaystyle
m^{\phi}_{t}=\mathds{E}\big{[}Z^{\phi}_{t}\big{]}=\mathds{E}\Big{[}\mathsf{e}_{k}^{\
\mathsf{T}}\exp({\lambda},t,k)W_{t}(\lambda,k)\mathsf{e}_{1}\Big{]}=\mathsf{e}_{k}^{\
\mathsf{T}}\exp({\lambda},t,k)\mathsf{e}_{1}.$

We now aim to apply either Lemma 7.6 in the lattice case or Lemma 7.1 in the
non-lattice case to the characteristic $\phi$. To do this, we need to verify
that $\phi$ satisfies either (2.19) or (7.3) respectively.

Using equations (4.16) and (4.17), we get

$\displaystyle\mathrm{V}\\!f_{x}(t)\leq|f_{x}(0)|\mathds{1}_{[0,\infty)}(t)+\int_{0}^{t\wedge
x}\big{|}\frac{\mathrm{d}}{\mathrm{d}\mathit{s}}f_{x}(s)\big{|}\,\mathrm{d}\mathit{s}\leq
C\times\begin{cases}0&\quad\text{for }t<0,\\\
e^{(\mathrm{Re}(\lambda)-\delta)(t-x)}&\quad\text{for }0\leq t<x,\\\
1&\quad\text{for }x\leq t\end{cases}$

for some $\delta\in(0,\mathrm{Re}(\lambda)-\vartheta)$ and a constant $C$ that
does not depend on $x\geq 0$. Thus,

$\displaystyle\int\mathrm{V}\\!f_{x}(t)\big{(}e^{-\vartheta t}+e^{-\alpha
t}\big{)}\,\mathrm{d}\mathit{t}\leq 2\int\mathrm{V}\\!f_{x}(t)e^{-\vartheta
t}\,\mathrm{d}\mathit{t}\leq C^{\prime}e^{-\vartheta x}$

for some other constant $C^{\prime}$ that also does not depend on $x$. Now
notice that we may apply (7.17) to $\phi$ since $\phi$ and $\mathds{E}[\phi]$
have càdlàg paths. Only the latter requires a proof. By the definition of
$\phi$ and (4.17),

$\displaystyle|\phi(t)|\leq C\mathds{1}_{[0,\infty)}(t)e^{\vartheta
t}\sum_{j=1}^{N}e^{-\vartheta X_{j}}$

for any $t\in\mathds{R}$. Hence, (A2.8) and the dominated convergence theorem
imply that $\mathds{E}[\phi]$ is càdlàg. Using the subadditivity of
$\mathrm{V}$, (7.17) and assumption (A2.8), we get

$\displaystyle\int\mathrm{V}\\!\phi(t)\big{(}e^{-\vartheta t}+e^{-\alpha
t}\big{)}\,\mathrm{d}\mathit{t}$
$\displaystyle=\int_{0}^{\infty}\mathrm{V}\\!\bigg{(}\mathds{E}\bigg{[}\sum_{j=0}^{N}f_{X_{j}}\bigg{]}\bigg{)}(t)\big{(}e^{-\vartheta
t}\\!+\\!e^{-\alpha t}\big{)}\,\mathrm{d}\mathit{t}$
$\displaystyle\leq\mathds{E}\bigg{[}\sum_{j=0}^{N}\int\mathrm{V}\\!f_{X_{j}}(t)\big{(}e^{-\vartheta
t}\\!+\\!e^{-\alpha t}\big{)}\,\mathrm{d}\mathit{t}\bigg{]}\leq
C^{\prime}\mathds{E}\bigg{[}\sum_{j=0}^{N}e^{-\vartheta
X_{j}}\bigg{]}<\infty.$

A similar argument gives (7.3) in the lattice case. An application of either
Lemma 7.6 or Lemma 7.1 (as mentioned in Remark 7.8), respectively, to the
characteristic $\phi$, yields

$\displaystyle m^{\phi}_{t}$
$\displaystyle=\sum_{\lambda^{\prime}\in\Lambda}\vec{b}_{\lambda^{\prime}}^{\;\mathsf{T}}\int\exp(\lambda^{\prime},t-x,k)\phi(x)\,\ell(\mathrm{d}\mathit{x})\mathsf{e}_{1}+O(e^{\theta
t})$
$\displaystyle=\sum_{\lambda^{\prime}\in\Lambda}\bigg{(}\bigg{(}\int\exp(\lambda^{\prime},-x,k)\phi(x)\,\ell(\mathrm{d}\mathit{x})\bigg{)}^{\mathsf{T}}\,\vec{b}_{\lambda^{\prime}}\bigg{)}^{\mathsf{T}}\exp(\lambda^{\prime},t,k)\mathsf{e}_{1}+O(e^{\theta
t}),$

as $t$ goes to infinity. Next take the difference of the two asymptotic
expansions for $m_{t}^{\phi}$ and then apply Lemma .1 to infer

$\bigg{(}\int\exp(\lambda,-x,k)\phi(x)\,\ell(\mathrm{d}\mathit{x})\bigg{)}^{\mathsf{T}}\,\vec{b}_{\lambda}=\mathsf{e}_{k}.$

It now suffices to evaluate the coefficients of the matrix

$\displaystyle
M^{\lambda}\vcentcolon=\int\exp(\lambda,-x,k)\phi(x)\,\ell(\mathrm{d}\mathit{x}).$

First, we deal with the non-lattice case. For this purpose, recalling basic
properties of the beta function $B$, we infer

$\displaystyle\int(-x)^{l}e^{-\lambda x}\phi(x)\,\mathrm{d}\mathit{x}$
$\displaystyle=(-1)^{l}\mathds{E}\bigg{[}\sum_{j=1}^{N}\int_{0}^{X_{j}}x^{l}(x-X_{j})^{k-1}e^{-\lambda
X_{j}}\,\mathrm{d}\mathit{x}\bigg{]}$
$\displaystyle=(-1)^{l}\iint_{0}^{y}x^{l}(x-y)^{k-1}\,\mathrm{d}\mathit{x}e^{-\lambda
y}\,\mu(\mathrm{d}\mathit{y})$ $\displaystyle=(-1)^{l+k-1}\int
B(l+1,k)y^{k+l}e^{-\lambda y}\,\mu(\mathrm{d}\mathit{y})$
$\displaystyle=-\frac{l!(k-1)!}{(l+k)!}(\mathcal{L}\mu)^{(k+l)}(\lambda),$

and therefore

$\displaystyle(M^{\lambda})_{i,j}$
$\displaystyle=\bigg{(}\int\exp(\lambda,-x,k)\phi(x)\mathrm{d}\mathit{x}\bigg{)}_{j,i}$
$\displaystyle=-\mathds{1}_{\\{j\geq
i\\}}\mathcal{L}\mu^{(k+j-i)}(\lambda)\cdot{j-1\choose
i-1}\frac{(j-i)!(k-1)!}{(j-i+k)!}$ $\displaystyle=-\mathds{1}_{\\{j\geq
i\\}}\frac{(j-1)!(k-1)!}{(i-1)!(j-i+k)!}(\mathcal{L}\mu)^{(k+j-i)}(\lambda).$

In the non-lattice case, invoking Faulhaber’s formula, we have

$\displaystyle\sum_{x\in\mathds{Z}}(-x)^{l}e^{-\lambda x}\phi(x)$
$\displaystyle=(-1)^{l}\mathds{E}\bigg{[}\sum_{j=1}^{N}\sum_{0\leq
x<X_{j}}x^{l}(x-X_{j})^{k-1}e^{-\lambda X_{j}}\bigg{]}$
$\displaystyle=(-1)^{l}\int\sum_{0\leq x<y}x^{l}(x-y)^{k-1}e^{-\lambda
y}\,\mu(\mathrm{d}\mathit{y})$
$\displaystyle=(-1)^{l}\int\sum_{m=1}^{k}{k-1\choose m-1}(-y)^{k-m}\sum_{0\leq
x<y}x^{l+m-1}e^{-\lambda y}\,\mu(\mathrm{d}\mathit{y})$
$\displaystyle=(-1)^{l}\int\sum_{m=1}^{k}{k-1\choose
m-1}(-y)^{k-m}\frac{B_{l+m}(y)-B_{l+m}(0)}{l+m}e^{-\lambda
y}\,\mu(\mathrm{d}\mathit{y})$
$\displaystyle=P_{k,l}\Big{(}\frac{\mathrm{d}}{\mathrm{d}\mathit{z}}\Big{)}\mathcal{L}\mu(z)\Big{|}_{z=\lambda},$

which gives

$\displaystyle(M^{\lambda})_{i,j}$
$\displaystyle=\Big{(}\int\exp(\lambda,-x,k)\phi(x)\mathrm{d}\mathit{x}\Big{)}_{j,i}$
$\displaystyle=\mathds{1}_{\\{j\geq i\\}}{j-1\choose
i-1}P_{k,j-i}\Big{(}\frac{\mathrm{d}}{\mathrm{d}\mathit{z}}\Big{)}\mathcal{L}\mu(z)\Big{|}_{z=\lambda}.$

∎

##  8 Discussion and open problems

In this section we formulate several open problems which are closely related
to the present framework.

######  Open problem 1.

Prove a corresponding limit theorem for the multitype CMJ process.

######  Open problem 2.

Provide functional versions of the theorems proved in this paper.

A drawback of our method in the non-lattice case is that, in order to find the
asymptotic of the mean $m^{\varphi}_{t}$ we need to assume that the measure
$\mu$ is absolutely continuous with respect to Lebesgue measure or that at
least (7.7) holds.

######  Open problem 3.

In the non-lattice case, work out a proof that does not require (7.7).

The Gaussian fluctuations appearing in our theorems are caused by the
finiteness of the second moment (2.9). In the case where the condition is not
satisfied one still may ask for a generalization.

######  Open problem 4.

Prove a version of the limit theorems with a stable limit.

One of the basic ingredients of the CMJ process is the underlying branching
random walk $(S(u))_{u\in\mathcal{T}}$ with positive increments. However, the
process $\mathcal{Z}^{\varphi}$ can also be defined for a branching random
walk with two-sided increments and suitable $\varphi$.

######  Open problem 5.

Investigate the behavior of $\mathcal{Z}^{\varphi}_{t}$ for a branching random
walk $(S(u))_{u\in\mathcal{T}}$ with two-sided increments.

A central limit theorem is usually complemented by a law of the iterated
logarithm (see, for instance, [22] for the central limit theorem and the law
of the iterated logarithm for Nerman’s martingale). This motivates the
following problem.

######  Open problem 6.

Prove a corresponding law of the iterated logarithm for
$\mathcal{Z}^{\varphi}_{t}$.

The martingale limits $W^{(j)}(\lambda)$ play an important role in the
asymptotic behavior of the general branching process. It is important to
obtain more information about their distributions. In particular, the
following problem seems to be quite relevant.

######  Open problem 7.

Derive the first-order asymptotic behavior of the tail probabilities
$\mathds{P}(|W^{(j)}(\lambda)|>t)$ as $t\to\infty$ for
$j=0,\ldots,k(\lambda)-1$.

We also believe that the approach developed in the present paper might be
useful for settling the following.

######  Open problem 8.

Find large deviation estimates for $\mathcal{Z}^{\varphi}_{t}$.

#### Acknowledgements

The authors thank two anonymous referees for exceptionally careful and
constructive reports whose consideration led to an improvement of the paper.
A. I. was supported by the Grant of the Ministry of Education and Science of
Ukraine for perspective development of a scientific direction “Mathematical
sciences and natural sciences” at Taras Shevchenko National University of
Kyiv. K. K. is also affiliated with the University of Wrocław, Poland. M. M.
was supported by DFG grant ME3625/4-1.

## An auxiliary result

######  Lemma .1.

Let $r\in\mathds{N}$, $k_{j}\in\mathds{N}_{0}$, $j=1,\ldots,r$,
$z_{1},\dots,z_{r}$ be distinct complex numbers with $|z_{j}|\geq\rho$ and
$b_{j,l}$, $j=1,\ldots,r$, $l=0,\ldots,k_{j}$ complex numbers. Then

$\displaystyle\sum_{j=1}^{r}\sum_{l=0}^{k_{j}}b_{j,l}n^{l}z_{j}^{n}=o(\rho^{n})\quad\text{as
}n\to\infty,\ n\in\mathds{N}$ (.1)

implies $b_{j,l}=0$ for all $j=1,\ldots,r$, $l=0,\ldots,k_{j}$.

###### Proof.

We use induction on $K\vcentcolon=k_{1}+\dots+k_{r}$. Suppose that $K=0$ and
denote by $V(z_{1},\ldots,z_{r})\vcentcolon=(z_{j}^{m-1})_{m,j=1,\ldots,r}$
the Vandermonde matrix associated with $z_{1},\ldots,z_{r}$. Then, putting
$b_{j}\vcentcolon=b_{j,0}$, each component of the vector

$V(z_{1},\ldots,z_{r})\begin{pmatrix}b_{1}z_{1}^{n}\\\ \vdots\\\
b_{r}z_{r}^{n}\end{pmatrix}=\bigg{(}\sum_{j=1}^{r}b_{j}z_{j}^{m-1}z_{j}^{n}\bigg{)}_{m=1,\ldots,r}$

is $o(\rho^{n})$ as $n\to\infty$. Since $z_{1},\ldots,z_{r}$ are distinct,
$\det V(z_{1},\ldots,z_{r})\not=0$, hence we may multiply the last displayed
equation by the inverse of $V(z_{1},\ldots,z_{r})$ from the left and conclude
that $b_{j}z_{j}^{n}=o(\rho^{n})$ as $n\to\infty$, which, in turn, gives
$b_{j}=0$ for $j=1,\ldots,r$. For the induction step, we assume that the
induction hypothesis holds whenever $k_{1}+\ldots+k_{r}\leq K$. If now
$k_{1}+\ldots+k_{r}=K+1$, then there is exists some $j_{0}\in\\{1,\ldots,r\\}$
with $k_{j_{0}}>0$. We define a linear operator $L$ by

$Lf(n)\vcentcolon=f(n)-z_{j_{0}}f(n-1)$

for any $f:\mathds{Z}\mapsto\mathds{C}$. If $f(n)=o(\rho^{n})$, then so is
$Lf(n)$. Moreover, if $f(n)=p(n)z^{n}$ for some polynomial $p$, then
$Lf(n)=\tilde{p}(n)z^{n}$ for another polynomial $\tilde{p}$ with
$\deg\tilde{p}\leq\deg p$ and if $f(n)=n^{l}z_{j_{0}}^{n}$, then
$Lf(n)=(ln^{l-1}+p(n))z_{j_{0}}^{n}$ for some polynomial $p$ with $\deg p\leq
l-2$. Applying $L$ to both sides of the relation (.1) we infer

$\displaystyle\sum_{j=1}^{r}\sum_{l=0}^{\tilde{k}_{j}}\tilde{b}_{j,l}n^{l}z_{j}^{n}=o(\rho^{n})\quad\text{as
}n\to\infty,\ n\in\mathds{N}$

for some $\tilde{k}_{j}\leq k_{j}$, $\tilde{k}_{j_{0}}=k_{j_{0}}-1$ and
$\tilde{b}_{j_{0},k_{j_{0}}-1}=k_{j_{0}}b_{j_{0},k_{j_{0}}}$. The induction
hypothesis gives that $\tilde{b}_{j_{0},k_{j_{0}}-1}=0$ which implies that
$b_{j_{0},k_{j_{0}}}=0$ as well. This allows us to replace $k_{j_{0}}$ by
$k_{j_{0}}-1$ in (.1). The claim now follows by induction. ∎

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