, and

# Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes

Alexander<EMAIL_ADDRESS>[ Konrad Koleskolabel=e2 [
<EMAIL_ADDRESS>Matthias Meinerslabel=e3 [
<EMAIL_ADDRESS>Faculty of Computer Science and
Cybernetics, Taras Shevchenko National University of Kyiv, Ukraine,
Mathematisches Institut, University of Gießen, Germany,

###### Abstract

Consider a supercritical Crump–Mode–Jagers process
$(\mathcal{Z}_{t}^{\varphi})_{t\geq 0}$ counted with a random characteristic
$\varphi$. Nerman’s celebrated law of large numbers [Z. Wahrsch. Verw. Gebiete
57, 365–395, 1981] states that, under some mild assumptions, $e^{-\alpha
t}\mathcal{Z}_{t}^{\varphi}$ converges almost surely as $t\to\infty$ to $aW$.
Here, $\alpha>0$ is the Malthusian parameter, $a$ is a constant and $W$ is the
limit of Nerman’s martingale, which is positive on the survival event. In this
general situation, under additional (second moment) assumptions, we prove a
central limit theorem for $(\mathcal{Z}_{t}^{\varphi})_{t\geq 0}$. More
precisely, we show that there exist a constant $k\in\mathds{N}_{0}$ and a
function $H(t)$, a finite random linear combination of functions of the form
$t^{j}e^{\lambda t}$ with $\alpha/2\leq\mathrm{Re}(\lambda)<\alpha$, such that
$(\mathcal{Z}_{t}^{\varphi}-ae^{\alpha t}W-H(t))/\sqrt{t^{k}e^{\alpha t}}$
converges in distribution to a normal random variable with random variance.
This result unifies and extends various central limit theorem-type results for
specific branching processes.

60J80,

60F05,

60G44,

Asymptotic fluctuations,

central limit theorem,

general branching process (counted with a random characteristic),

Laplace transform,

Nerman’s martingale,

###### keywords:

[class=MSC]

###### keywords:

###### Contents

  1. 1 Introduction
  2. 2 Setup, preliminaries and main results
    1. 2.1 The general branching process counted with a random characteristic
    2. 2.2 Assumptions
    3. 2.3 Main results
  3. 3 Applications
    1. 3.1 The Galton-Watson process
    2. 3.2 Nerman’s martingales
    3. 3.3 Epidemic models
    4. 3.4  Crump-Mode-Jagers processes with homogeneous Poisson offspring process and general lifetimes
    5. 3.5 The conservative fragmentation model
  4. 4 Preliminaries for the proofs of the main results
    1. 4.1 Change of measure and the connection to renewal theory
    2. 4.2 The expectation of the general branching process
    3. 4.3 Matrix notation
  5. 5 Nerman’s martingales as general branching processes
    1. 5.1 Nerman’s martingales with complex parameters
    2. 5.2 Convergence of Nerman’s martingales
    3. 5.3 Limits of Nerman’s martingales as general branching processes
  6. 6 Proofs of the main results
    1. 6.1 Centered characteristics
    2. 6.2 Deterministic characteristics
    3. 6.3 Slowly growing mean process with signed characteristics
    4. 6.4 Proof of Theorem 2.14
  7. 7 Asymptotic expansion of the mean
    1. 7.1 The lattice case
    2. 7.2 The non-lattice case
    3. 7.3  Proofs of Theorems 2.7, 2.8 and 2.9
    4. 7.4 Determining the coefficients
  8. 8 Discussion and open problems

##  1 Introduction

A general (Crump-Mode-Jagers) branching process starts at time $0$ with a
single individual, the ancestor, who is alive in the random time interval
$[0,\zeta)$ for a random variable $\zeta$, the life span, taking values in
$[0,\infty]$. The ancestor produces offspring born at the points of a
reproduction point process $\xi$ on $[0,\infty)$. No particular assumption
about the dependence structure between $\xi$ and $\zeta$ is made. For each
individual $u$ that is ever born there is an independent copy
$(\xi_{u},\zeta_{u})$ of the pair $(\xi,\zeta)$ that determines the birth
times of the individual’s offspring relative to $u$’s time of birth and its
life span.

The general branching process encompasses e.g. the Bienaymé-Galton-Watson
process, the Yule process, the continuous-time Markov branching process, the
Sevastyanov process, and the Bellman-Harris process. We refer to [23] for a
more detailed account of the history of the general branching process and its
predecessors.

The general branching process counted with a random characteristic at time $t$
is the sum over all individuals ever born where the contribution of each
individual to the sum is determined by some random characteristic that may
take into account all aspects of the individual’s life such as its age at time
$t$, its life span, etc. This formulation makes it possible to treat at one go
various quantities of interest derived from the general branching process such
as the number of births up to time $t$, the number of individuals alive at
time $t$, the number of individuals alive at time $t$ which are younger than a
given threshold $a>0$, etc. A formal description of the model will be given in
Section 2.

General branching processes serve as models of biological populations such as
humans, cells or plants [16, 23, 31, 38], as models for tumor growth [14, 31],
but also for neutron chain reactions [3] or fragmentation [28] (after a change
of time) to name but a few. The general branching process is also an important
tool within related fields of applied probability or theoretical computer
science. In fact, its applications in these fields are numerous and any
attempt to give a complete survey here is hopeless. We confine ourselves to
mentioning its successful application in the study of asymptotic properties of
random graph growth models driven by preferential attachment dynamics [6, 9,
34, 42] and particularly random tree growth models [13, 20, 21, 29, 32, 39].
It is also used as an approximation for epidemic models [10, 44] and as a
model of the initial phase of epidemics such as SARS, Ebola and SARS-CoV-2 [8,
11, 12], during which the disease spreads exponentially fast but the impact of
population structure and preventive measures is still small [44].

The laws of large numbers of the supercritical general branching process
counted with a random characteristic are due to Nerman [35, 36] in the single-
type, non-lattice case, that is, when the reproduction point process is not
concentrated on any lattice. There were earlier results for special cases, but
here we refrain from sketching the history and instead refer to the
introduction of [36]. The lattice version of Nerman’s law of large numbers was
proved by Gatzouras [15].

In view of the relevance of the general branching process in applications and
the fact that the laws of large numbers date back as far as 1981, it is
remarkable, and rather surprising, that there is no comprehensive central
limit theorem for the general process counted with a random characteristic in
the literature. However, there are results for related models indicating the
intricate nature of the fluctuations that can occur. For the multi-type
continuous-time Markov branching process with finite type space where
individuals give birth only at the time of their death Athreya [4, 5] proved a
central limit theorem and Janson [26] proved a functional central limit
theorem. Asmussen and Hering [3, Section VIII.3] provide results for the
asymptotic fluctuations of multi-type Markov branching processes with rather
general type space. In principle, these results contain the single-type case
of the general branching process since such a process can be seen as a Markov
process in which the type of an individual at time $t$ is its entire life
history up to time $t$. However, this type space is large, and the assumptions
of [3] are typically not satisfied except in special cases such as the case of
the Galton-Watson process. Recently, Janson studied the asymptotic
fluctuations of single-type supercritical general branching processes in the
lattice case [27]. For the non-lattice case, there is a second-order result by
Janson and Neininger [28] for Kolmogorov’s conservative fragmentation model
that may be translated into the language of general branching processes. It
gives a central limit theorem for the number of individuals born up to time
$t$, but it requires that the offspring variable
$N\vcentcolon=\xi([0,\infty))$ be bounded and the additional assumption that
$\int e^{-x}\,\xi(\mathrm{d}\mathit{x})=1$ almost surely, a rather restrictive
assumption in the context of general branching processes. Another related
result is the central limit theorem for Nerman’s martingale [22].

In the present paper, we close the gap in the literature and present a central
limit theorem for the general branching process counted with a random
characteristic. Our main result, Theorem 2.14, contains and extends all
results for single-type processes summarized above. A non-exhaustive list of
applications given in Section 3 contains Galton-Watson processes, Nerman’s
martingale and its complex-valued counterparts, epidemic models, Crump-Mode-
Jagers processes with homogeneous Poisson offspring process and general
lifetimes, and conservative fragmentation models.

### Organization of the paper

The paper is organized as follows. In Section 2 we formally introduce the
general branching process counted with a random characteristic. We further
state and discuss the assumptions we are working with. In Section 2.3, we
state the main result, Theorem 2.14, and its corollaries. We then apply our
general results to some specific models in Section 3. Section 4 contains some
preliminaries for the proofs. Nerman’s martingale and further related
martingales play a crucial role in our theory. All these martingales are
introduced and discussed in Section 5. Section 6 is devoted to proving our
main result Theorem 2.14. Our central limit theorem is based on an asymptotic
expansion of the mean of a general branching process counted with a random
characteristic. Such asymptotic expansions are derived in Section 7. We close
the paper with Section 8, in which possible future research directions are
outlined.

##  2 Setup, preliminaries and main results

We continue with a formal description of the general branching process.

###  2.1 The general branching process counted with a random characteristic

We introduce the general (Crump-Mode-Jagers) branching process following
Jagers [23, 24]. The process starts with a single individual, the ancestor,
born at time $0$. The ancestor produces offspring born at the points of a
reproduction point process $\xi=\sum_{j=1}^{N}\delta_{X_{j}}$ on $[0,\infty)$
where $N=\xi([0,\infty))$ takes values in $\mathds{N}_{0}\cup\\{\infty\\}$
with $\mathds{N}_{0}\vcentcolon=\\{0,1,2,\ldots\\}$ and
$X_{j}\vcentcolon=\inf\\{t\geq 0:\xi([0,t])\geq j\\}$. Here and throughout the
paper, the infimum of the empty set is defined to be $\infty$. The ancestor
has a random lifetime $\zeta$, which may be dependent on $\xi$. Formally,
$\zeta$ is a random variable assuming values in $[0,\infty]$.

Individuals are indexed by
$u\in\mathcal{I}=\bigcup_{n\in\mathds{N}_{0}}\mathds{N}^{n}$ according to
their genealogy. Here, $\mathds{N}=\\{1,2,\ldots\\}$ and
$\mathds{N}^{0}\vcentcolon=\\{\varnothing\\}$ is the singleton set containing
only the empty tuple $\varnothing$. We use the usual Ulam-Harris notation. We
abbreviate a tuple $u=(u_{1},\ldots,u_{n})\in\mathds{N}^{n}$ by $u_{1}\ldots
u_{n}$ and refer to $n$ as the length or generation of $u$; we write $|u|=n$.
In this context, any $u=u_{1}\ldots u_{n}\in\mathcal{I}$ is called (potential)
individual. Its ancestral line is encoded by

$\varnothing\to u_{1}\to u_{1}u_{2}\to\ldots\to u_{1}\ldots u_{n}=u$

where $u_{1}$ is the $u_{1}^{\mathrm{th}}$ child of the ancestor, $u_{1}u_{2}$
the $u_{2}^{\mathrm{th}}$ child of $u_{1}$, etc. If $v=v_{1}\ldots
v_{m}\in\mathcal{I}$, then $uv$ is short for $u_{1}\ldots u_{n}v_{1}\ldots
v_{m}$. For $u\in\mathcal{I}$ and $i\in\mathds{N}$, the individuals $ui$ will
be called children of $u$. Conversely, $u$ will be called parent of $ui$. More
generally, $w$ will be called descendant of $u$ (short: $u\preceq w$) iff
$uv=w$ for some $v\in\mathcal{I}$. Conversely, $u$ will be called an
ancestor/progenitor of $w$. We write $u\prec w$ if $u\preceq w$ and $u\not=w$.
Often, we shall refer to $\mathds{N}^{n}$ as the (potential) $n^{\mathrm{th}}$
generation ($n\in\mathds{N}_{0}$). With these notations, we have

$|u|=n\quad\text{iff}\quad u\in\mathds{N}^{n}\quad\text{iff}\quad u\text{ is
an $n^{\mathrm{th}}$ generation (potential) individual}.$

For $u=u_{1}\ldots u_{n}\in\mathds{N}^{n}$ and $k\in\mathds{N}_{0}$, let
$u|_{k}$ denote the ancestor of $u$ in the $k^{\mathrm{th}}$ generation.
Formally, $u|_{k}$ is the restriction of the vector $u$ to its first $k$
components:

$u|_{k}=\begin{cases}\varnothing&\text{if $k=0$},\\\ u_{1}\ldots
u_{k}&\text{if $1\leq k\leq|u|$},\\\ u&\text{if $k>|u|$}.\end{cases}$ (2.1)

For typographical reasons, we may sometimes write $v|k$ instead of $v|_{k}$.
For $u\in\mathcal{I}$ let $u\mathcal{I}$ denote the subtree of $\mathcal{I}$
emanating from $u$, that is,

$u\mathcal{I}\vcentcolon=\\{uv:\,v\in\mathcal{I}\\}=\\{w\in\mathcal{I}:\,w|_{|u|}=u\\}.$

For each $u\in\mathcal{I}$ there is an independent copy $(\xi_{u},\zeta_{u})$
of the pair $(\xi,\zeta)$ that determines the birth times of $u$’s offspring
relative to its time of birth, and the duration of its life. Quantities
derived from $(\xi_{u},\zeta_{u})$ are indexed by $u$. For instance, $N_{u}$
is the number of offspring of $u$ and $X_{u,k}$ is the difference between the
birth-time of the $k^{\mathrm{th}}$ child of $u$ and $u$ itself, etc. The
birth-times $S(u)$ for $u\in\mathcal{I}$ are defined recursively. We set
$S(\varnothing)\vcentcolon=0$ and, for $n\in\mathds{N}_{0}$,

$S(uj)\vcentcolon=S(u)+X_{u,j}\quad\text{for }u\in\mathds{N}^{n}\text{ and
}j\in\mathds{N}.$

The family tree of all individuals ever born is denoted by
$\mathcal{T}\vcentcolon=\\{u\in\mathcal{I}:S(u)<\infty\\}.$ We call

$\mathcal{S}\vcentcolon=\bigcap_{n\in\mathds{N}}\\{\\#(\mathcal{T}\cap\mathds{N}^{n})\geq
1\\}$

the survival set and its complement
$\mathcal{S}^{\mathsf{c}}=\cup_{n\in\mathds{N}}\\{\\#(\mathcal{T}\cap\mathds{N}^{n})=0\\}$
the extinction set. The time of death of individual $u$ is $S(u)+\zeta_{u}$.
An individual $u$ is alive at time $t\geq 0$ if it is born, but not yet dead
at time $t$, i.e., if

$S(u)\leq t<S(u)+\zeta_{u}.$

We now construct the canonical space for the general branching process. For
$u\in\mathcal{I}$, let $(\Omega_{u},\mathcal{A}_{u},P_{u})$ be a copy of a
given probability space
$(\Omega_{\varnothing},\mathcal{A}_{\varnothing},P_{\varnothing})$, the _life
space_ of the ancestor. An element $\omega\in\Omega_{u}$ is a possible life
career for individual $u$ and any property of interest of $u$ like its mass at
some age or its life span is viewed as a measurable function on the life
space. In particular, $\xi$ and $\zeta$, the reproduction point process and
the life span, are measurable functions defined on
$(\Omega_{\varnothing},\mathcal{A}_{\varnothing})$.

From the life space, we construct the population space:

$\textstyle(\Omega,\mathcal{F},\mathds{P})\vcentcolon=\big{(}\bigtimes_{u\in\mathcal{I}}\Omega_{u},\bigotimes_{u\in\mathcal{I}}\mathcal{A}_{u},\bigotimes_{u\in\mathcal{I}}P_{u}\big{)}.$

For $u\in\mathcal{I}$, we write $\pi_{u}$ for the projection
$\pi_{u}:\bigtimes_{v\in\mathcal{I}}\Omega_{v}\to\Omega_{u}$ and $\theta_{u}$
for the shift
$\theta_{u}((\omega_{v})_{v\in\mathcal{I}})=(\omega_{uv})_{v\in\mathcal{I}}$.
To formally lift an entity $\chi$ defined on the life space, i.e. a function
$\chi$ on $\Omega_{u}$, to the population space, we define
$\chi_{u}\vcentcolon=\chi\circ\pi_{u}$. In particular,
$\xi_{u}=\xi\circ\pi_{u}$ and $\zeta_{u}=\zeta\circ\pi_{u}$. In slight abuse
of notation, if $\chi$ is defined on the life space, when working on the
population space, we write $\chi$ instead of $\chi\circ\pi_{\varnothing}$. For
instance, we sometimes write $\mathds{P}(\zeta\leq t)$ for $P_{u}(\zeta\leq
t)=\mathds{P}(\zeta_{\varnothing}\leq t)$. A technical remark is in order.
Sometimes random variables independent of $\mathcal{F}$ appear. This means
that when required, we work on a suitable extension of the space
$(\Omega,\mathcal{F},\mathds{P})$.

We are interested in the general branching process counted with a random
characteristic. A _random characteristic_ $\varphi$ is a random process on
$(\Omega_{\varnothing},\mathcal{A}_{\varnothing},P_{\varnothing})$ taking
values in the Skorokhod space of right-continuous functions
$f:\mathds{R}\to\mathds{R}^{d}$ with existing left limits at every point in
$\mathds{R}$. Such functions are called càdlàg for short. The characteristic
$\varphi$ may also be viewed as a stochastic process
$\varphi:\Omega_{\varnothing}\times\mathds{R}\to\mathds{R}^{d}$,
$(\omega,t)\mapsto\varphi(\omega,t)$ with right-continuous paths and existing
left limits. Notice that unlike in some important references [15, 36], we
allow, and actually need at some places, that $\varphi(t)\not=0$ for some
$t<0$. It is known that such a process is product-measurable. Define
$\varphi_{u}=\varphi\circ\pi_{u}$. By product measurability,
$\varphi_{u}(t-S(u))$ is a random variable. Note that, for given
$u\in\mathcal{I}$, $\varphi_{u}$ is independent of $S(u)$. However,
$\varphi_{u}$ and $S(v)$ can be dependent, when $u$ is an ancestor of $v$. The
general branching process counted with characteristic $\varphi$ is
$\mathcal{Z}^{\varphi}=(\mathcal{Z}_{t}^{\varphi})_{t\in\mathds{R}}$ where
$\mathcal{Z}_{t}^{\varphi}$ is defined by

$\mathcal{Z}_{t}^{\varphi}\vcentcolon=\sum_{u\in\mathcal{I}}\varphi_{u}(t-S(u)),\quad
t\in\mathds{R}.$ (2.2)

Here, we use the convention $\varphi(-\infty)\vcentcolon=0$ and so the above
sum involves only terms associated with individuals that are eventually born.
In the special case $\varphi=\mathds{1}_{[0,\zeta)}$,

$\mathcal{Z}^{\mathds{1}_{[0,\zeta)}}_{t}=\sum_{u\in\mathcal{I}}\mathds{1}_{[0,\zeta_{u})}(t-S(u))=\sum_{u\in\mathcal{I}}\mathds{1}_{\\{S(u)\leq
t<S(u)+\zeta_{u}\\}},$ (2.3)

i.e., $\mathcal{Z}^{\mathds{1}_{[0,\zeta)}}_{t}$ is the number of individuals
alive at time $t$. Similarly,

$N((t,t+a])\vcentcolon=\mathcal{Z}^{\mathds{1}_{[0,a)}}_{t+a}=\sum_{u\in\mathcal{I}}\mathds{1}_{[0,a)}(t+a-S(u))=\sum_{u\in\mathcal{I}}\mathds{1}_{\\{t<S(u)\leq
t+a\\}}$ (2.4)

is the number of individuals born strictly after time $t$ and up to and
including time $t+a$, $a>0$. The setup covers a wide range of possible
applications. Some special cases and specific examples are covered in Section
3.

Notice that $\mathcal{Z}^{\varphi}$ is not well-defined a priori. Conditions
for the finiteness of the general branching process are given in [23, Section
6.2]. For instance, the existence of a Malthusian parameter, a condition
formally stated as (A1) below and assumed throughout this paper, implies that
the number of individuals born up to and including time $t$ is finite for all
$t\geq 0$ almost surely, see [23, Theorem 6.2.3]. In particular, if (A1)
holds, then $\mathcal{Z}_{t}^{\varphi}$ is well-defined whenever the
characteristic $\varphi$ vanishes on the negative half-line since in this
case, the sum on the right-hand side of (2.2) has only finitely many non-
vanishing summands almost surely. As, in general, we allow the characteristic
$\varphi$ to be real-valued and do not require that it vanishes on the
negative half-line, we need a finiteness result that goes beyond [23, Theorem
6.2.3]. Jagers and Nerman [25] work under their assumption (6.1), which
corresponds to our condition (A4) for $|\varphi|$ below. However, in our
proofs, we shall require the well-definedness of
$\mathcal{Z}_{t}^{\chi_{\lambda}}$ for a specific centered characteristic
$\chi_{\lambda}$, defined in Section 5, with $|\chi_{\lambda}|$ not satisfying
(A4). Instead, we work under (A4) and (A5) to ensure the general branching
process counted with a random characteristic to be well-defined. The
corresponding result is Proposition 2.2 below.

###  2.2 Assumptions

We write $\mu(\cdot)$ for the intensity measure $\mathds{E}[\xi(\cdot)]$ of
the point process $\xi(\cdot)$, and $\mathcal{L}\mu$ for its Laplace
transform, i.e.,

$\mathcal{L}\mu(z)\vcentcolon=\int
e^{-zx}\,\mu(\mathrm{d}\mathit{x})=\mathds{E}\bigg{[}\sum_{j=1}^{N}e^{-zX_{j}}\bigg{]}$
(2.5)

for all $z\in\mathds{C}$ for which the above integral converges absolutely.

Throughout this paper we distinguish between the lattice and the non-lattice
case. Here, we say that $\xi$ is _lattice_ if $\mu([0,\infty)\setminus
h\mathds{N}_{0})=0$ for some $h>0$, and we say that $\xi$ is _non-lattice_ ,
otherwise. In the lattice case, without loss of generality, we assume that the
lattice span is $1$, $\mu([0,\infty)\setminus\mathds{N}_{0})=0$ and
$\mu([0,\infty)\setminus h\mathds{N}_{0})>0$ for all $h>1$. We set
$\mathds{G}\vcentcolon=\mathds{Z}$ in the lattice case and
$\mathds{G}\vcentcolon=\mathds{R}$ in the non-lattice case. We use the symbol
$\ell$ to denote the counting measure on $\mathds{Z}$ in the lattice case and
the Lebesgue measure in the non-lattice case, respectively.

For a function $f:\mathds{G}\mapsto\mathds{C}$ we define the bilateral Laplace
transform $\mathcal{L}f$ of $f$ at $z\in\mathds{C}$ by

$\mathcal{L}f(z)\vcentcolon=\int_{\mathds{G}}e^{-zx}f(x)\,\ell(\mathrm{d}\mathit{x})$

whenever the integral converges absolutely.

The following assumption is essential in the law of large numbers [15, 36]
and, therefore, also for the central limit theorem studied here.

  1. (A1)

There exists a _Malthusian parameter_ $\alpha>0$, i.e., an $\alpha>0$
satisfying

$\displaystyle\mathcal{L}\mu(\alpha)=\int e^{-\alpha
x}\,\mu(\mathrm{d}\mathit{x})=1\quad\text{and}$ (2.6)
$\displaystyle\mathds{E}\bigg{[}\sum_{j=1}^{N}X_{j}e^{-\alpha
X_{j}}\bigg{]}=-(\mathcal{L}\mu)^{\prime}(\alpha)=\vcentcolon\beta\in(0,\infty).$
(2.7)

Notice that (A1) implies the supercriticality of the general branching
process, that is, $\mathds{E}[N]=\mu([0,\infty))\in(1,\infty]$, which, in
turn, ensures that the underlying branching process survives with positive
probability meaning that $\mathds{P}(\mathcal{S})>0$. We stress that the case
$\mathds{P}(N=\infty)>0$ is allowed. For the rest of the paper, we assume that
(A1) is satisfied.

In our main results, we further assume that the Laplace transform
$\mathcal{L}\mu$ is finite on an open half-space $\mathrm{Re}(z)>\vartheta$
for some $\vartheta<\frac{\alpha}{2}$:

  1. (A2)

There exists $\vartheta\in(0,\frac{\alpha}{2})$ such that

$\mathcal{L}\mu(\vartheta)=\mathds{E}\bigg{[}\sum_{j=1}^{N}e^{-\vartheta
X_{j}}\bigg{]}<\infty.$ (2.8)

For the central limit theorem, we need a second moment assumption for the
point process $\xi$. Before we state it, we set $k^{*}$ to be the maximum of
all multiplicities of the roots of $\mathcal{L}\mu(z)=1$ on the critical line
$\mathrm{Re}(z)=\frac{\alpha}{2}$ or $k^{*}\vcentcolon=\frac{1}{2}$ if there
is no such root.

  1. (A3)

The random variable

$\int(1+x^{k^{*}-\frac{1}{2}})e^{-\frac{\alpha}{2}x}\,\xi(\mathrm{d}\mathit{x})=\sum_{j=1}^{N}\big{(}1+X_{j}^{k^{*}-\frac{1}{2}}\big{)}e^{-\frac{\alpha}{2}X_{j}}$
(2.9)

has finite second moment.

######  Remark 2.1.

Notice that Condition (A6) in [27], namely, the existence of a
$\vartheta<\frac{\alpha}{2}$ such that

$\mathds{E}\bigg{[}\bigg{(}\sum_{j=1}^{N}e^{-\vartheta
X_{j}}\bigg{)}^{\\!\\!2}\,\bigg{]}<\infty,$

implies both our conditions (A2.8) and (A2.9). Janson’s condition (A6) may be
easier to check in cases where it holds.

The existence of the Malthusian parameter allows us to define a nonnegative
martingale, called _Nerman’s martingale_ , namely,

$\displaystyle W_{t}=W_{t}(\alpha)=\sum_{u\in\mathcal{C}_{t}}e^{-\alpha
S(u)},\quad t\geq 0$ (2.10)

where

$\displaystyle\mathcal{C}_{t}\vcentcolon=\\{uj\in\mathcal{T}:S(u)\leq
t<S(uj)\\}$ (2.11)

is the coming generation at time $t$. For the proof of the martingale property
under (A1) see [36, Proposition 2.4]. We denote the almost sure limit of
Nerman’s martingale by $W$. Martingale theory implies that $\mathds{E}[W]=1$
iff $(W_{t})_{t\geq 0}$ is uniformly integrable. Sufficient conditions for the
latter can be found in [36, Corollary 3.3], [37, Theorem 2.1] and [15,
Theorems 2.1 and 3.3]. In the given situation, $(W_{t})_{t\geq 0}$ is
uniformly integrable iff

$\mathds{E}[Z_{1}\log_{+}Z_{1}]<\infty$ ($Z\log Z$)

holds where

$\displaystyle Z_{n}=\sum_{|u|=n}e^{-\alpha S(u)},\quad n\in\mathds{N}_{0}.$
(2.12)

The process $(Z_{n})_{n\in\mathds{N}_{0}}$ is also a nonnegative martingale,
called Biggins’ martingale, and it has the same almost sure limit $W$ as
Nerman’s martingale $(W_{t})_{t\geq 0}$ [15, Theorem 3.3]. Since (A2.9)
immediately implies ($Z\log Z$), we infer that validity of (A1) and (A2.9)
implies that both martingales, $(W_{t})_{t\geq 0}$ and
$(Z_{n})_{n\in\mathds{N}_{0}}$, converge almost surely and in $L^{1}$ to the
same limit $W\geq 0$. Hence, in our theorems, ($Z\log Z$) will not be imposed
explicitly, but will hold automatically whenever (A1) and (A2.9) are assumed
to hold.

We continue with assumptions concerning the random characteristic $\varphi$.
These assumptions are not made throughout the paper, but in certain results
only. It will be explicitly stated, when this is the case.

Throughout the paper, if $\varphi$ is a nonnegative or integrable
characteristic (meaning that $\mathds{E}[|\varphi(t)|]$ is finite for every
$t\in\mathds{R}$), then we write $\mathds{E}[\varphi]$ for the (measurable)
function that maps
$t\mapsto\mathds{E}[\varphi](t)\vcentcolon=\mathds{E}[\varphi(t)]$. This
notation has the advantage that if $X$ is a random variable, then we can write
$\mathds{E}[\varphi](X)$, which is again a random variable. Similarly, we
write $\operatorname{\mathrm{Var}}[\varphi]$ for the variance function
$\mathds{E}[(\varphi-\mathds{E}[\varphi])^{2}]$, so
$\operatorname{\mathrm{Var}}[\varphi](t)=\operatorname{\mathrm{Var}}[\varphi(t)]$.
We start with an assumption regarding the mean of the characteristic.

  1. (A4)

$\varphi(t)\in L^{1}$ for every $t\in\mathds{R}$ and
$t\mapsto\mathds{E}[\varphi](t)e^{-\alpha t}$ is directly Riemann integrable.

If (A1) is fulfilled, and if $\varphi$ is a real-valued characteristic such
that $|\varphi|$ satisfies (A4), then, in the non-lattice case, the law of
large numbers by Nerman (see [25, Theorem 6.1]) states that

$e^{-\alpha
t}\mathcal{Z}^{\varphi}_{t}\to\beta^{-1}\mathcal{L}\mathds{E}[\varphi](\alpha)W=\beta^{-1}\int
e^{-\alpha x}\mathds{E}[\varphi](x)\,\mathrm{d}\mathit{x}\cdot W\quad\text{as
}t\to\infty$ (2.13)

in probability. If, additionally, ($Z\log Z$) holds, then the convergence in
(2.13) holds in $L^{1}$. To see this, first recall that ($Z\log Z$) implies
$\mathds{E}[W]=1$ and hence $e^{-\alpha
t}\mathds{E}[\mathcal{Z}^{\varphi}_{t}]$ converges by the two-sided version of
the key renewal theorem [2, Satz 2.5.3] to $\beta^{-1}\int e^{-\alpha
x}\mathds{E}[\varphi](x)\,\mathrm{d}\mathit{x}$, which is the expectation of
the random variable on the right-hand side of (2.13). If $\varphi$ is
nonnegative, then the convergence of the first moment in combination with
convergence in probability gives the convergence in $L^{1}$ by Proposition
4.12 in [30]. The case of general $\varphi$ can be reduced to the case of
nonnegative $\varphi$ using the decomposition
$\varphi=\varphi_{+}-\varphi_{-}$ of $\varphi$ into its positive part minus
its negative part.

What is more, (A2.9) implies $\mathcal{L}\mu(\frac{\alpha}{2})<\infty$ and
hence the holomorphy of $\mathcal{L}\mu$ on the half-space
$\mathrm{Re}(z)>\frac{\alpha}{2}$, which implies that all higher derivatives
of $\mathcal{L}\mu$ in the point $z=\alpha$ exist. This in turn implies (5.4)
in [36] (for instance with $g(t)=1\wedge t^{-2}$ there). Hence, Conditions 5.1
of [36], 3.2 of [15] and (3.2) and (3.4) of [33] are satisfied. This ensures
that the convergence in (2.13) holds in the almost sure sense and in $L^{1}$
provided that

  * •

$\varphi$ vanishes on $(-\infty,0)$ and satisfies Condition 5.2 of [36] in the
non-lattice case;

  * •

$\varphi$ vanishes on $(-\infty,0)$ and satisfies Condition 3.1 in [15] in the
lattice case or

  * •

satisfies Eq. (3.3) in [33] in the case that $\varphi$ does not vanish on the
negative half-line.

The next two assumptions are conditions on the second moments of the
characteristic $\varphi$.

  1. (A5)

$\varphi(t)\in L^{2}$ for every $t\in\mathds{R}$ and
$t\mapsto\operatorname{\mathrm{Var}}[\varphi](t)e^{-\alpha t}$ directly
Riemann integrable.

  1. (A6)

For any $t\in\mathds{R}$ there is an $\varepsilon>0$ such that the family

$\textstyle(|\varphi(x)|^{2})_{|x-t|\leq\varepsilon}\quad\text{is uniformly
integrable.}$

Notice that if $\varphi$ is deterministic real-valued, then (A6) holds since
$\varphi$ is càdlàg, in particular, locally bounded.

In the lattice case, if $t\in\mathds{Z}$, then $t-S(u)\in\mathds{Z}$ for all
individuals $u$ with $S(u)<\infty$. Then $\mathcal{Z}_{t}^{\varphi}$ depends
only on the values $\varphi_{u}(x)$ for $x\in\mathds{Z}$ ($u\in\mathcal{I}$).
In particular, the values of $\varphi$ on $\mathds{R}\setminus\mathds{Z}$ are
irrelevant for our purposes. Therefore, in the lattice case, we make the
assumption that $\varphi$ has paths that are constant on intervals of the form
$[n,n+1)$, $n\in\mathds{Z}$. With this assumption, condition (A6) is
meaningful also in the lattice case, but reduces to the condition that
$\varphi(x)\in L^{2}$ for all $x\in\mathds{Z}$, a condition implicitly
contained in (A5).

We continue with a proposition giving sufficient conditions for the general
branching process counted with characteristic $\varphi$ to be well-defined.
Before this, we introduce the notion of _admissible ordering_ of
$\mathcal{I}$. We call a sequence $u_{1},u_{2},\ldots\in\mathcal{I}$ an
admissible ordering of $\mathcal{I}$ if

  * •

$\mathcal{I}_{n}\vcentcolon=\\{u_{1},\ldots,u_{n}\\}$ is a subtree of the
Ulam-Harris tree $\mathcal{I}$ of cardinality $n$,

  * •

$\mathcal{I}=\bigcup_{n\in\mathds{N}}\mathcal{I}_{n}$.

Admissible orderings exist. Indeed, we can construct
$(u_{n})_{n\in\mathds{N}}$ recursively. First, let $u_{1}=\varnothing$. If we
have constructed $u_{i}$ for $i=1,\ldots,2^{k}$ where $k\in\mathds{N}_{0}$,
then, for any $2^{k}<i\leq 2^{k+1}$, we set $u_{i}\vcentcolon=u_{i-2^{k}}j$
with the smallest $j\in\mathds{N}$ such that
$u_{i-2^{k}}j\not\in\\{v_{1},\ldots,u_{2^{k}}\\}$, see Figure 1.

Figure 1: The subtree $\mathcal{I}_{16}$ for the particular admissible
ordering of $\mathcal{I}$ given above.

Recall that a series $\sum_{n\in\mathds{N}}x_{n}$ in a Banach space
$(X,\|\cdot\|)$ is said to converge unconditionally if, for any
$\varepsilon>0$, there is a finite $I\subseteq\mathds{N}$ such that
$\|\sum_{n\in J}x_{n}\|<\varepsilon$ for any finite
$J\subseteq\mathds{N}\setminus I$. An equivalent definition is that the series
converges for any rearrangement. For this and other characterizations we refer
the reader to [19].

######  Proposition 2.2.

Suppose that (A1) holds and that $\varphi$ is a random characteristic
satisfying (A4) and (A5). Then, for every $t\in\mathds{R}$,

$\mathcal{Z}_{t}^{\varphi}\vcentcolon=\sum_{u\in\mathcal{I}}\varphi_{u}(t-S(u))$

converges unconditionally in $L^{1}$ and almost surely over every admissible
ordering of $\mathcal{I}$.

The proof of the proposition will be given in Section 4.2.

######  Remark 2.3.

Notice that
$\mathcal{Z}_{t}^{\varphi}=\mathcal{Z}_{t}^{\varphi\mathds{1}_{(-\infty,t]}}$
and thus from the proposition, we infer that $\mathcal{Z}_{t}^{\varphi}$
converges unconditionally in $L^{1}$ and almost surely for every
$t\in\mathds{R}$ if (A1) holds and, for every $t\in\mathds{R}$,
$\varphi\mathds{1}_{(-\infty,t]}$ satisfies (A4) and (A5).

######  Remark 2.4.

Notice that if the random characteristics $\varphi$ and $\psi$ satisfy
condition (A6), then so does any linear combination of them. Further, by the
dominated convergence theorem, both the expectation function and the variance
function of any linear combination of $\varphi$ and $\psi$ are càdlàg. This
particularly implies that these functions are locally bounded and a.e.
continuous. Consequently, if, in addition to (A6), also (A5) holds for
$\varphi$ and $\psi$, then (A5) also holds for any linear combination of
$\varphi$ and $\psi$. Indeed, for $\beta_{1},\beta_{2}\in\mathds{R}$,

$\operatorname{\mathrm{Var}}[\beta_{1}\varphi(t)+\beta_{2}\psi(t)]e^{-\alpha
t}\leq 2\beta_{1}^{2}\operatorname{\mathrm{Var}}[\varphi(t)]e^{-\alpha
t}+2\beta_{2}^{2}\operatorname{\mathrm{Var}}[\psi(t)]e^{-\alpha t},\quad
t\in\mathds{R}.$

By [40, Remark 3.10.5 on p. 237], the function in focus is directly Riemann
integrable as a locally Riemann integrable function dominated by a directly
Riemann integrable function.

The next proposition gives sufficient conditions for (A4), (A5) and (A6). To
formulate it, we introduce the following notation. If
$f:\mathds{R}\to\mathds{R}$ is a function, we set
$f^{*}(t)\vcentcolon=\sup_{|x-t|\leq 1}|f(x)|$. This notation extends
immediately (pathwise) to random càdlàg functions such as random
characteristics $\varphi$.

######  Proposition 2.5.

Suppose that (A1) holds.

  1. (a)

If $f:\mathds{R}\to\mathds{R}$ is càdlàg and $\int
f^{*}(x)\,\mathrm{d}\mathit{x}<\infty$, then $f$ is directly Riemann
integrable. Conversely, if $f:\mathds{R}\to\mathds{R}$ is directly Riemann
integrable, then so is $f^{*}$. ((A1) is not needed.)

  2. (b)

If a random characteristic $\varphi$ satisfies

$\int\mathds{E}[\varphi^{*}](x)e^{-\alpha x}\,\mathrm{d}\mathit{x}<\infty,$
(2.14)

then (A4) holds for $\varphi$.

  3. (c)

If a random characteristic $\varphi$ satisfies

$\int\mathds{E}\big{[}(\varphi^{*})^{2}\big{]}(x)e^{-\alpha
x}\,\mathrm{d}\mathit{x}<\infty,$ (2.15)

then $\varphi$ also satisfies (A5) and (A6).

By $\Lambda$ we denote the set of solutions to the equation

$\displaystyle\mathcal{L}\mu(\lambda)=1$ (2.16)

such that $\mathrm{Re}(\lambda)>\frac{\alpha}{2}$ and by $\partial\Lambda$ we
denote the set of roots on the _critical line_
$\mathrm{Re}(\lambda)=\frac{\alpha}{2}$. In the lattice case, $\mathcal{L}\mu$
is $2\pi\mathrm{i}$-periodic, and we define $\Lambda$ to be the set of
$\lambda$ with $\mathrm{Re}(\lambda)>\frac{\alpha}{2}$ satisfying (2.16) and
$\mathrm{Im}(\lambda)\in(-\pi,\pi]$. Analogously, in this case,
$\partial\Lambda$ denotes the set of roots $\lambda$ with
$\mathrm{Re}(\lambda)=\frac{\alpha}{2}$ satisfying
$\mathrm{Im}(\lambda)\in(-\pi,\pi]$. Finally, in both the non-lattice and the
lattice case, we set $\Lambda_{\geq}\vcentcolon=\Lambda\cup\partial\Lambda$.
Notice that $\alpha\in\Lambda$ and that every other element
$\lambda\in\Lambda_{\geq},\lambda\not=\alpha$ satisfies
$\mathrm{Re}(\lambda)\in[\frac{\alpha}{2},\alpha)$ and
$\mathrm{Im}(\lambda)\not=0$. Further,
$\lambda=\theta+\mathrm{i}\eta\in\Lambda_{\geq}$ implies that the complex
conjugate $\overline{\lambda}=\theta-\mathrm{i}\eta\in\Lambda_{\geq}$ except
if $\eta=\pi$ in the lattice case.

Although one may consider cases where $\Lambda_{\geq}$ contains infinitely
many elements, in all relevant examples $\Lambda_{\geq}$ is finite. Therefore,
and for simplicity, we assume throughout the paper the following:

  1. (A7)

The set of roots $\Lambda_{\geq}$ is finite.

We stress that if $\mu$ has a density with respect to the Lebesgue measure and
(A2.8) holds, then also (A7) holds. This is justified in the proof of Lemma
7.3.

###  2.3 Main results

For each root $\lambda\in\mathds{C}$ of the function $\mathcal{L}\mu-1$, we
denote its multiplicity by $k(\lambda)\in\mathds{N}$. Then, for any
$j=0,\ldots,k(\lambda)-1$, we can define

$W_{t}^{(j)}(\lambda)\vcentcolon=(-1)^{j}\sum_{u\in\mathcal{C}_{t}}S(u)^{j}e^{-\lambda
S(u)},\qquad t\in\mathds{R},$ (2.17)

where $\mathcal{C}_{t}$ is the coming generation at time $t$ formally defined
in (2.11). The Malthusian parameter $\alpha>0$ is a root of multiplicity $1$
and gives rise to one martingale, namely, Nerman’s martingale
$(W_{t})_{t\in\mathds{R}}=(W_{t}(\alpha))_{t\in\mathds{R}}$ defined in (2.10),
which is of great importance in the law of large numbers for the general
branching process. On the other hand, the martingales corresponding to
$\lambda\in\Lambda$ are relevant in the central limit theorem.

######  Theorem 2.6.

Suppose that (A1) through (A2.9) hold. Then, for any $\lambda\in\Lambda$ and
$j=0,\ldots,k(\lambda)-1$, the process $(W^{(j)}_{t}(\lambda))_{t\geq 0}$ is a
martingale and there is a random variable $W^{(j)}(\lambda)\in L^{2}$ such
that

$W_{t}^{(j)}(\lambda)\to W^{(j)}(\lambda)\quad\text{a.\,s.\ and in
}L^{2}\text{ as }t\to\infty.$

There are more technicalities to deal with before the main result can be
stated in its most general form but special cases are given now as
illustration. We start with the non-lattice case. To this end, we need one
more piece of notation. For a function $f:\mathds{R}\mapsto\mathds{R}$ we
define the total variation function $\mathrm{V}\\!f$ by

$\mathrm{V}\\!f(x)\vcentcolon=\sup\bigg{\\{}\sum_{j=1}^{n}|f(x_{j})-f(x_{j-1})|:-\infty<x_{0}<\ldots<x_{n}\leq
x,\ n\in\mathds{N}\bigg{\\}}$ (2.18)

for $x\in\mathds{R}$. In some theorems related to the non-lattice case we
require the following additional assumption on the characteristic $\varphi$:

$\displaystyle\int(\mathrm{V}\\!\mathds{E}[\varphi])(x)\big{(}e^{-\vartheta
x}+e^{-\alpha x}\big{)}\,\mathrm{d}\mathit{x}<\infty$ (2.19)

for some $\vartheta<\frac{\alpha}{2}$. We use the symbol $\vartheta$ both in
(A2.8) and in (2.19) to denote some parameter $<\frac{\alpha}{2}$ at which the
corresponding condition is satisfied. If we make both assumptions at the same
time, there is no harm in assuming that the $\vartheta$’s coincide, which is
why we do not distinguish them by our notation.

######  Theorem 2.7.

Suppose that (A1) through (A2.9) hold and that the intensity measure $\mu$ has
a density with respect to the Lebesgue measure. Further, suppose that there
are no roots of the equation $\mathcal{L}\mu(z)=1$ in the strip
$\vartheta<\mathrm{Re}(z)<\alpha$. Then, for any characteristic $\varphi$
satisfying (A5), (A6) and (2.19), there exists $\sigma\geq 0$ such that, for
$a_{\alpha}\vcentcolon=\beta^{-1}\int\mathds{E}[\varphi(x)]e^{-\alpha
x}\,\mathrm{d}\mathit{x}$ and a standard normal random variable $\mathcal{N}$
independent of $W$,

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

The constant $\sigma$ can be explicitly computed, see the formula given in
Theorem 2.14 below.

######  Theorem 2.8.

Suppose that (A1) through (A2.9) hold and that the intensity measure $\mu$ has
a density with respect to the Lebesgue measure. Then (A7) holds and there are
$b_{\lambda,l}$, $l=0,\ldots,k(\lambda)-1$, $\lambda\in\Lambda_{\geq}$
satisfying $b_{\overline{\lambda},l}=\overline{b_{\lambda,l}}$ such that for
any characteristic $\varphi$ satisfying (A5), (A6) and (2.19) there exists
$\sigma\geq 0$ such that, for a standard normal random variable $\mathcal{N}$
independent of $W$, the following assertions hold.

  1. (i)

If there are no roots of $\mathcal{L}\mu(z)=1$ on the critical line
$\mathrm{Re}(z)=\frac{\alpha}{2}$, then

$\displaystyle
e^{-\frac{\alpha}{2}t}\bigg{(}\mathcal{Z}_{t}^{\varphi}-\sum_{\lambda\in\Lambda}e^{\lambda
t}\sum_{l=0}^{k(\lambda)-1}b_{\lambda,l}\sum_{j=0}^{l}\binom{l}{j}W^{(j)}(\lambda)\int(t\\!-\\!x)^{l-j}\mathds{E}[\varphi(x)]e^{-\lambda
x}\,\mathrm{d}\mathit{x}\bigg{)}$
$\displaystyle\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sigma\sqrt{\tfrac{W}{\beta}}\mathcal{N}$

as $t\to\infty$.

  2. (ii)

Otherwise, let $k\in\mathds{N}$ be the maximal multiplicity $k(\lambda)$ of a
root $\lambda\in\partial\Lambda$. Then

$\displaystyle
e^{-\frac{\alpha}{2}t}t^{-k+\frac{1}{2}}\bigg{(}\mathcal{Z}_{t}^{\varphi}-\sum_{\lambda\in\Lambda}e^{\lambda
t}\sum_{l=0}^{k(\lambda)-1}b_{\lambda,l}\sum_{j=0}^{l}\binom{l}{j}W^{(j)}(\lambda)\int(t\\!-\\!x)^{l-j}\mathds{E}[\varphi(x)]e^{-\lambda
x}\,\mathrm{d}\mathit{x}\bigg{)}$
$\displaystyle\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sigma\sqrt{\tfrac{W}{\beta}}\mathcal{N}$

as $t\to\infty$.

The lattice analogue of Theorem 2.8 is given next.

######  Theorem 2.9.

Suppose that (A1) through (A2.9) hold and that the intensity measure $\mu$ is
lattice with span $1$. Then there are $b_{\lambda,l}$,
$l=0,\ldots,k(\lambda)-1$, $\lambda\in\Lambda_{\geq}$ satisfying
$b_{\overline{\lambda},l}=\overline{b_{\lambda,l}}$ such that for any
characteristic $\varphi$ satisfying

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

for some $\vartheta<\frac{\alpha}{2}$ and

$\displaystyle\sum_{n\in\mathds{Z}}\operatorname{\mathrm{Var}}[\varphi](n)e^{-\alpha
n}<\infty$

there exists $\sigma\geq 0$ such that, for a standard normal random variable
$\mathcal{N}$ independent of $W$, the following assertions hold.

  1. (i)

If there are no roots of $\mathcal{L}\mu(z)=1$ on the critical line
$\mathrm{Re}(z)=\frac{\alpha}{2}$, then

$\displaystyle
e^{-\frac{\alpha}{2}t}\bigg{(}\mathcal{Z}_{t}^{\varphi}-\sum_{\lambda\in\Lambda}e^{\lambda
t}\sum_{l=0}^{k(\lambda)-1}b_{\lambda,l}\sum_{j=0}^{l}\binom{l}{j}W^{(j)}(\lambda)\sum_{n\in\mathds{Z}}(t\\!-\\!n)^{l-j}\mathds{E}[\varphi(n)]e^{-\lambda
n}\ \bigg{)}$
$\displaystyle\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sigma\sqrt{\tfrac{W}{\beta}}\mathcal{N}$

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

  2. (ii)

Otherwise, let $k\in\mathds{N}$ be the maximal multiplicity $k(\lambda)$ of
the roots $\lambda\in\partial\Lambda$. Then

$\displaystyle
e^{-\frac{\alpha}{2}t}t^{-k+\frac{1}{2}}\bigg{(}\mathcal{Z}_{t}^{\varphi}-\sum_{\lambda\in\Lambda}e^{\lambda
t}\sum_{l=0}^{k(\lambda)-1}b_{\lambda,l}\sum_{j=0}^{l}\binom{l}{j}W^{(j)}(\lambda)\sum_{n\in\mathds{Z}}(t\\!-\\!n)^{l-j}\mathds{E}[\varphi(n)]e^{-\lambda
n}\bigg{)}$
$\displaystyle\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sigma\sqrt{\tfrac{W}{\beta}}\mathcal{N}$

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

######  Remark 2.10.

In Theorems 2.8 and 2.9, we do not exclude the case $\sigma=0$. There, a more
precise limit theorem can be derived with the help of Theorem 2.14. In
particular, the expression in parentheses in (i) is just a deterministic
function of the order $O(e^{\theta t})$ for some $\theta<\frac{\alpha}{2}$
(cf. Theorem 2.14(i)). In case (ii), we need to additionally subtract a linear
combination of $e^{\lambda t}t^{l}$, where $\lambda$ runs over the roots on
the critical line and $l<k(\lambda)$ with $k(\lambda)$ denoting the order of
the root, and to use a different normalization in order to get a nontrivial
limit.

######  Remark 2.11.

The constants $b_{\lambda,l}$, $l=0,\ldots,k(\lambda)-1$,
$\lambda\in\Lambda_{\geq}$ can be computed using Proposition 7.9.

######  Remark 2.12.

Note that the fluctuations in Theorem 2.9 and 2.8 are very similar to
fluctuations obtain by Janson [26] in the multi-type case where the birth
times constitutes a homogeneous Poisson point process. This indicates that
there may be a general theorem describing the fluctuations of multitype CMJ
process, which covers the aforementioned models (c.f. Open Problem 1 in
Section 8).

In order to obtain the asymptotic expansion of $\mathcal{Z}^{\varphi}_{t}$ as
presented in Theorems 2.7, 2.8 and 2.9, we first need an expansion for the
mean $m^{\varphi}_{t}\vcentcolon=\mathds{E}[\mathcal{Z}^{\varphi}_{t}]$,
$t\in\mathds{G}$. It turns out that, under suitable assumptions, the following
holds:

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

for some constants $a_{\lambda,l}\in\mathds{C}$ and a function $r$ satisfying
$|r(t)|\leq Ce^{\alpha t/2}/(1+t^{2})$ for all $t\in\mathds{G}$ and some
finite constant $C\geq 0$.

We shall provide three different sets of sufficient conditions for (2.20) to
hold. The first case is when the characteristic $\varphi$ is chosen in such a
way that $\mathcal{Z}^{\varphi}_{t}$ is a rescaled martingale (see Section 5).
In this case, (2.20) holds trivially. The second case is when
$\mathds{G}=\mathds{Z}$. Then expansion (2.20) is obtained in Lemma 7.1 via
generating functions. The third case is the non-lattice case where under the
additional (technical) assumption (7.7), expansion (2.20) is derived in Lemma
7.6. There might be more examples of $\mathcal{Z}^{\varphi}_{t}$ that are not
covered by any of the three sufficient conditions, even though the
corresponding expansion of $\mathds{E}[\mathcal{Z}^{\varphi}_{t}]$ is of form
(2.20). For this reason, we formulate our main result, Theorem 2.14, for
processes $\mathcal{Z}_{t}^{\varphi}$ for which
$\mathds{E}[\mathcal{Z}^{\varphi}_{t}]$ satisfies (2.20). What is more, in
some examples, it is in fact more convenient to directly check that (2.20)
holds rather than checking the assumptions of Theorem 2.8 or 2.9, see e. g.
Sections 3.1 and 3.2.

From (2.20) we can obtain an asymptotic expansion of
$\mathcal{Z}_{t}^{\varphi}$ where the principal terms are of the form a
constant times $e^{\lambda t}t^{j}W^{(j)}(\lambda)$ for $\lambda\in\Lambda$
and $j=0,\ldots,k(\lambda)-1$. More precisely, the principal terms are given
by the expression

$\displaystyle H_{\Lambda}(t)\vcentcolon=\sum_{\lambda\in\Lambda}e^{\lambda
t}\sum_{l=0}^{k(\lambda)-1}\sum_{j=0}^{l}a_{\lambda,l}\binom{l}{j}t^{j}W^{(l-j)}(\lambda).$
(2.21)

If, additionally, there are roots $\lambda\in\partial\Lambda$, then the next
terms in the expansion are given by the following deterministic sum

$\displaystyle
H_{\partial\Lambda}(t)\vcentcolon=\sum_{\lambda\in\partial\Lambda}e^{\lambda
t}\sum_{l=0}^{k(\lambda)-1}a_{\lambda,l}t^{l},\qquad t\in\mathds{R}.$ (2.22)

(Of course, if $\Lambda_{\geq}=\partial\Lambda\cup\\{\alpha\\}$, the terms
from $H_{\partial\Lambda}(t)$ are the subleading terms.) We set
$H(t)\vcentcolon=H_{\Lambda}(t)+H_{\partial\Lambda}(t)$, $t\in\mathds{R}$.
Further, for any $\lambda\in\partial\Lambda$ and $l=0,\ldots,k(\lambda)-1$, we
define a random variable $R_{\lambda,l}$ by

$\displaystyle
R_{\lambda,l}\vcentcolon=\sum_{j=l}^{k(\lambda)-1}a_{\lambda,j}\binom{j}{l}\sum_{k=1}^{N}(-X_{k})^{j-l}e^{-\lambda
X_{k}}.$ (2.23)

Assumption (A2.9) guarantees that $R_{\lambda,l}\in L^{2}$ for all
$\lambda\in\partial\Lambda$ and $l=0,\ldots,k(\lambda)-1$. We may thus define

$\rho_{l}^{2}\vcentcolon=\sum_{\begin{subarray}{c}\lambda\in\partial\Lambda:\\\
k(\lambda)>l\end{subarray}}\operatorname{\mathrm{Var}}[R_{\lambda,l}]$

where
$\operatorname{\mathrm{Var}}[R_{\lambda,l}]=\mathds{E}[|R_{\lambda,l}|^{2}]-|\mathds{E}[R_{\lambda,l}]|^{2}$.
In general, throughout the paper, if $Y$ is a complex-valued random variable
with finite mean, we set
$\operatorname{\mathrm{Var}}[Y]\vcentcolon=\mathds{E}[|Y-\mathds{E}[Y]|^{2}]$.

As a final preparation for our main result, we recall the fact that if a
sequence of random variables $(Y_{n})_{n\in\mathds{N}_{0}}$ defined on
$(\Omega,\mathcal{F},\mathds{P})$ converges in distribution to some random
variable $Y$, this convergence is said to be _stable_ if for all continuity
points $y$ of the distribution function of $Y$ and all $E\in\mathcal{F}$, the
limit $\lim_{n\to\infty}\mathds{P}(\\{Y_{n}\leq y\\}\cap E)$ exists. In this
case, we write
$Y_{n}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{st}$\cr\kern-0.21529pt\cr$\to$\cr}}}Y$.
An alternative characterization is the following. There is a copy $Y^{*}$ of
$Y$ defined on some extension of the probability space
$(\Omega,\mathcal{F},\mathds{P})$ such that, for every
$\mathcal{F}$-measurable random variable $X$, it holds that

$(Y_{n},X)\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}(Y^{*},X)\text{
as }n\to\infty,$

see [1, Condition (B’)]. Without loss of generality we may and will assume
that $Y=Y^{*}$ whenever we write
$Y_{n}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{st}$\cr\kern-0.21529pt\cr$\to$\cr}}}Y$.
The second definition is more convenient as it allows to manipulate the
sequence $Y_{n}$ (e.g. by multiplying with another random variable) without
loosing the convergence in distribution. In Remark 2.16 below we prove (2.28)
using such an argument.

######  Remark 2.13.

Although it is not stated explicitly but it follows from the proofs that the
convergences in Theorems 2.7, 2.8 and 2.9 are in fact stable.

For a measurable function $f$ and a measure $\nu$ (possibly random) on the
Borel $\sigma$-field, we write $f*\nu=\nu*f$ for the Lebesgue-Stieltjes
convolution of $f$ and $\nu$, i.e., $f*\nu(t)=\nu*f(t)=\int
f(t-x)\,\nu(\mathrm{d}\mathit{x})$ whenever the integral exists. In
particular,

$f*\xi(t)=\int
f(t-x)\,\xi(\mathrm{d}\mathit{x})=\sum_{j=1}^{N}f(t-X_{j}),\quad
t\in\mathds{R}.$

######  Theorem 2.14.

Suppose that $\xi$ satisfies (A1) through (A2.9) and (A7) and that the real-
valued characteristic $\varphi$ satisfies (A4) through (A6). Further, assume
that $m^{\varphi}_{t}$ satisfies (2.20) with
$\sup_{t\in\mathds{G}}(1+t^{2})e^{-\frac{\alpha}{2}t}r(t)<\infty$, and let
$n\vcentcolon=\max\\{l\in\mathds{N}_{0}:\rho_{l}>0\\}$ with $n=-1$ if the set
is empty; in that case we set $\rho_{-1}\vcentcolon=0$. Then there exists a
finite constant $\sigma\geq 0$ such that, with

$\textstyle
a_{t}^{2}\vcentcolon=\sigma^{2}+\frac{\rho_{n}^{2}}{2n+1}t^{2n+1},\qquad t>0,$

it holds

  1. (i)

if $\sigma^{2}=\rho_{n}^{2}=0$, then $\mathcal{Z}^{\varphi}_{t}=H(t)+r(t)$ a.
s. for all $t\geq 0$,

  2. (ii)

if $\sigma^{2}>0$ or $\rho_{n}^{2}>0$, then

$\displaystyle
a_{t}^{-1}e^{-\frac{\alpha}{2}t}\big{(}\mathcal{Z}^{\varphi}_{t}-H(t)\big{)}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{st}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sqrt{\tfrac{W}{\beta}}\mathcal{N}$
(2.26)

as $t\to\infty$, $t\in\mathds{G}$ where $\mathcal{N}$ is a standard normal
random variable independent of $\mathcal{F}$ and $\beta$ is as defined in
(2.7).

If $n=-1$ the constant $\sigma$ can be explicitly computed, namely,

$\displaystyle\sigma^{2}=\int\operatorname{\mathrm{Var}}\big{[}\varphi(x)+h^{\varphi}*\xi(x)\big{]}e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x})$ (2.27)

where

$\displaystyle
h^{\varphi}(t)\vcentcolon=m^{\varphi}_{t}-\sum_{\lambda\in\Lambda}e^{\lambda
t}\sum_{l=0}^{k(\lambda)-1}a_{\lambda,l}t^{l}.$

In the situation of Theorem 2.14, the following remarks are in order.

######  Remark 2.15.

Observe that, in the non-lattice case, if $\lambda\in\Lambda_{\geq}$, then so
is $\overline{\lambda}$ and $k(\overline{\lambda})=k(\lambda)$. Moreover, as
$m_{t}^{\varphi}$ is real for any $t\in\mathds{R}$, we have

$\displaystyle m_{t}^{\varphi}$
$\displaystyle=\sum_{\lambda\in\Lambda_{\geq}}\sum_{l=0}^{k(\lambda)-1}a_{\lambda,l}t^{l}e^{\lambda
t}+r(t)=\sum_{\lambda\in\Lambda_{\geq}}\sum_{l=0}^{k(\lambda)-1}a_{{\overline{\lambda}},l}t^{l}e^{\overline{\lambda}t}+r(t)$
$\displaystyle=\sum_{\lambda\in\Lambda_{\geq}}\sum_{l=0}^{k(\lambda)-1}\overline{a_{\overline{\lambda},l}}t^{l}e^{\lambda
t}+\overline{r(t)}=\overline{m_{t}^{\varphi}},\quad t\in\mathds{R},$

whence

$\displaystyle
0=m_{t}^{\varphi}-\overline{m_{t}^{\varphi}}=\sum_{\lambda\in\Lambda_{\geq}}\sum_{l=0}^{k(\lambda)-1}(a_{\lambda,l}-\overline{a_{\overline{\lambda},l}})t^{l}e^{\lambda
t}+o(e^{\frac{\alpha}{2}t})\quad\text{as }t\to\infty,\ t\in\mathds{R}.$

Next, we can choose $h>0$ such that the $e^{\lambda h}$,
$\lambda\in\Lambda_{\geq}$ are distinct. Recalling that
$\mathrm{Re}(\lambda)\geq\frac{\alpha}{2}$ for $\lambda\in\Lambda_{\geq}$,
Lemma .1 gives that $a_{\lambda,l}-\overline{a_{\overline{\lambda},l}}=0$,
that is, $\overline{a_{\lambda,l}}=a_{\overline{\lambda},l}$. In particular,
$r(t)$ is real for any $t\in\mathds{R}$.

A similar reasoning in the lattice case gives
$\overline{a_{\lambda,l}}=a_{\overline{\lambda},l}$ for all
$\lambda\in\Lambda$ with $|\mathrm{Im}(\lambda)|<\pi$ and
$a_{\lambda,l}\in\mathds{R}$ if $\mathrm{Im}(\lambda)=\pi$.

######  Remark 2.16.

Observe that for $N(t)\vcentcolon=\mathcal{Z}_{t}^{\mathds{1}_{[0,\infty)}}$,
the number of individuals born up to and including time $t$, by (2.13), we
have

$e^{-\alpha t}N(t)\to\frac{1}{\beta}\int\limits_{[0,\infty)}e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x})\,W=\frac{c_{\alpha}}{\beta}W\quad\text{a.\,s.~{}as}~{}t\to\infty,~{}t\in\mathds{G},$

where $c_{\alpha}=(1-e^{-\alpha})^{-1}$ in the lattice case,
$c_{\alpha}=\alpha^{-1}$ in the non-lattice case. The stable convergence in
(2.26) yields

$a_{t}^{-1}\sqrt{\frac{c_{\alpha}}{N(t)}}\big{(}\mathcal{Z}^{\varphi}_{t}-H(t)\big{)}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\mathcal{N}\quad\text{as
}t\to\infty\quad\text{conditionally given }\mathcal{S}.$ (2.28)

Indeed, with
$G(t)\vcentcolon=a_{t}^{-1}e^{-\frac{\alpha}{2}t}\big{(}\mathcal{Z}^{\varphi}_{t}-H(t)\big{)}$,

(2.28) is equivalent to

$\mathds{P}(\mathcal{S})^{-1}\cdot\mathds{E}\Big{[}g\Big{(}\sqrt{\tfrac{c_{\alpha}e^{\alpha
t}}{N(t)}}G(t)\Big{)}\mathds{1}_{\mathcal{S}}\Big{]}\to\mathds{E}[g(\mathcal{N})]$

as $t\to\infty$, for any continuous, nonnegative function $g$ bounded by $1$.
With $F(t)\vcentcolon=\sqrt{\frac{c_{\alpha}We^{\alpha t}}{\beta
N(t)}}\mathds{1}_{\mathcal{S}}+\mathds{1}_{\mathcal{S}^{c}}$, which is well-
defined since $N(t)>0$ on $\mathcal{S}$, the above convergence can be
rewritten as

$\displaystyle\mathds{P}(\mathcal{S})^{-1}\cdot\mathds{E}\Big{[}g\Big{(}\sqrt{\beta}\tfrac{F(t)G(t)}{\sqrt{W}}\Big{)}\mathds{1}_{\mathcal{S}}\Big{]}\to\mathds{E}[g(\mathcal{N})]\quad\text{as
}t\to\infty.$ (2.29)

For any fixed $\varepsilon>0$, there is some $\delta>0$ such that
$\mathds{P}(\mathcal{S}\cap\\{W<\delta\\})=\mathds{P}(0<W<\delta)\leq\varepsilon$
and consequently

$\displaystyle\Big{|}\mathds{E}\Big{[}g\Big{(}\sqrt{\beta}\tfrac{F(t)G(t)}{\sqrt{W}}\Big{)}\mathds{1}_{\mathcal{S}}\Big{]}-\mathds{E}\Big{[}g\Big{(}\sqrt{\beta}\tfrac{F(t)G(t)}{\sqrt{W\vee\delta}}\Big{)}\mathds{1}_{\mathcal{S}}\Big{]}\Big{|}\leq\varepsilon$

as well as

$\displaystyle\Big{|}\mathds{E}[g(\mathcal{N})\mathds{1}_{\mathcal{S}}]-\mathds{E}\Big{[}g\Big{(}\tfrac{\sqrt{W}\mathcal{N}}{\sqrt{W\vee\delta}}\Big{)}\mathds{1}_{\mathcal{S}}\Big{]}\Big{|}\leq\varepsilon.$

On the other hand, (2.26) yields

$\displaystyle(G(t),W,\mathds{1}_{\mathcal{S}})\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}(\sqrt{\tfrac{W}{\beta}}\mathcal{N},W,\mathds{1}_{\mathcal{S}})\quad\text{as
}t\to\infty.$

Since $F(t)\to 1$ almost surely, Slutsky’s theorem implies

$\displaystyle(G(t),W,\mathds{1}_{\mathcal{S}},F(t))\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}(\sqrt{\tfrac{W}{\beta}}\mathcal{N},W,\mathds{1}_{\mathcal{S}},1)\quad\text{as
}t\to\infty.$

Taking advantage of the fact that the function $(u,v,x,y)\mapsto
g(\sqrt{\beta}\frac{uy}{\sqrt{v\vee\delta}})(|x|\wedge 1)$ is bounded and
continuous, we conclude

$\displaystyle\mathds{E}\Big{[}g\Big{(}\sqrt{\beta}\tfrac{F(t)G(t)}{\sqrt{W\vee\delta}}\Big{)}\mathds{1}_{\mathcal{S}}\Big{]}\to\mathds{E}\Big{[}g\Big{(}\tfrac{\sqrt{W}\mathcal{N}}{\sqrt{W\vee\delta}}\Big{)}\mathds{1}_{\mathcal{S}}\Big{]}$

as $t\to\infty$, and consequently

$\displaystyle\limsup_{t\to\infty}\Big{|}\mathds{E}\Big{[}g\Big{(}\sqrt{\beta}\tfrac{F(t)G(t)}{\sqrt{W}}\Big{)}\mathds{1}_{\mathcal{S}}\Big{]}-\mathds{E}[g(\mathcal{N})\mathds{1}_{\mathcal{S}}]\Big{|}\leq
2\varepsilon.$

Letting now $\varepsilon$ to 0 and using the independence of $\mathcal{N}$ and
$\mathds{1}_{\mathcal{S}}$ we get (2.29) and thereby (2.28).

######  Remark 2.17.

(i) Notice that formula (2.27) is not well-defined in the case $n\geq 0$ as
then the integral diverges. However, in this case, the exact value of $\sigma$
is irrelevant and can be set to $\sigma\vcentcolon=0$.

(ii) In the non-lattice case the variance $\sigma^{2}$ given by (2.27) can be
calculated using the bilateral Laplace transform $\mathcal{L}h^{\varphi}$ of
$h^{\varphi}$. Indeed, by Plancherel’s theorem,

$\displaystyle\sigma^{2}$
$\displaystyle=\int\operatorname{\mathrm{Var}}[\varphi(x)+h^{\varphi}*\xi(x)]e^{-\alpha
x}\,\mathrm{d}\mathit{x}$
$\displaystyle=\mathds{E}\bigg{[}\int\big{(}\big{(}\varphi(x)\\!-\\!\mathds{E}[\varphi](x)+(h^{\varphi}*(\xi\\!-\\!\mu))(x)\big{)}e^{-\frac{\alpha}{2}x}\big{)}^{2}\,\mathrm{d}\mathit{x}\bigg{]}$
$\displaystyle=\frac{1}{2\pi}\mathds{E}\bigg{[}\int\big{|}\mathcal{L}((\varphi(\cdot)\\!-\\!\mathds{E}[\varphi](\cdot))e^{-\alpha\cdot/2})(\mathrm{i}\eta)+\mathcal{L}\big{(}((\xi\\!-\\!\mu)*h^{\varphi})(\cdot)e^{-\frac{\alpha}{2}\cdot}\big{)}(\mathrm{i}\eta)\big{|}^{2}\,\mathrm{d}\eta\bigg{]}$
$\displaystyle=\frac{1}{2\pi}\mathds{E}\bigg{[}\int\big{|}\mathcal{L}(\varphi\\!-\\!\mathds{E}[\varphi])(\tfrac{\alpha}{2}+\mathrm{i}\eta)+\mathcal{L}\big{(}((\xi\\!-\\!\mu)*h^{\varphi})(\cdot)\big{)}(\tfrac{\alpha}{2}+\mathrm{i}\eta)\big{|}^{2}\,\mathrm{d}\eta\bigg{]}$
$\displaystyle=\frac{1}{2\pi}\int\limits_{\mathrm{Re}(z)=\frac{\alpha}{2}}\operatorname{\mathrm{Var}}[\mathcal{L}\varphi(z)+\mathcal{L}\xi(z)\mathcal{L}h^{\varphi}(z)]\,|\mathrm{d}z|,$

where the variance of a complex random variable is defined in terms of
absolute squares. An analogous formula holds in the lattice case, see [27]. We
refrain from giving further details.

(iii) Suppose now that $\varphi$ vanishes on $(-\infty,0)$. Then so do
$m_{t}^{\varphi}$ and the remainder function $r$ from the expansion (2.20).
Additionally, assume that $r(t)=O(e^{(\frac{\alpha}{2}-\varepsilon)t})$ for
some $\varepsilon>0$, which holds in typical cases (see Section 7). In the
non-lattice case of Theorem 2.14, if all the roots in $\Lambda$ are simple and
there are no roots on the critical line
$\\{\mathrm{Re}(z)=\frac{\alpha}{2}\\}$, the bilateral Laplace transform
$\mathcal{L}h^{\varphi}$ of $h^{\varphi}$ coincides on a neighborhood of
$\\{\mathrm{Re}(z)=\frac{\alpha}{2}\\}$ with the function

$z\mapsto\frac{\mathcal{L}(\mathds{E}[\varphi])(z)}{1-\mathcal{L}\mu(z)}.$

To see this, notice that
$r(t)=m^{\varphi}_{t}-\sum_{\lambda\in\Lambda}a_{\lambda,0}e^{\lambda
t}\mathds{1}_{[0,\infty)}(t)$ and let
$h_{r}(t)\vcentcolon=h^{\varphi}(t)-r(t)$. The bilateral Laplace transform
$\mathcal{L}r$ is well-defined on
$\\{\mathrm{Re}(z)>\frac{\alpha}{2}-\varepsilon\\}$. Moreover, for
$\mathrm{Re}(z)>\alpha$,

$\displaystyle\mathcal{L}r(z)=\mathcal{L}m^{\varphi}(z)-\sum_{\lambda\in\Lambda}\frac{a_{\lambda,0}}{z-\lambda}=\frac{\mathcal{L}(\mathds{E}[\varphi])(z)}{1-\mathcal{L}\mu(z)}-\sum_{\lambda\in\Lambda}\frac{a_{\lambda,0}}{z-\lambda}.$

The right-hand side, being holomorphic on
$\\{\mathrm{Re}(z)>\frac{\alpha}{2}-\varepsilon\\}$ (all the singularities are
removable), coincides with $\mathcal{L}r$ on that domain. On the other hand,
decreasing $\varepsilon$ if needed we can and do assume that
$\Lambda\subseteq\\{\mathrm{Re}(z)>\frac{\alpha}{2}+\varepsilon\\}$. In
particular, $\mathcal{L}h_{r}(z)$ is well-defined on
$\\{\mathrm{Re}(z)<\frac{\alpha}{2}+\varepsilon\\}$ and equal to
$\sum_{\lambda\in\Lambda}\frac{a_{\lambda,0}}{z-\lambda}$. As a result, on the
set
$\\{\frac{\alpha}{2}-\varepsilon<\mathrm{Re}(z)<\frac{\alpha}{2}+\varepsilon\\}$
we have
$\mathcal{L}h^{\varphi}(z)=\mathcal{L}r(z)+\mathcal{L}h_{r}(z)=\frac{\mathcal{L}(\mathds{E}[\varphi])(z)}{1-\mathcal{L}\mu(z)}$.

######  Remark 2.18.

Suppose that $\xi$ satisfies (A1) through (A4) and that the real-valued
characteristics $\varphi_{1},\ldots,\varphi_{d}$ satisfy (A5) and (A6).
Further, assume that each $m^{\varphi_{j}}_{t}$ satisfies (2.20) (with
coefficients $a_{\lambda,l}^{j}$ and remainder $r_{j}$ depending on $j$). Then
Theorem 2.14 gives joint convergence in distribution of the vector
$(\mathcal{Z}_{t}^{\varphi_{1}},\ldots,\mathcal{Z}_{t}^{\varphi_{d}})$.
Indeed, by the Cramér-Wold device, convergence in distribution of the vector
is equivalent to convergence of all linear combinations of the form

$\sum_{j=1}^{d}c_{j}\mathcal{Z}_{t}^{\varphi_{j}}=\mathcal{Z}_{t}^{\sum_{j=1}^{d}c_{j}\varphi_{j}}.$

A routine verification shows that the characteristic
$\sum_{j=1}^{d}c_{j}\varphi_{j}$ satisfies the assumptions of Theorem 2.14.

As a particular case of Remark 2.18 with $\varphi_{j}(\cdot)=\varphi(\cdot-
s_{j})$ for $-\infty<s_{1}<\ldots<s_{d}<\infty$ and a given random
characteristic $\varphi$, we get the following result for the finite-
dimensional distributions:

######  Corollary 2.19.

In the situation of Theorem 2.14 suppose that $\sigma^{2}\not=0$ or
$\rho_{n}\not=0$. Then for $d\vcentcolon=(2n+1)\vee 0$,

$t^{-\frac{d}{2}}e^{-\frac{\alpha}{2}t}\big{(}\mathcal{Z}^{\varphi}_{t-s}-H(t-s)\big{)}_{s\in\mathds{R}}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{f.d.}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sqrt{\tfrac{W}{\beta}}(G_{s})_{s\in\mathds{R}}$

where $(G_{s})_{s\in\mathds{R}}$ is a centered Gaussian process with the
covariance function

$\displaystyle\mathds{E}[G_{s}G_{u}]=\int\operatorname{\mathrm{Cov}}\big{[}\varphi(x-s)-h^{\varphi}*\xi(x-s),\varphi(x-u)-h^{\varphi}*\xi(x-u)\big{]}e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x})$

for any $s,u\in\mathds{R}$ if $d=0$, i.e., $n=-1$ and

$\displaystyle\mathds{E}[G_{s}G_{u}]$
$\displaystyle=\frac{1}{d}\sum_{\begin{subarray}{c}\lambda\in\partial\Lambda:\\\
k(\lambda)\geq
n+1\end{subarray}}\operatorname{\mathrm{Cov}}\bigg{[}\sum_{j=n}^{k(\lambda)-1}a_{\lambda,j}\binom{j}{n}\sum_{k=1}^{N}(-X_{k}-s)^{j-n}e^{-\lambda(X_{k}+s)},$
$\displaystyle\hphantom{=\sum_{\begin{subarray}{c}\lambda\in\partial\Lambda:\\\
k(\lambda)\geq
n+1\end{subarray}}\operatorname{\mathrm{Cov}}\bigg{[}}~{}\sum_{j=n}^{k(\lambda)-1}a_{\lambda,j}\binom{j}{n}\sum_{k=1}^{N}(-X_{k}-u)^{j-n}e^{-\lambda(X_{k}+u)}\bigg{]}$

if $d=2n+1$ with $n\geq 0$.

We close this section with a figure displaying the way to the proofs of our
main results.

Figure 2: Diagram representing the dependence of the results.

##  3 Applications

###  3.1 The Galton-Watson process

Consider a supercritical Galton-Watson branching process, i.e.,
$\xi=\sum_{k=1}^{N}\delta_{1}=N\delta_{1}$ where $N$ is a random variable
taking values in $\mathds{N}_{0}$ with
$m\vcentcolon=\mathds{E}[N]\in(1,\infty)$ and $\mathds{E}[N^{2}]<\infty$. Then
$\xi$ is lattice with span $1$. Further,

$\mathcal{L}\mu(\lambda)=\mathds{E}\bigg{[}\sum_{k=1}^{N}e^{-\lambda}\bigg{]}=me^{-\lambda},\quad\lambda\in\mathds{C}.$

The equation $\mathcal{L}\mu(z)=1$ is equivalent to $e^{z}=m$ and has only one
solution in the strip $\mathrm{Im}(z)\in(-\pi,\pi]$. In particular, (A1)
holds, i.e., there is a Malthusian parameter $\alpha>0$, namely, $\alpha=\log
m$, and $\Lambda_{\geq}=\\{\alpha\\}$. In this case the parameter $\beta$
defined by (2.7) is equal to 1. Then

$\mathds{E}\bigg{[}\bigg{(}\sum_{k=1}^{N}e^{-\theta}\bigg{)}^{\\!2}\bigg{]}=e^{-2\theta}\mathds{E}[N^{2}]<\infty\quad\text{for
all }\theta\in\mathds{R}.$

By Remark 2.1, this implies that (A2.8) and (A2.9) hold.

Consider the characteristic
$\phi_{\alpha}(t)\vcentcolon=e^{-\alpha}\mathds{1}_{[0,1)}(t)$. Then for any
$n\in\mathds{N}_{0}$, $\mathcal{Z}_{n}^{\phi_{\alpha}}$ is the number of
individuals in the $n^{\mathrm{th}}$ generation and the corresponding Nerman’s
martingale (2.10) is the size of $n^{\mathrm{th}}$ generation normalized by
its expectation $m^{n}$, i.e., $W_{n}=e^{-\alpha
n}\mathcal{Z}_{n}^{\phi_{\alpha}}$. Clearly, $\phi_{\alpha}$ satisfies (A4),
(A5) and (A6), and $r=0$. Therefore, we may apply the lattice version of
Theorem 2.14, which yields

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

where $\mathcal{N}$ is standard normal and independent of $W$. To calculate
$a_{\alpha}\vcentcolon=a_{\alpha,0}$, we use (2.20):
$m^{n}=\mathds{E}[\mathcal{Z}_{n}^{\phi_{\alpha}}]=a_{\alpha}m^{n}$, i.e.,
$a_{\alpha}=1$. Further, $\sigma>0$ is given by (2.27), i.e.,

$\displaystyle\sigma^{2}=\sum_{n\in\mathds{Z}}\operatorname{\mathrm{Var}}\big{[}\phi_{\alpha}(n)+h*\xi(n)\big{]}e^{-\alpha
n}=\sum_{n\in\mathds{Z}}\operatorname{\mathrm{Var}}\big{[}h*\xi(n)\big{]}m^{-n},$

where

$\displaystyle h(n)=m^{\phi_{\alpha}}_{n}-a_{\alpha}e^{\alpha
n}=m^{n}\mathds{1}_{\\{n\geq 0\\}}-m^{n}=-m^{n}\mathds{1}_{\\{n<0\\}}.$

Now, since $h*\xi(n)=Nh(n-1)$ we infer

$\displaystyle\sigma^{2}=\sum_{n<1}\operatorname{\mathrm{Var}}[N]m^{2n-2}m^{-n}=\frac{\operatorname{\mathrm{Var}}[N]}{m^{2}-m}.$

Consequently,

$\displaystyle
m^{\frac{n}{2}}\big{(}W_{n}-W\big{)}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\Big{(}\frac{\operatorname{\mathrm{Var}}[N]W}{m^{2}-m}\Big{)}^{\frac{1}{2}}\mathcal{N}\quad\text{as
}n\to\infty.$

We have thus just rediscovered Heyde’s classical central limit theorem for the
martingale in the Galton-Watson process [18].

We can also deal with the total number of individuals in the generations
$0,\ldots,n$. Indeed, this number is $\mathcal{Z}_{n}^{f}$ for
$f(t)\vcentcolon=\mathds{1}_{[0,\infty)}(t)$, $t\in\mathds{R}$, which
satisfies (A4), (A5) and (A6). Thus, again as $r=0$, Theorem 2.14 gives

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

This time $a_{\alpha}$ can be computed with the help of (2.20) as follows. We
have the asymptotic expansion

$m_{n}^{f}=\mathds{E}[\mathcal{Z}_{n}^{f}]=\sum_{k=0}^{n}m^{k}=\frac{m^{n+1}-1}{m-1}=\frac{m}{m-1}e^{\alpha
n}-\frac{1}{m-1},$

for $n\geq 0$ and 0 otherwise. Consequently, $a_{\alpha}=\frac{m}{m-1}$ and
thereupon

$\displaystyle
m^{-\frac{n}{2}}\Big{(}\mathcal{Z}_{n}^{f}-\frac{m^{n+1}}{m-1}W\Big{)}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sigma\sqrt{W}\mathcal{N}\quad\text{as
}n\to\infty.$

This time $\sigma>0$ is given by
$\sigma^{2}=\sum_{n\in\mathds{Z}}\operatorname{\mathrm{Var}}[N]|h(n-1)|^{2}m^{-n}$
with

$\displaystyle h(n)=m_{n}^{f}-\frac{m}{m-1}e^{\alpha n}.$

Therefore,

$\displaystyle\sigma^{2}$
$\displaystyle=\operatorname{\mathrm{Var}}[N]\Big{(}\sum_{n<1}\Big{(}\frac{m}{m-1}m^{n-1}\Big{)}^{2}m^{-n}+\sum_{n\geq
1}\Big{(}\frac{1}{m-1}\Big{)}^{2}m^{-n}\Big{)}$
$\displaystyle=\frac{1}{(m-1)^{2}}\operatorname{\mathrm{Var}}[N]\bigg{(}\sum_{n\leq
0}m^{n}+\sum_{n>0}m^{-n}\bigg{)}=\frac{m+1}{(m-1)^{3}}\operatorname{\mathrm{Var}}[N].$

###  3.2 Nerman’s martingales

Suppose that $\xi$ is non-lattice and satisfies (A1) through (A2.9), and let
$\lambda=\theta+\mathrm{i}\eta$ be a root to $\mathcal{L}\mu(z)=1$ with
$0\leq\mathrm{Re}(\lambda)=\theta<\frac{\alpha}{2}$. Further, suppose that

$\displaystyle\mathds{E}\bigg{[}\bigg{(}\sum_{k=1}^{N}e^{-\theta
X_{k}}\bigg{)}^{\\!\\!2}\bigg{]}<\infty.$ (3.1)

For simplicity let $Z_{1}(\lambda)\vcentcolon=\sum_{k=1}^{N}e^{-\lambda
X_{k}}$. We can view the complex variable $Z_{1}(\lambda)$ as a random
variable taking values in $\mathds{R}^{2}$. We denote by $\Sigma^{\lambda}$
the corresponding covariance matrix. The aforementioned condition guarantees
that $\Sigma^{\lambda}$ is well-defined.

Let $(W_{t}(\lambda))_{t\geq 0}$ be defined by (2.17) for $j=0$. But, since
$\mathrm{Re}(\lambda)<\frac{\alpha}{2}$, we cannot apply Theorem 2.6, though
one can still wonder what the long-term behavior of the process is. To analyze
this, we shall apply a special case of our main result, Theorem 2.14. Let

$\phi(t)\vcentcolon=e^{\lambda
t}\sum_{j=1}^{N}\mathds{1}_{[0,X_{j})}(t)e^{-\lambda X_{j}},\quad
t\in\mathds{R}.$

Then

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

and, by (3.1), we conclude that the functions

$\displaystyle t\mapsto
e^{-(\alpha-p\theta)t}\mathds{1}_{[0,\infty)}(t)\mathds{E}\bigg{[}\bigg{|}\sum_{j=1}^{N}e^{-\theta
X_{j}}\bigg{|}^{p}\bigg{]}\quad\text{with }p=1,2$

are directly Riemann integrable. By [40, Remark 3.10.5], characteristics
$\phi,\mathrm{Re}(\phi)$ and $\mathrm{Im}(\phi)$ fulfill (A4), (A5) and (A6).
Remark 5.3 below (applied to
$\phi=\mathsf{e}_{1}^{\mathsf{T}}\phi_{\lambda}\mathsf{e}_{1}$, where
$\mathsf{e}_{1}^{\mathsf{T}}$ is the transpose of $\mathsf{e}_{1}$) gives that

$\mathcal{Z}^{\phi}_{t}=e^{\lambda t}W_{t}(\lambda)$

and $\mathds{E}[\mathcal{Z}^{\phi}_{t}]=e^{\lambda
t}\mathds{1}_{[0,\infty)}(t)$.

By Theorem 6.6, we deduce

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

or equivalently

$(\mathrm{Re}(e^{(\lambda-\frac{\alpha}{2})t}W_{t}(\lambda)),\mathrm{Im}(e^{(\lambda-\frac{\alpha}{2})t}W_{t}(\lambda)))\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sqrt{\tfrac{W}{\beta}}\mathcal{N}$

where $\mathcal{N}$ is a $2$-dimensional centered Gaussian vector with
covariance matrix $\Sigma$, which can be explicitly computed. Indeed, we have

$\displaystyle\Sigma_{11}$
$\displaystyle\vcentcolon=\int\operatorname{\mathrm{Var}}\Big{[}\mathrm{Re}(\phi_{\lambda}(x))+\sum_{j=1}^{N}\mathrm{Re}(e^{\lambda(x-X_{j})})\mathds{1}_{[0,\infty)}(x-X_{j})\Big{]}e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x})$
$\displaystyle=\int\limits_{0}^{\infty}\operatorname{\mathrm{Var}}\big{[}\mathrm{Re}(e^{\lambda
x}Z_{1}(\lambda))\big{]}e^{-\alpha x}\,\mathrm{d}\mathit{x}$
$\displaystyle=\int\limits_{0}^{\infty}e^{2\theta x}\Big{(}\cos^{2}(\eta
x)\Sigma_{11}^{\lambda}-2\sin(\eta x)\cos(\eta
x)\Sigma_{12}^{\lambda}+\sin^{2}(\eta x)\Sigma_{22}^{\lambda}\Big{)}e^{-\alpha
x}\,\mathrm{d}\mathit{x}$ $\displaystyle=$
$\displaystyle\frac{\Sigma_{11}^{\lambda}}{\alpha-2\theta}\frac{2\eta^{2}+(\alpha-2\theta)^{2}}{4\eta^{2}+(\alpha-2\theta)^{2}}-{\Sigma_{12}^{\lambda}}\frac{2\eta}{4\eta^{2}+(\alpha-2\theta)^{2}}+\frac{\Sigma_{22}^{\lambda}}{\alpha-2\theta}\frac{2\eta^{2}}{4\eta^{2}+(\alpha-2\theta)^{2}}.$

Similarly,

$\displaystyle\Sigma_{22}$
$\displaystyle=\int\limits_{0}^{\infty}\operatorname{\mathrm{Var}}\big{[}\mathrm{Im}(e^{\lambda
x}Z_{1}(\lambda))\big{]}e^{-\alpha x}\,\mathrm{d}\mathit{x}$
$\displaystyle=\frac{\Sigma_{22}^{\lambda}}{\alpha-2\theta}\frac{2\eta^{2}+(\alpha-2\theta)^{2}}{4\eta^{2}+(\alpha-2\theta)^{2}}+{\Sigma_{12}^{\lambda}}\frac{2\eta}{4\eta^{2}+(\alpha-2\theta)^{2}}+\frac{\Sigma_{11}^{\lambda}}{\alpha-2\theta}\frac{2\eta^{2}}{4\eta^{2}+(\alpha-2\theta)^{2}},$

and

$\displaystyle\Sigma_{12}$
$\displaystyle=\int\limits_{0}^{\infty}\operatorname{\mathrm{Cov}}\big{[}\mathrm{Re}(e^{\lambda
x}Z_{1}(\lambda)),\mathrm{Im}(e^{\lambda x}Z_{1}(\lambda))\big{]}e^{-\alpha
x}\,\mathrm{d}\mathit{x}$
$\displaystyle=\frac{(\Sigma_{11}^{\lambda}-\Sigma_{22}^{\lambda})\eta}{4\eta^{2}+(\alpha-2\theta)^{2}}+{\Sigma_{12}^{\lambda}}\frac{\alpha-2\theta}{4\eta^{2}+(\alpha-2\theta)^{2}}.$

###  3.3 Epidemic models

In this section, we consider the epidemic model discussed in [12]. In this
model, the role of the ancestor is that of the first person in a community
infected by an infectious disease. Birth events become infection events etc.

Suppose that $\xi$ is a Poisson point process on $[0,\infty)$ with intensity
measure $R_{0}g(x)\,\mathrm{d}\mathit{x}$ where

$g(x)=\mathds{1}_{(0,\infty)}(x)\frac{b^{a}x^{a-1}}{\Gamma(a)}e^{-bx},\quad
x\in\mathds{R}$

is the density of the Gamma distribution with parameters $a,b>0$ and $R_{0}>1$
is the basic reproduction mean. (No additional difficulties would occur if
$R_{0}$ was replaced by a positive random variable $N$ with mean $R_{0}$ and
finite variance.) The function $g$ is the infection rate scaled to become a
probability density. It models the time delay between the infection of a
person and a random person infected by that person. Characteristics of
interest are $I(t)=R_{0}g(t)$ and $f(t)=\mathds{1}_{[0,\infty)}(t)$ with
$\mathcal{Z}_{t}^{I}$ being the incidence at time $t$ and
$\mathcal{Z}_{t}^{f}$ the number of infections up to time $t$. In the given
situation, the Laplace transform $\mathcal{L}\mu$ can be calculated explicitly
in terms of $a,b$ and $R_{0}$, namely,

$\displaystyle\mathcal{L}\mu(\lambda)$
$\displaystyle=\int_{0}^{\infty}e^{-\lambda
x}\,\mu(\mathrm{d}\mathit{x})=R_{0}\Big{(}\frac{b}{b+\lambda}\Big{)}^{\\!a},\quad\mathrm{Re}(\lambda)>-b.$

Hence the equation $\mathcal{L}\mu(\lambda)=1$ takes the form

$R_{0}\Big{(}\frac{b}{b+\lambda}\Big{)}^{a}=1.$

Write $\frac{b}{b+\lambda}=re^{\mathrm{i}\varphi}$ with $r>0$ and
$|\varphi|<\pi$. Then $R_{0}(re^{\mathrm{i}\varphi})^{a}=1$ is equivalent to

$e^{-\mathrm{i}a\varphi}=r^{a}R_{0}.$

This implies $r=R_{0}^{-1/a}$ and
$\varphi\in(2\pi/a)\mathds{Z}\cap(-\pi,\pi)$. Solving for $\lambda$ yields

$\lambda=b(R_{0}^{1/a}e^{-\mathrm{i}\varphi}-1)$ (3.2)

with $\varphi\in(2\pi/a)\mathds{Z}\cap(-\pi,\pi)$, cf. Figure 3. The
Malthusian parameter is obtained by setting $\varphi=0$, i.e.,

$\alpha=b(R_{0}^{1/a}-1).$

The real part of a root $\lambda$ as in (3.2) is given by

$\mathrm{Re}(\lambda)=b(R_{0}^{1/a}\cos\varphi-1).$ (3.3)

A second root exists only if $a>2$ (otherwise
$(2\pi/a)\mathds{Z}\cap(-\pi,\pi)=\\{0\\}$), in which case the root
$\lambda\not=\alpha$ with largest real part is
$\lambda=b(R_{0}^{1/a}e^{\mathrm{i}2\pi/a}-1)$ with

$\mathrm{Re}(\lambda)=b(R_{0}^{1/a}\cos(\tfrac{2\pi}{a})-1).$

This is clearly negative unless $a>4$ since for $a\in(2,4]$, we have
$\cos(\tfrac{2\pi}{a})\leq 0$. In the case $a>4$, we have

$\mathrm{Re}(\lambda)=b(R_{0}^{1/a}\cos(\tfrac{2\pi}{a})-1)\geq\frac{\alpha}{2}=\frac{b}{2}(R_{0}^{1/a}-1)$

if and only if $a>6$ and $R_{0}\geq
R_{0}(a)\vcentcolon=(2\cos(\frac{2\pi}{a})-1)^{-a}$. Notice that
$R_{0}(a)\to\infty$ for $a\downarrow 6$ and $R_{0}(a)\to 1$ for $a\to\infty$.
If $R_{0}<R_{0}(a)$, Theorem 2.7 applies and yields Gaussian fluctuations of
$\mathcal{Z}_{t}^{I}$ and $\mathcal{Z}_{t}^{f}$. That is, for
$\mathcal{Z}_{t}^{f}$ we have

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

with $\beta\vcentcolon=R_{0}ab^{a}(b+\alpha)^{-a-1}$ and
$a_{\alpha}\vcentcolon=(\alpha\beta)^{-1}$. Left with calculating $\sigma$ we
obtain with the help of Remark 2.17

$\displaystyle\sigma^{2}$
$\displaystyle=\frac{1}{2\pi}\int\limits_{\mathrm{Re}(z)=\frac{\alpha}{2}}\operatorname{\mathrm{Var}}\Big{[}\mathcal{L}f(z)+\mathcal{L}\xi(z)\frac{\mathcal{L}f(z)}{1-\mathcal{L}\mu(z)}\Big{]}\,|\mathrm{d}z|$
$\displaystyle=\frac{1}{2\pi}\int\limits_{\mathrm{Re}(z)=\frac{\alpha}{2}}\Big{|}\frac{1}{z(1-\mathcal{L}\mu(z))}\Big{|}^{2}\operatorname{\mathrm{Var}}[\mathcal{L}\xi(z)]\,|\mathrm{d}z|$
$\displaystyle=\frac{1}{2\pi}\int\limits_{\mathrm{Re}(z)=\frac{\alpha}{2}}\Big{|}\frac{1}{z(1-\mathcal{L}\mu(z))}\Big{|}^{2}\mathcal{L}\mu(\alpha)\,|\mathrm{d}z|$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{4}{(\alpha^{2}+4t^{2})g(t)}\,\mathrm{d}t,$

with
$g(t)\vcentcolon=1+R_{0}^{2}\Big{(}\frac{4b^{2}}{(2b+\alpha)^{2}+4t^{2}}\Big{)}^{a}-2R_{0}\Big{(}\frac{4b^{2}}{(2b+\alpha)^{2}+4t^{2}}\Big{)}^{\frac{a}{2}}\cos\Big{(}a\arctan\Big{(}\frac{2t}{2b+\alpha}\Big{)}\Big{)}$.

If $R_{0}\geq R_{0}(a)$, the more general Theorem 2.14 applies and gives
additional periodic fluctuations of greater magnitude than the Gaussian
fluctuations. We refrain from providing further details.

$\theta$$\eta$$\alpha$$\tfrac{\alpha}{2}$

$\theta$$\eta$$\alpha$$\tfrac{\alpha}{2}$

Figure 3: Solutions to $\mathcal{L}\mu(\lambda)=1$ in the cases $a=18$, $b=1$,
$R_{0}=10$ (left figure) and $a=18$, $b=1$, $R_{0}=12$. In the left figure,
the root $\lambda\not=\alpha$ with largest real part has
$\mathrm{Re}(\lambda)<\frac{\alpha}{2}$, in the right figure
$\mathrm{Re}(\lambda)>\frac{\alpha}{2}$.

###  3.4  Crump-Mode-Jagers processes with homogeneous Poisson offspring
process and general lifetimes

In this section, we assume that

$\xi\vcentcolon=\sum_{j\geq 1}\mathds{1}_{[0,\zeta]}(P_{j})\delta_{P_{j}}$

where $(P_{j})_{j\geq 1}$ are the arrival times of a homogeneous Poisson
process with intensity $b>0$, independent of the $[0,\infty]$-valued random
variable $\zeta$. We are interested in
$\mathcal{Z}^{\mathds{1}_{[0,\zeta)}}_{t}$ the number of individuals alive at
time $t$, see (2.3). Thus, the corresponding characteristic $\phi$ is given by
$\phi(t)\vcentcolon=\mathds{1}_{[0,\zeta)}(t)$ for $t\geq 0$.

We put $\mathcal{L}_{\zeta}(z)\vcentcolon=\mathds{E}[e^{-z\zeta}]$ for
$\mathrm{Re}(z)\geq 0$ and start by noting that

$\displaystyle\mathcal{L}\mu(z)$ $\displaystyle=\mathds{E}\bigg{[}\sum_{j\geq
1}\mathds{1}_{[0,\zeta]}(P_{j})e^{-zP_{j}}\bigg{]}=\mathds{E}\bigg{[}\int\mathds{1}_{[0,\zeta]}(x)e^{-zx}b\mathrm{d}\mathit{x}\bigg{]}=\int\limits_{0}^{\infty}e^{-zx}b\mathds{P}(\zeta\geq
x)\,\mathrm{d}\mathit{x}$
$\displaystyle=\frac{b(1-\mathcal{L}_{\zeta}(z))}{z},\quad\mathrm{Re}(z)>0.$

The Malthusian parameter $\alpha$ is the unique real number that satisfies

$\displaystyle 1-\mathcal{L}_{\zeta}(\alpha)=\frac{\alpha}{b},$

and the parameter $\beta$ is given by

$\beta=\frac{1}{\alpha}(1+b\mathcal{L}_{\zeta}^{\prime}(\alpha))=\frac{1}{\alpha}(1-b\mathds{E}[\zeta
e^{-\alpha\zeta}]).$

Now we shall show that $\alpha$ is the only solution to $\mathcal{L}\mu(z)=1$
with $\mathrm{Re}(z)>0$. Indeed, for positive $\theta$ and $\eta$

$\displaystyle\mathrm{Im}$
$\displaystyle\big{(}\mathcal{L}\mu(\theta+\mathrm{i}\eta)\big{)}=\int_{0}^{\infty}\sin(\eta
x)e^{-\theta x}b\mathds{P}(\zeta\geq x)\mathrm{d}\mathit{x}$
$\displaystyle=b\sum_{l\geq 0}\int\limits_{2\pi
l/\eta}^{\pi(2l+1)/\eta}\sin(\eta x)\Big{(}e^{-\theta x}\mathds{P}(\zeta\geq
x)-e^{-\theta(x+\pi/\eta)}\mathds{P}(\zeta\geq
x+\pi/\eta)\Big{)}\mathrm{d}\mathit{x}>0$

which, together with
$\mathcal{L}\mu(\overline{\lambda})=\overline{\mathcal{L}\mu(\lambda)}$, shows
that $\Lambda_{\geq}\cap\\{z:\mathrm{Re}(z)>0\\}=\\{\alpha\\}$. Since
$\mathcal{L}(\mathds{E}[\phi])(\alpha)=b^{-1}\mathcal{L}\mu(\alpha)=b^{-1}$,
an application of Theorem 2.7 yields

$\displaystyle
e^{-\frac{\alpha}{2}t}\Big{(}\mathcal{Z}^{\mathds{1}_{[0,\zeta)}}_{t}-e^{\alpha
t}\frac{W}{b\beta}\Big{)}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sigma\sqrt{\tfrac{W}{\beta}}\mathcal{N}.$

Next, we express the variance $\sigma^{2}$ in terms of the parameters $b$,
$\alpha$ and $\beta$. By Remark 2.17

$\displaystyle\sigma^{2}=\frac{1}{2\pi}\int\limits_{\mathrm{Re}(z)=\frac{\alpha}{2}}\operatorname{\mathrm{Var}}\Big{[}\mathcal{L}\phi(z)+\mathcal{L}\xi(z)\frac{\mathcal{L}(\mathds{E}[\phi])(z)}{1-\mathcal{L}\mu(z)}\Big{]}\,|\mathrm{d}z|.$

For $\mathrm{Re}(z)>0$

$\displaystyle\operatorname{\mathrm{Var}}\Big{[}\mathcal{L}\phi(z)+\mathcal{L}\xi(z)\frac{\mathcal{L}(\mathds{E}[\phi])(z)}{1-\mathcal{L}\mu(z)}\Big{]}$
$\displaystyle=\mathds{E}\Big{[}\operatorname{\mathrm{Var}}\Big{[}\mathcal{L}\phi(z)+\mathcal{L}\xi(z)\frac{\mathcal{L}\mu(z)}{b(1-\mathcal{L}\mu(z))}\Big{|}{\zeta}\Big{]}\Big{]}$
$\displaystyle\hphantom{=}+\operatorname{\mathrm{Var}}\Big{[}\mathds{E}\Big{[}\mathcal{L}\phi(z)+\mathcal{L}\xi(z)\frac{\mathcal{L}\mu(z)}{b(1-\mathcal{L}\mu(z))}\Big{|}\zeta\Big{]}\Big{]}=\vcentcolon\mathit{I}+\mathit{II}$

with

$\displaystyle\mathit{I}$
$\displaystyle=\mathds{E}\Big{[}\operatorname{\mathrm{Var}}\Big{[}\mathcal{L}\xi(z)\frac{\mathcal{L}\mu(z)}{b(1-\mathcal{L}\mu(z))}\Big{|}\zeta\Big{]}\Big{]}$
$\displaystyle=\mathds{E}\big{[}\operatorname{\mathrm{Var}}\big{[}\mathcal{L}\xi(z)\big{|}\zeta\big{]}\big{]}\Big{|}\frac{\mathcal{L}\mu(z)}{b(1-\mathcal{L}\mu(z))}\Big{|}^{2}=\mathds{E}\Big{[}\int_{0}^{\zeta}e^{-2\mathrm{Re}(z)x}b\,\mathrm{d}x\Big{]}\Big{|}\frac{\mathcal{L}\mu(z)}{b(1-\mathcal{L}\mu(z))}\Big{|}^{2},$

where we have used properties of the Poisson process, and

$\displaystyle\mathit{II}$
$\displaystyle=\operatorname{\mathrm{Var}}\Big{[}\frac{1-e^{-z\zeta}}{z}+\frac{b(1-e^{-z\zeta})}{z}\frac{\mathcal{L}\mu(z)}{b(1-\mathcal{L}\mu(z))}\Big{]}=\Big{|}\frac{1}{z(1-\mathcal{L}\mu(z))}\Big{|}^{2}\operatorname{\mathrm{Var}}[e^{-z\zeta}].$

Assuming now that $\mathrm{Re}(z)=\frac{\alpha}{2}$ we arrive at

$\displaystyle\mathit{I}=\Big{|}\frac{\mathcal{L}\mu(z)}{b(1-\mathcal{L}\mu(z))}\Big{|}^{2}$

because

$\displaystyle\mathds{E}\Big{[}\int_{0}^{\zeta}e^{-2\mathrm{Re}(z)x}b\,\mathrm{d}x\Big{]}=\mathds{E}\Big{[}\int_{0}^{\zeta}e^{-\alpha
x}b\,\mathrm{d}x\Big{]}=1,$

and

$\mathit{II}=\Big{|}\frac{1}{z(1-\mathcal{L}\mu(z))}\Big{|}^{2}\big{(}\mathcal{L}_{\zeta}(\alpha)-|\mathcal{L}_{\zeta}(z)|^{2}\big{)}.$

Observing that $\overline{z}=\alpha-z$ and
$\overline{\mathcal{L}\mu(z)}=\mathcal{L}\mu(\alpha-z)$ whenever
$\mathrm{Re}(z)=\frac{\alpha}{2}$, we further infer

$\displaystyle\Big{|}\frac{z\mathcal{L}\mu(z)}{b}\Big{|}^{2}+\mathcal{L}_{\zeta}(\alpha)-|\mathcal{L}_{\zeta}(z)|^{2}$
$\displaystyle=|1-\mathcal{L}_{\zeta}(z)|^{2}+1-\frac{\alpha}{b}-|\mathcal{L}_{\zeta}(z)|^{2}$
$\displaystyle=1-\mathcal{L}_{\zeta}(z)+\overline{1-\mathcal{L}_{\zeta}(z)}-\tfrac{\alpha}{b}$
$\displaystyle=\tfrac{1}{b}\big{(}z(\mathcal{L}\mu(z)-1)+(\alpha-z)(\mathcal{L}\mu(\alpha-z)-1)\big{)},$

which in turn gives

$\displaystyle\mathit{I}+\mathit{II}$
$\displaystyle=\Big{|}\frac{1}{z(1-\mathcal{L}\mu(z))}\Big{|}^{2}\Big{(}\Big{|}\frac{z\mathcal{L}\mu(z)}{b}\Big{|}^{2}+\mathcal{L}_{\zeta}(\alpha)-|\mathcal{L}_{\zeta}(z)|^{2}\Big{)}$
$\displaystyle=\frac{z(\mathcal{L}\mu(z)-1)+(\alpha-z)(\mathcal{L}\mu(\alpha-z)-1)}{bz(\mathcal{L}\mu(z)-1)(\alpha-z)(\mathcal{L}\mu(\alpha-z)-1)}$
$\displaystyle=\frac{1}{b(\alpha-z)(\mathcal{L}\mu(\alpha-z)-1)}+\frac{1}{bz(\mathcal{L}\mu(z)-1)}$
$\displaystyle=\frac{2}{b}\mathrm{Re}\Big{(}\frac{1}{z(\mathcal{L}\mu(z)-1)}\Big{)}.$

We can now compute the variance as follows

$\displaystyle\sigma^{2}=\lim_{R\to\infty}\frac{1}{b\pi}\mathrm{Re}\bigg{(}\int\limits_{\frac{\alpha}{2}-\mathrm{i}R}^{\frac{\alpha}{2}+\mathrm{i}R}\frac{1}{z(\mathcal{L}\mu(z)-1)}\,|\mathrm{d}z|\bigg{)}=\frac{1}{b\pi}\lim_{R\to\infty}\mathrm{Im}\bigg{(}\int\limits_{\frac{\alpha}{2}-\mathrm{i}R}^{\frac{\alpha}{2}+\mathrm{i}R}\frac{1}{z(\mathcal{L}\mu(z)-1)}\,\mathrm{d}z\bigg{)}.$

To calculate the limit, we use the residue theorem. For $R>\alpha$,

$\displaystyle\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{Re^{\mathrm{i}\theta}\mathrm{d}\theta}{(\frac{\alpha}{2}+Re^{\mathrm{i}\theta})(\mathcal{L}\mu(\frac{\alpha}{2}+Re^{\mathrm{i}\theta})-1)}-\int\limits_{\frac{\alpha}{2}-\mathrm{i}R}^{\frac{\alpha}{2}+\mathrm{i}R}\frac{\mathrm{d}z}{z(\mathcal{L}\mu(z)-1)}=2\pi\mathrm{i}\,\underset{z=\alpha}{\operatorname{\mathrm{Res}}}\frac{1}{z(\mathcal{L}\mu(z)-1)}.$

It suffices to show that the imaginary part of the integrand of the first
integral decays to zero uniformly in $\theta$ as $R$ goes to infinity. In view
of the inequality $|\mathcal{L}\mu(z)|\leq 2b|z|^{-1}$ and its consequence
$|\mathcal{L}\mu(z)-1|\geq 1-2b|z|^{-1}$ (both hold true for
$\mathrm{Re}(z)\geq 0$) we conclude that

$\displaystyle\frac{Re^{\mathrm{i}\theta}}{(\frac{\alpha}{2}+Re^{\mathrm{i}\theta})(\mathcal{L}\mu(\frac{\alpha}{2}+Re^{\mathrm{i}\theta})-1)}=\frac{{(\frac{\alpha}{2}+Re^{-\mathrm{i}\theta})(\mathcal{L}\mu(\frac{\alpha}{2}+Re^{-\mathrm{i}\theta})-1)}Re^{\mathrm{i}\theta}}{\big{|}(\frac{\alpha}{2}+Re^{\mathrm{i}\theta})(\mathcal{L}\mu(\frac{\alpha}{2}+Re^{\mathrm{i}\theta})-1)\big{|}^{2}},$

whence

$\displaystyle\mathrm{Im}\Big{(}\frac{Re^{\mathrm{i}\theta}}{(\frac{\alpha}{2}+Re^{\mathrm{i}\theta})(\mathcal{L}\mu(\frac{\alpha}{2}+Re^{\mathrm{i}\theta})-1)}\Big{)}=O(R^{-1})$

as $R\to\infty$ uniformly in $\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]$.
Finally,

$\displaystyle\sigma^{2}=\frac{2}{\alpha b\beta}$

and thereupon

$\displaystyle
e^{-\frac{\alpha}{2}t}\Big{(}\mathcal{Z}^{\mathds{1}_{[0,\zeta)}}_{t}-e^{\alpha
t}\frac{W}{b\beta}\Big{)}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sqrt{\frac{2W}{\alpha
b\beta^{2}}}\mathcal{N}.$

###  3.5 The conservative fragmentation model

In this section we consider the conservative fragmentation model as discussed
in [28]. Let $b\geq 2$ be integer and $(V_{1},V_{2},\ldots,V_{b})$ a vector of
nonnegative random variables such that $\sum_{j=1}^{b}V_{j}=1$ a. s. For the
sake of simplicity, we assume that $V_{1},\ldots,V_{b}$ have Lebesgue
densities except possible atoms at $0$. (Our theory would allow to cover more
general cases, too.) Starting with an object of mass $x\geq 1$, we break it
into pieces with masses $(V_{1}x,V_{2}x,\ldots,V_{b}x)$. Continue recursively
with each piece of mass $\geq 1$, using new and independent copies of the
random vector $(V_{1},V_{2},\ldots,V_{b})$ each time. Once a fragment has mass
$<1$, it is not further crumbled. The process terminates a. s. after a finite
number of steps, leaving a finite set of fragments of masses $<1$.

Denote by $n(x)$ the random number of fragmentation events, i.e., the number
of pieces of mass $\geq 1$ that appear during the process. Further, let
$n_{e}(x)$ be the final number of fragments, i.e., the number of pieces of
mass $<1$ that appear. A limit theorem for $n(x)$ has been proved in [28],
where it was shown that the asymptotic behavior of $n(x)$ as $x$ goes to
infinity depends on the position of the roots of the function
$z\mapsto\sum_{j\geq 1}\mathds{E}[V_{j}^{z}]$.

Letting $\xi\vcentcolon=\sum_{j=1}^{b}\mathds{1}_{\\{V_{j}>0\\}}\delta_{-\log
V_{j}}$, we conclude that the corresponding Malthusian parameter is 1, i.e.,
$\alpha=1$ and the limit of Nerman’s martingale satisfies $W=1$ a. s. Note
also that $n(x)=N(\log x)$ corresponds to the number of individuals born up to
and including time $\log x$ and similarly, we can represent $n_{e}(x)$ as a
general branching process, namely, $n_{e}(x)=\mathcal{Z}^{\varphi}_{\log x}$,
with

$\varphi(t)\vcentcolon=\sum_{j=1}^{b}\mathds{1}_{\\{V_{j}>0\\}}\mathds{1}_{[0,-\log
V_{j})}(t)\quad\text{for }t\in\mathds{R}.$

Hence, our main result provides (precise) limit theorems for both $n$ and
$n_{e}$. For instance, in the case when all root from $\Lambda$ are simple, we
infer from Theorem 2.8 (the constants $b_{\lambda,0}$ in the theorem can
easily be seen to equal $b_{\lambda,0}=-1/(\mathcal{L}\mu)^{\prime}(\lambda)$,
$\lambda\in\Lambda$ by Proposition 7.9)

$\displaystyle x^{-1/2}$
$\displaystyle\Big{(}n(x)+\sum_{\lambda\in\Lambda}\tfrac{W(\lambda)}{\lambda(\mathcal{L}\mu)^{\prime}(\lambda)}x^{\lambda}\Big{)}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\sigma\mathcal{N}\quad\text{if
}\partial\Lambda\text{ is empty and}$ $\displaystyle x^{-1/2}\big{(}\log
x\big{)}^{-k+1/2}$
$\displaystyle\Big{(}n(x)+\sum_{\lambda\in\Lambda}\tfrac{W(\lambda)}{\lambda(\mathcal{L}\mu)^{\prime}(\lambda)}x^{\lambda}\Big{)}\mathrel{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\mathrm{d}$\cr\kern-0.21529pt\cr$\to$\cr}}}\rho_{k-1}\mathcal{N}\quad\text{if
}\partial\Lambda\text{ is non-empty},$

where $k$ is the largest multiplicity of a root on the critical line
$\mathrm{Re}(z)=\frac{\alpha}{2}$.

##  4 Preliminaries for the proofs of the main results

In this section we gather facts from the literature, introduce some notation
used throughout the paper and perform some basic calculations.

###  4.1 Change of measure and the connection to renewal theory

The existence of the Malthusian parameter (i.e., (2.6)) enables us to use a
change-of-measure argument as follows. We define a random walk
$(S_{n})_{n\in\mathds{N}_{0}}$ with $S_{0}=0$ on some probability space with
underlying probability measure $\mathsf{P}$ and increment distribution given
by

$\displaystyle\mathsf{P}(S_{1}\in B)=\mathds{E}\bigg{[}\sum_{|u|=1}e^{-\alpha
S(u)}\mathds{1}_{B}(S(u))\bigg{]}=\int\limits_{B}e^{-\alpha
x}\,\mu(\mathrm{d}\mathit{x}),\qquad B\in\mathcal{B}(\mathds{R}).$ (4.1)

With this definition, the many-to-one formula (see, e.g., [43, Theorem 1.1])
holds:

$\displaystyle\mathsf{E}[f(S_{1},\ldots,S_{n})]$
$\displaystyle=\mathds{E}\bigg{[}\sum_{|u|=n}e^{-\alpha
S(u)}f(S(u|_{1}),\ldots,S(u))\bigg{]}$ (4.2)

for all Borel measurable $f:\mathds{R}^{n}\to\mathds{R}$ such that the
expectation on the left- or right-hand side of (4.2) is well-defined, possibly
infinite. In particular, under (2.7)

$\displaystyle\mathsf{E}[S_{1}]$ $\displaystyle=\beta\in(0,\infty).$ (4.3)

In other words, the increments of the random walk
$(S_{n})_{n\in\mathds{N}_{0}}$ have positive, finite mean. As a consequence,
the associated renewal measure

$\mathsf{U}(\cdot)=\sum_{n\in\mathds{N}_{0}}\mathsf{P}(S_{n}\in\cdot)$

is uniformly locally finite in the sense that

$\mathsf{U}([t,t+h])\leq\mathsf{U}([0,h])<\infty\qquad\text{for all }t,h\geq
0.$ (4.4)

Indeed, if $\tau\vcentcolon=\inf\\{n\in\mathds{N}_{0}:S_{n}\geq t\\}$, then

$\sum_{n\geq 0}\mathds{1}_{[t,t+h]}(S_{n})=\sum_{n\geq
0}\mathds{1}_{[t,t+h]}(S_{\tau+n})\leq\sum_{n\geq
0}\mathds{1}_{[0,h]}(S_{\tau+n}-S_{\tau}).$

Now take expectations and use the strong Markov property at $\tau$ to infer
(4.4).

By the many-to-one formula, (A2.8) implies that the increments of the
associated random walk $(S_{n})_{n\in\mathds{N}_{0}}$ have a finite
exponential moment of order $\alpha-\vartheta>\alpha/2$ since

$\displaystyle\mathsf{E}[e^{(\alpha-\vartheta)S_{1}}]$
$\displaystyle=\mathds{E}\bigg{[}\sum_{j=1}^{N}e^{-\vartheta
X_{j}}\bigg{]}=\mathcal{L}\mu(\vartheta)<\infty.$ (4.5)

###  4.2 The expectation of the general branching process

There is a connection between the renewal measure $\mathsf{U}$ and the
expectation $m_{t}^{\varphi}=\mathds{E}[\mathcal{Z}_{t}^{\varphi}]$ of the
general branching process counted with characteristic $\varphi$ provided that
$\varphi$ satisfies suitable assumptions. For instance, if $\varphi$ is
nonnegative and $t\mapsto\mathds{E}[\varphi(t)]e^{-\alpha t}$ is a directly
Riemann integrable function, then we infer from the many-to-one formula

$\displaystyle m_{t}^{\varphi}e^{-\alpha t}$
$\displaystyle\vcentcolon=\mathds{E}[\mathcal{Z}_{t}^{\varphi}]e^{-\alpha
t}=\sum_{n=0}^{\infty}\mathds{E}\bigg{[}\sum_{|u|=n}e^{-\alpha
S(u)}\varphi_{u}(t-S(u))e^{-\alpha(t-S(u))}\bigg{]}$
$\displaystyle=\sum_{n=0}^{\infty}\mathds{E}\bigg{[}\sum_{|u|=n}e^{-\alpha
S(u)}\mathds{E}[\varphi](t-S(u))e^{-\alpha(t-S(u))}\bigg{]}$
$\displaystyle=\sum_{n=0}^{\infty}\mathsf{E}\big{[}\mathds{E}[\varphi](t-S_{n})e^{-\alpha(t-S_{n})}\big{]}$
$\displaystyle=\int\mathds{E}[\varphi](t-x)e^{-\alpha(t-x)}\,\mathsf{U}(\mathrm{d}\mathit{x}).$
(4.6)

By the direct Riemann integrability of
$t\mapsto\mathds{E}[\varphi(t)]e^{-\alpha t}$ and (4.4), the function
$t\mapsto m_{t}^{\varphi}e^{-\alpha t}$ is bounded and, moreover,

$\lim_{\begin{subarray}{c}t\to\infty\\\
t\in\mathds{G}\end{subarray}}e^{-\alpha
t}m_{t}^{\varphi}=\frac{1}{\beta}\int\mathds{E}[\varphi](x)e^{-\alpha
x}\,\ell(\mathrm{d}\mathit{x})=\frac{1}{\beta}(\mathcal{L}\mathds{E}[\varphi])(\alpha)$
(4.7)

by the key renewal theorem, see [7, Theorem 4.2] in the non-lattice case and
[2, Theorem 2.5.3] in the lattice (and non-lattice) case. Recall that, in the
lattice case, $\mathcal{L}\mathds{E}[\varphi]$ denotes the ‘discrete’
bilateral Laplace transform of $\mathds{E}[\varphi]$, $\mathds{G}=\mathds{Z}$
and $\ell$ is the counting measure on $\mathds{Z}$, whereas in the non-lattice
case, $\mathds{E}[\varphi]$ is the ‘continuous’ bilateral Laplace transform,
$\mathds{G}=\mathds{R}$ and $\ell$ is the Lebesgue measure on $\mathds{R}$.

We need a lemma in preparation for the proof of Proposition 2.2.

######  Lemma 4.1.

Suppose that (A1) holds and that $\chi$ is a centered characteristic, i.e.,
$\mathds{E}[\chi(t)]=0$ for all $t\in\mathds{R}$. Fix $t\in\mathds{R}$ and
suppose that

$\displaystyle\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}}\operatorname{\mathrm{Var}}[\chi](t-S(u))\bigg{]}<\infty.$
(4.8)

Let $(u_{n})_{n\in\mathds{N}}$ be an admissible ordering of $\mathcal{I}$ (see
the paragraph before Proposition 2.2 for the definition). Define

$M_{n}(t)\vcentcolon=\sum_{j=1}^{n}\chi_{u_{j}}(t-S(u_{j}))$

for $n\in\mathds{N}_{0}$. Then $(M_{n}(t))_{n\in\mathds{N}_{0}}$ is a centered
martingale and bounded in $L^{2}$. In particular,

$\displaystyle Z^{\chi}_{t}\vcentcolon=\sum_{u\in\mathcal{I}}\chi_{u}(t-S(u))$
(4.9)

converges unconditionally in $L^{2}$ and it is also the almost sure limit of
$M_{n}(t)$ as $n\to\infty$. Further, for any (deterministic) sequence
$(\mathcal{I}_{n})_{n\in\mathds{N}_{0}}$ with
$\mathcal{I}_{n}\uparrow\mathcal{I}$,

$M_{\mathcal{I}_{n}}(t)=\sum_{u\in\mathcal{I}_{n}}\chi_{u}(t-S(u))\to\mathcal{Z}_{t}^{\chi}\quad\text{in
$L^{2}$ as }n\to\infty.$

Moreover,

$\displaystyle\operatorname{\mathrm{Var}}[\mathcal{Z}_{t}^{\chi}]=\mathds{E}[(\mathcal{Z}_{t}^{\chi})^{2}]=\mathds{E}[\mathcal{Z}_{t}^{\chi^{2}}]<\infty.$
(4.10)

Finally, (A5) is sufficient for (4.8) to hold for every $t\in\mathds{R}$.

###### Proof.

Let $\mathcal{G}_{n}=\sigma(\pi_{u_{j}}:j\leq n)$, where it should be recalled
that $\pi_{u}$ is the projection onto the life space of individual $u$, in
particular, $(\xi_{u},\zeta_{u},\chi_{u})$ is $\pi_{u}$-measurable. Then
$(M_{n}(t))_{n\in\mathds{N}_{0}}$ is adapted with respect to
$(\mathcal{G}_{n})_{n\in\mathds{N}_{0}}$ as, for any $u\in\mathcal{I}_{n}$,
both $S(u)$ and $\chi_{u}$ are $\mathcal{G}_{n}$-measurable. Moreover, (4.8)
implies that for any $u\in\mathcal{I}$,
$\mathds{E}[|\chi_{u}(t-S(u))|]<\infty$. Hence $M_{n}(t)$ is integrable for
any $n\in\mathds{N}_{0}$. The martingale property then follows since
$S(u_{n+1})$ is $\mathcal{G}_{n}$-measurable whereas $\chi_{u_{n+1}}$ is
independent of $\mathcal{G}_{n}$ and since
$\mathds{E}[\chi](x)=\mathds{E}[\chi(x)]=0$ for all $x\in\mathds{R}$, so

$\mathds{E}[\chi_{u_{n+1}}(t-S(u_{n+1}))|\mathcal{G}_{n}]=\mathds{E}[\chi](t-S(u_{n+1}))=0\quad\text{almost
surely.}$

Next, we observe that, since the increments of $L^{2}$-martingales are
uncorrelated,

$\displaystyle\mathds{E}[M_{n}(t)^{2}]$
$\displaystyle=\mathds{E}\bigg{[}\sum_{j=1}^{n}\chi_{u_{j}}^{2}(t-S(u_{j}))\bigg{]}=\mathds{E}\bigg{[}\sum_{j=1}^{n}\mathds{E}[\chi^{2}](t-S(u_{j}))\bigg{]}$
$\displaystyle\leq\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}}\mathds{E}[\chi^{2}](t-S(u))\bigg{]}=\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}}\operatorname{\mathrm{Var}}[\chi](t-S(u))\bigg{]}.$
(4.11)

By (4.8), the martingale $(M_{n}(t))_{n\in\mathds{N}_{0}}$ is bounded in
$L^{2}$ and thus converges in $L^{2}$ and almost surely. We denote the limit
by $\mathcal{Z}_{t}^{\chi}$ and view it as the limit of the series on the
right-hand side of (4.9). This is justified by the following argument. For any
subset $\mathcal{J}\subseteq\mathcal{I}$, finite or infinite, since for any
$u\in\mathcal{J}$ there is a unique $j\in\mathds{N}$ with $u=u_{j}$ and again
since martingale increments are uncorrelated, we have

$\mathds{E}\bigg{[}\bigg{|}\sum_{u\in\mathcal{J}}\chi_{u}(t-S(u))\bigg{|}^{2}\bigg{]}=\mathds{E}\bigg{[}\sum_{u\in\mathcal{J}}\operatorname{\mathrm{Var}}[\chi](t-S(u))\bigg{]}.$

From this and the Cauchy criterion, on the one hand, we infer the
unconditional convergence in $L^{2}$ of the series in (4.9), thereby
justifying to write $Z_{t}^{\chi}$ for the limit. On the other hand, we
conclude the convergence of $M_{\mathcal{I}_{n}}(t)$ to
$\mathcal{Z}_{t}^{\chi}$. Since convergence in $L^{2}$ implies convergence in
$L^{1}$, $\mathcal{Z}_{t}^{\chi}$ is centered. Using this and again the
convergence in $L^{2}$, we deduce

$\displaystyle\operatorname{\mathrm{Var}}[\mathcal{Z}_{t}^{\chi}]$
$\displaystyle=\mathds{E}[(\mathcal{Z}_{t}^{\chi})^{2}]=\lim_{n\to\infty}\mathds{E}[M_{n}(t)^{2}]$
$\displaystyle=\lim_{n\to\infty}\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}_{n}}\chi_{u}^{2}(t-S(u))\bigg{]}=\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}}\chi_{u}^{2}(t-S(u))\bigg{]},$

i.e., (4.10) holds. Finally, (A5) implies (4.8) since, for any
$t\in\mathds{R}$, by (4.6) and the subsequent arguments,

$\displaystyle\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}}\operatorname{\mathrm{Var}}[\chi](t-S(u))\bigg{]}$
$\displaystyle=\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}}\mathds{E}[\chi^{2}](t-S(u))\bigg{]}$
$\displaystyle=e^{\alpha
t}\int\mathds{E}[\chi^{2}](t-x)e^{-\alpha(t-x)}\,\mathsf{U}(\mathrm{d}\mathit{x})\leq
Ce^{\alpha t}$

where as before $\mathsf{U}$ is the renewal measure of the associated random
walk $(S_{n})_{n\in\mathds{N}_{0}}$ and $C>0$ is some finite constant. ∎

We are now ready to prove Proposition 2.2.

###### Proof of Proposition 2.2.

By (A4), $\mathds{E}[\varphi](t)$ is finite for every $t\in\mathds{R}$ and we
may write

$\displaystyle\varphi_{u}(t-S(u))=\mathds{E}[\varphi](t-S(u))+\big{(}\varphi_{u}(t-S(u))-\mathds{E}[\varphi](t-S(u))\big{)}$

for every $u\in\mathcal{I}$. It is therefore enough to check that both series

$\displaystyle\sum_{u\in\mathcal{I}}\mathds{E}[\varphi](t-S(u))\quad\text{and}\quad\sum_{u\in\mathcal{I}}\big{(}\varphi_{u}(t-S(u))-\mathds{E}[\varphi](t-S(u))\big{)}$
(4.12)

converge almost surely over admissible orderings and unconditionally in
$L^{1}$. For the first series, note that by (A4) the function
$f(t)\vcentcolon=|\mathds{E}[\varphi](t)|e^{-\alpha t}$ is directly Riemann
integrable as well and by (4.2), we have

$\displaystyle\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}}|\mathds{E}[\varphi](t-S(u))|\bigg{]}=\mathsf{E}\bigg{[}\sum_{n\geq
0}e^{\alpha S_{n}}|\mathds{E}[\varphi](t-S_{n})|\bigg{]}=e^{\alpha t}\cdot
f*\mathsf{U}(t),$

which is finite by (4.4) and the direct Riemann integrability of $f$. Hence,
the series converges unconditionally in $L^{1}$ and absolutely almost surely.
The same argument as above gives

$\displaystyle\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}}\operatorname{\mathrm{Var}}[\varphi](t-S(u))\bigg{]}<\infty,$

i.e., $\chi(t)\vcentcolon=\varphi(t)-\mathds{E}[\varphi](t)$ is a centered
characteristic satisfying (4.8). We may thus apply Lemma 4.1 to conclude that
the second series in (4.12) converges almost surely over admissible orderings
of $\mathcal{I}$ and unconditionally in $L^{2}$. ∎

We close this subsection with the proof of Proposition 2.5.

###### Proof of Proposition 2.5.

(a) Since $f$ is càdlàg, it is locally bounded and continuous Lebesgue-almost
everywhere. By (a slightly extended version of) [40, Remark 3.10.4 on p. 236],
this together with

$\displaystyle\sum_{n\in\mathds{Z}}\sup_{x\in[n,n+1]}|f(x)|\leq\int
f^{*}(x)\,\mathrm{d}\mathit{x}<\infty$

ensures the direct Riemann integrability of $f$.

Conversely, if $f$ is directly Riemann integrable, then it is locally bounded
and continuous Lebesgue-almost everywhere. Local boundedness of $f$ entails
that of $f^{*}$. Since $f^{*}$ is continuous on $\\{x\in\mathds{R}:f\text{ is
continuous at }x-1\text{ and }x+1\\}$, this implies that also $f^{*}$ is
continuous Lebesgue-almost everywhere. Furthermore, for every
$x\in\mathds{R}$, we have

$\displaystyle\sum_{n\in\mathds{Z}}$ $\displaystyle\sup_{n\leq
x<n+1}f^{*}(x)=\sum_{n\in\mathds{Z}}\sup_{n-1\leq x<n+2}|f(x)|\leq
3\sum_{n\in\mathds{Z}}\sup_{n\leq x<n+1}|f(x)|<\infty$

since $f$ is directly Riemann integrable. Thus, again by [40, Remark 3.10.4 on
p. 236], $f^{*}$ is directly Riemann integrable.

We prove (b) and (c) at one go. To this end, let $p=1$ in the situation of (b)
and $p=2$ in the situation (c). Define $\phi(t)\vcentcolon=\varphi(t)^{p}$ for
$t\in\mathds{R}$. Then we infer

$\int\mathds{E}[\phi^{*}](x)e^{-\alpha x}\,\mathrm{d}\mathit{x}<\infty,$
(4.13)

from (2.14) or (2.15), respectively, where we have used that
$(\varphi^{2})^{*}=(\varphi^{*})^{2}$ in the situation of (c). From (4.13) we
deduce that $\mathds{E}[\phi^{*}](x)<\infty$ for Lebesgue-almost all
$x\in\mathds{R}$ and hence

$\mathds{E}\Big{[}\sup_{|t-x|\leq\frac{1}{2}}|\phi(t)|\Big{]}<\infty\quad\text{for
all }x\in\mathds{R}.$ (4.14)

In the case of (c), this implies the validity of (A6). In both cases, (4.14)
together with the dominated convergence theorem imply that $\mathds{E}[\phi]$
has càdlàg paths and thus also $f$ defined by
$f(t)\vcentcolon=\mathds{E}[\phi(t)]e^{-\alpha t}$. Further, $\int
f^{*}(x)\,\mathrm{d}\mathit{x}<\infty$ by (2.14) and (2.15), respectively,
since $(\mathds{E}[\phi](t)e^{-\alpha t})^{*}\leq
e^{\alpha}\mathds{E}[\phi]^{*}(t)e^{-\alpha t}\leq
e^{\alpha}\mathds{E}[\phi^{*}](t)e^{-\alpha t}$. Part (b) now follows from
(a). In the situation of (c), we deduce from (a) that
$t\mapsto\mathds{E}[\varphi^{2}](t)e^{-\alpha t}$ is directly Riemann
integrable. Also, $\mathds{E}[\varphi]$ has càdlàg paths by (4.14) and the
dominated convergence theorem. Therefore,
$\operatorname{\mathrm{Var}}[\varphi](t)=\mathds{E}[\varphi^{2}](t)-\mathds{E}[\varphi(t)]^{2}$
is càdlàg and, in particular, locally bounded and continuous Lebesgue-almost
everywhere. Since

$\displaystyle\sum_{n\in\mathds{Z}}\sup_{x\in[n,n+1]}\operatorname{\mathrm{Var}}[\varphi(x)]e^{-\alpha
x}$
$\displaystyle\leq\sum_{n\in\mathds{Z}}\sup_{x\in[n,n+1]}\mathds{E}[\varphi(x)^{2}]e^{-\alpha
x}<\infty$

the direct Riemann integrability of
$\operatorname{\mathrm{Var}}[\varphi](t)e^{-\alpha t}$ follows from [40,
Remark 3.10.4], i.e., (A5) holds.

∎

###  4.3 Matrix notation

For any $s\in\mathds{R}$ and $\gamma\in\mathds{C}$ we define the following
lower triangular $k\times k$ matrix

$\displaystyle\exp(\gamma,s,k)\vcentcolon=e^{\gamma
s}\times\begin{pmatrix}1&0&0&\dots\qquad&0\\\ s&1&0&\dots\qquad&0\\\
s^{2}&2s&1&\dots\qquad&0\\\ \vdots&\vdots&\vdots&\ddots\qquad&\vdots\\\
s^{k-1}\quad&{k-1\choose 1}s^{k-2}\quad&{k-1\choose
2}s^{k-3}\quad&\dots\qquad&1\end{pmatrix}.$ (4.15)

The $(i,j)^{\mathrm{th}}$ entry of the matrix is $e^{\gamma
s}\binom{i-1}{j-1}s^{i-j}$, $i,j=1,\ldots,k$, where $\binom{i-1}{j-1}=0$ for
$j>i$ should be recalled. Matrices of this form will be very useful since they
simplify the notation and allow us to deal with polynomial terms with relative
ease. Indeed, for any $s,t\in\mathds{R}$ and $\gamma\in\mathds{C}$,

$\exp(\gamma,s,k)\cdot\exp(\gamma,t,k)=\exp(\gamma,s+t,k).$

This can be seen from elementary but tedious calculations. Alternatively,
notice that

$\displaystyle\exp(\gamma,s,k)=\exp(sJ_{\gamma,k})$

where the matrix $J_{\gamma,k}$ is defined by

$J_{\gamma,k}\vcentcolon=\begin{pmatrix}\gamma&&\\\ 1&\ddots&\mbox{0}\\\
&\ddots&\ddots&\\\ \mbox{0}&&k\\!-\\!1&\gamma\end{pmatrix}.$

This leads to

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\mathit{x}}\exp(\gamma,x,k)=J_{\gamma,k}\exp(\gamma,x,k).$
(4.16)

With $\|\cdot\|$ denoting the operator norm and $\|\cdot\|_{\mathsf{HS}}$
denoting the Hilbert-Schmidt norm, the following (crude) bound holds for every
$\delta>0$:

$\displaystyle\|\exp(\gamma,s,k)\|$
$\displaystyle\leq\|\exp(\gamma,s,k)\|_{\mathsf{HS}}\leq
C^{\prime}(1+|s|)^{k-1}e^{\mathrm{Re}(\gamma)s}\leq
Ce^{\mathrm{Re}(\gamma)s+\delta|s|}$ (4.17)

for some constant $C^{\prime}>0$ depending on $k$ only and another constant
$C>0$ depending on $k$ and $\delta>0$. For a vector $x$, we write
$x^{\mathsf{T}}$ for its transpose. Further, we write
$\mathsf{e}_{1},\mathsf{e}_{2},\ldots$ for the canonical base vectors in
Euclidean space. Here, for ease of notation, we are slightly sloppy as we do
not specify the dimension of that space (formally, all Euclidean spaces may be
embedded into an appropriate infinite-dimensional space such as $\ell^{2}$).
Then, for instance,

$\exp(\gamma,s,k)\cdot\mathsf{e}_{1}=e^{\gamma s}\begin{pmatrix}1\\\ s\\\
s^{2}\\\ \vdots\\\ s^{k-1}\end{pmatrix}.$

Throughout the paper, for $\mathrm{Re}(\lambda)>\vartheta$,
$n\in\mathds{N}_{0}$ and $k\in\mathds{N}$, we denote by $Z_{n}(\lambda,k)$ the
following random matrix

$Z_{n}(\lambda,k)\vcentcolon=\sum_{|u|=n}\exp(\lambda,-S(u),k).$ (4.18)

We set $Z_{n}(\lambda)\vcentcolon=Z_{n}(\lambda,1)$ for
$\mathrm{Re}(\lambda)\geq\vartheta$. In particular,
$\mu(\theta)=\mathds{E}[Z_{1}(\theta)]$ and (A2.8) becomes
$\mathds{E}[Z_{1}(\vartheta)]<\infty$.

##  5 Nerman’s martingales as general branching processes

Nerman’s martingale and its complex counterparts are crucial for the paper as
they constitute the building blocks for the asymptotic expansion of
$\mathcal{Z}^{\varphi}$. In the present section, we demonstrate how these
martingales can be represented in terms of Crump-Mode-Jagers processes and
which characteristics come into play.

Suppose that (A1) holds and that $\mathcal{L}\mu(\vartheta)<\infty$ for some
$0<\vartheta<\alpha$. (Notice that the last condition is implied by (A2.8).)
Further, let $\lambda\in\mathds{C}$ with $\mathrm{Re}(\lambda)>\vartheta$ be a
root of multiplicity $k=k(\lambda)\in\mathds{N}$ of the mapping
$z\mapsto\mathcal{L}\mu(z)-1$, i.e.,

$\displaystyle\mathcal{L}\mu(\lambda)=\mathds{E}\bigg{[}\sum_{j=1}^{N}e^{-\lambda
X_{j}}\bigg{]}=1,$ (5.1)
$\displaystyle\mathcal{L}\mu^{(l)}(\lambda)=(-1)^{l}\mathds{E}\bigg{[}\sum_{j=1}^{N}X_{j}^{l}e^{-\lambda
X_{j}}\bigg{]}=0\quad\text{for }l=1,\ldots,k(\lambda)-1,$ (5.2)
$\displaystyle\mathcal{L}\mu^{(k(\lambda))}(\lambda)\neq 0.$ (5.3)

Conditions (5.1) and (5.2) are equivalent to

$\displaystyle\mathds{E}[Z_{1}(\lambda,k)]=\mathds{E}\bigg{[}\sum_{j=1}^{N}\exp(\lambda,-X_{j},k)\bigg{]}=I_{k}$
(5.4)

where $I_{k}$ is the $k\times k$ identity matrix.

Define the random matrix

$Y_{u}\vcentcolon=Z_{u,1}(\lambda,k)-I_{k}=\int\exp(\lambda,-x,k)\,\xi_{u}(\mathrm{d}\mathit{x})-I_{k}.$

Notice that $\mathds{E}[Y_{u}]$ is the $k\times k$ zero matrix by (5.4).
Moreover, if $\mathrm{Re}(\lambda)\geq\frac{\alpha}{2}$ and (A2.9) is
satisfied then, by the penultimate inequality in (4.17), we have
$\mathds{E}[\|Y_{u}\|^{2}]\leq C_{\lambda,k}<\infty$ for some constant
$C_{\lambda,k}$ that depends only on $\lambda$ and $k$.

Now for such $\lambda$ we define matrix-valued characteristics
$\phi_{\lambda}$ and $\chi_{\lambda,I}$, which play a crucial role in the
proof of the main theorem. For $t\in\mathds{R}$, we set

$\displaystyle\phi_{\lambda}(t)$
$\displaystyle\vcentcolon=\sum_{j=1}^{N}\mathds{1}_{[0,X_{j})}(t)\exp({\lambda,t-X_{j}},k),$
(5.5)

and for any interval $I=[a,b)\cap\mathds{R}$ with $-\infty\leq a<b<\infty$

$\displaystyle\chi_{\lambda,I}(t)\vcentcolon=\mathds{1}_{I}(t)\exp({\lambda,t},k)Y_{\varnothing}\quad\text{
and }\quad\chi_{\lambda}(t)\vcentcolon=\chi_{\lambda,(-\infty,0)}(t).$ (5.6)

By definition, characteristics take values in $\mathds{R}^{d}$ for some
$d\in\mathds{N}$, but here we use an obvious extension to $\mathds{C}$ by
splitting into real and imaginary part. Note also that both $\phi_{\lambda}$
and $\chi_{\lambda,I}$ are $\sigma(\xi)$-measurable and, in particular, the
tuples $(\xi_{u},\phi_{\lambda,u},\chi_{\lambda,u})$, $u\in\mathcal{I}$ are i.
i. d., where
$\chi_{\lambda,u}=\mathds{1}_{(-\infty,0)}(t)\exp({\lambda,t},k)Y_{u},t\in\mathds{R}$.

######  Lemma 5.1.

Suppose that (A1) through (A2.9) hold. Let $\lambda\in\mathds{C}$ with
$\mathrm{Re}(\lambda)\geq\vartheta$ be such that $\mathcal{L}\mu(\lambda)=1$
and $\mathds{E}[\|Z_{1}(\lambda)\|^{2}]<\infty$. Denote by $k$ the
multiplicity of $\lambda$ and fix $x,y\in\mathds{R}^{k}$.

  1. (a)

The characteristic $x^{\mathsf{T}}\phi_{\lambda}y$ satisfies (A4), (A5) and
(A6).

  2. (b)

Let $\mathrm{Re}(\lambda)>\frac{\alpha}{2}$, and $I=[a,b)\cap\mathds{R}$ be an
interval with $-\infty\leq a<b<\infty$. Then the characteristic
$x^{\mathsf{T}}\chi_{\lambda,I}y$ satisfies (A4), (A5) and (A6).

###### Proof.

Clearly, both characteristics $x^{\mathsf{T}}\phi_{\lambda}y$ and
$x^{\mathsf{T}}\chi_{\lambda}y$ have càdlàg paths. Without loss of generality,
we may assume that $|x|,|y|\leq 1$. In view of Proposition 2.5 it suffices to
verify the integrability conditions (2.14) and (2.15). Let us first assume
that $\mathrm{Re}(\lambda)>\frac{\alpha}{2}$. Then we take
$\gamma\in(\frac{\alpha}{2},\mathrm{Re}(\lambda))$ and from (4.17) we infer
the existence of a constant $C$ that depends only on $\lambda,\gamma$ and $k$
such that, for $s\leq 0$ and $|t-s|\leq 1$, we have

$\big{|}x^{\mathsf{T}}\exp(\lambda,s,k)y\big{|}\leq\|\exp(\lambda,s-t,k)\|_{\mathsf{HS}}\|\exp(\lambda,t,k)\|_{\mathsf{HS}}\leq
Ce^{\gamma t}.$

We can thus write for $t\in\mathds{R}$

$\displaystyle(x^{\mathsf{T}}\phi_{\lambda}y)^{*}(t)$
$\displaystyle=\sup_{|s-t|\leq
1}\big{|}x^{\mathsf{T}}\phi_{\lambda}(s)y\big{|}$
$\displaystyle\leq\sup_{|s-t|\leq
1}\sum_{j=1}^{N}\mathds{1}_{[0,X_{j})}(s)\big{|}x^{\mathsf{T}}\exp(\lambda,s-X_{j},k)y\big{|}$
$\displaystyle\leq
C\sum_{j=1}^{N}\mathds{1}_{[-1,X_{j}+1)}(t)e^{\gamma(t-X_{j})}.$

Hence, we have

$\displaystyle\int\mathds{E}\Big{[}(x^{\mathsf{T}}\phi_{\lambda}y)^{*}(t)\Big{]}e^{-\alpha
t}\,\mathrm{d}\mathit{t}$ $\displaystyle\leq
C\int\limits_{-\infty}^{\infty}\mathds{E}\bigg{[}\sum_{j=1}^{N}\mathds{1}_{[-1,X_{j}+1)}(t)e^{\gamma(t-X_{j})}\bigg{]}e^{-\alpha
t}\,\mathrm{d}\mathit{t}$ $\displaystyle\leq
C\mathds{E}\bigg{[}\sum_{j=1}^{N}e^{-\gamma
X_{j}}\bigg{]}\int\limits_{-1}^{\infty}e^{(\gamma-\alpha)t}\,\mathrm{d}\mathit{t}<\infty$

by (A2.8). Further,

$\displaystyle\int$
$\displaystyle\mathds{E}\Big{[}\Big{(}(x^{\mathsf{T}}\phi_{\lambda}y)^{*}(t)\Big{)}^{2}\Big{]}e^{-\alpha
t}\,\mathrm{d}\mathit{t}$ $\displaystyle\leq
C^{2}\int\limits_{-\infty}^{\infty}\mathds{E}\bigg{[}\sum_{1\leq i,j\leq
N}\mathds{1}_{[-1,X_{i}+1)}(t)e^{\gamma(t-X_{i})}\mathds{1}_{[-1,X_{j}+1)}(t)e^{\gamma(t-X_{j})}\bigg{]}e^{-\alpha
t}\,\mathrm{d}\mathit{t}$ $\displaystyle\leq
C^{2}\mathds{E}\bigg{[}\sum_{1\leq i,j\leq N}\int_{-1}^{(X_{i}\wedge
X_{j})+1}e^{(2\gamma-\alpha)t}\,\mathrm{d}\mathit{t}\ e^{-\gamma
X_{i}}e^{-\gamma X_{j}}\bigg{]}$
$\displaystyle\leq\frac{e^{2\gamma-\alpha}C^{2}}{2\gamma-\alpha}\mathds{E}\bigg{[}\sum_{1\leq
i,j\leq N}e^{(2\gamma-\alpha)(X_{i}\wedge X_{j})}e^{-\gamma X_{i}}e^{-\gamma
X_{j}}\bigg{]}$
$\displaystyle\leq\frac{e^{2\gamma-\alpha}C^{2}}{2\gamma-\alpha}\mathds{E}\bigg{[}\sum_{1\leq
i,j\leq N}e^{(\gamma-\frac{\alpha}{2})(X_{i}+X_{j})}e^{-\gamma
X_{i}}e^{-\gamma X_{j}}\bigg{]}$
$\displaystyle=\frac{e^{2\gamma-\alpha}C^{2}}{2\gamma-\alpha}\mathds{E}\bigg{[}\Big{(}\sum_{j=1}^{N}e^{-\frac{\alpha}{2}X_{j}}\Big{)}^{2}\bigg{]}<\infty$

by (A2.9). Now assume that $\mathrm{Re}(\lambda)=\frac{\alpha}{2}$. Then for
$s\leq 0$ and $|t-s|\leq 1$, we have, again by (4.17),

$\big{|}x^{\mathsf{T}}\exp(\lambda,s,k)y\big{|}\leq
C(1+|t|)^{k-1}e^{\frac{\alpha}{2}t}$

for some $C$ depending on $\lambda$ and $k$ (not necessarily the exact
constant $C$ from (4.17), but a larger, finite one). This, in turn, gives, for
arbitrary $t\in\mathds{R}$,

$\displaystyle(x^{\mathsf{T}}\phi_{\lambda}y)^{*}(t)$
$\displaystyle=\sup_{|s-t|\leq
1}\big{|}x^{\mathsf{T}}\phi_{\lambda}(s)y\big{|}$
$\displaystyle\leq\sup_{|s-t|\leq
1}\sum_{j=1}^{N}\mathds{1}_{[0,X_{j})}(s)\big{|}x^{\mathsf{T}}\exp(\lambda,s-X_{j},k)y\big{|}$
$\displaystyle\leq
C\sum_{j=1}^{N}\mathds{1}_{[-1,X_{j}+1)}(t)(1+|t-X_{j}|)^{k-1}e^{\frac{\alpha}{2}(t-X_{j})},$

and, consequently,

$\displaystyle\int\mathds{E}\big{[}(x^{\mathsf{T}}\phi_{\lambda}y)^{*}(t)\big{]}e^{-\alpha
t}\,\mathrm{d}\mathit{t}\leq\frac{C}{\sqrt{2}}\mathds{E}\bigg{[}\sum_{j=1}^{N}(2+X_{j})^{k-\frac{1}{2}}e^{-\frac{\alpha}{2}X_{j}}\bigg{]}\int\limits_{-1}^{\infty}e^{-\frac{\alpha}{2}t}\,\mathrm{d}\mathit{t}<\infty$

by (A2.9) and

$\displaystyle\int$
$\displaystyle\mathds{E}\big{[}\big{(}(x^{\mathsf{T}}\phi_{\lambda}y)^{*}(t)\big{)}^{2}\big{]}e^{-\alpha
t}\,\mathrm{d}\mathit{t}$ $\displaystyle\leq
C^{2}\int\limits_{-\infty}^{\infty}\mathds{E}\bigg{[}\sum_{1\leq i,j\leq
N}\mathds{1}_{[-1,X_{i}+1)}(t)(1+|t-X_{i}|)^{k-1}e^{\frac{\alpha}{2}(t-X_{i})}$
$\displaystyle\hphantom{\leq
C\int\limits_{-\infty}^{\infty}\mathds{E}\bigg{[}\sum_{1\leq i,j\leq
N}}\cdot\mathds{1}_{[-1,X_{j}+1)}(t)(1+|t-X_{j}|)^{k-1}e^{\frac{\alpha}{2}(t-X_{j})}\bigg{]}e^{-\alpha
t}\,\mathrm{d}\mathit{t}$ $\displaystyle\leq
C^{2}\mathds{E}\bigg{[}\sum_{1\leq i,j\leq N}\int\limits_{-1}^{X_{i}\wedge
X_{j}+1}\,\mathrm{d}\mathit{t}\,(2+X_{i})^{k-1}e^{-\frac{\alpha}{2}X_{i}}(2+X_{j})^{k-1}e^{-\frac{\alpha}{2}X_{j}}\bigg{]}$
$\displaystyle\leq C^{2}\mathds{E}\bigg{[}\sum_{1\leq i,j\leq N}(X_{i}\wedge
X_{j}+2)(2+X_{i})^{k-1}e^{-\frac{\alpha}{2}X_{i}}(2+X_{j})^{k-1}e^{-\frac{\alpha}{2}X_{j}}\bigg{]}$
$\displaystyle\leq
2^{2k-1}C^{2}\mathds{E}\bigg{[}\Big{(}\sum_{j=1}^{N}(1+\tfrac{X_{j}}{2})^{k-\frac{1}{2}}e^{-\frac{\alpha}{2}X_{j}}\Big{)}^{2}\bigg{]}<\infty$

again by (A2.9), which finish the proof of (a). Regarding part (b), notice
that (A4) holds trivially as $\chi_{\lambda,I}$ is centered. Further, observe
that

$\displaystyle(x^{\mathsf{T}}\chi_{\lambda,I}y)^{*}(t)$
$\displaystyle=\sup_{|s-t|\leq
1}\mathds{1}_{[a,b)\cap\mathds{R}}(s)\big{|}x^{\mathsf{T}}\exp({\lambda,s},k)(Z_{1}(\lambda,k)-I_{k})y\big{|}$
$\displaystyle\leq
C\mathds{1}_{(-\infty,b+1)}(t)\|\exp({\lambda,t},k)\|\|Z_{1}(\lambda,k)-I_{k}\|$
$\displaystyle=C^{2}\mathds{1}_{(-\infty,b+1)}(t)e^{\gamma
t}\bigg{(}1+\sum_{j=1}^{N}e^{-\frac{\alpha}{2}X_{j}}\bigg{)}$

where, as before, $\gamma\in(\frac{\alpha}{2},\mathrm{Re}(\lambda))$. Thus

$\displaystyle\int\mathds{E}\big{[}\big{(}(x^{\mathsf{T}}\chi_{\lambda,I}y)^{*}(t)\big{)}^{2}\big{]}e^{-\alpha
t}\,\mathrm{d}\mathit{t}\leq
C^{4}\mathds{E}\bigg{[}\Big{(}1+\sum_{j=1}^{N}e^{-\frac{\alpha}{2}X_{j}}\Big{)}^{2}\bigg{]}\int\limits_{-\infty}^{b+1}e^{(2\gamma-\alpha)t}\,\mathrm{d}\mathit{t}<\infty$

by (A2.9), which completes the proof of (b). ∎

As a consequence of the above lemma we conclude that for any $t\in\mathds{R}$
both $\mathcal{Z}^{\phi_{\lambda}}_{t}$ and
$\mathcal{Z}^{\chi_{\lambda,I}}_{t}$ are well-defined. The first one is so as
an unconditional limit in $L^{1}$ by Proposition 2.2 and the second as an
unconditional limit in $L^{2}$ by (the first part of) Lemma 4.1 (by Lemma 5.1,
the characteristic $\chi_{\lambda,I}$ satisfies (A5); according to the last
part of Lemma 4.1, (A5) entails (4.8), the principal assumption of the first
part of Lemma 4.1). In particular,

$\displaystyle\mathcal{Z}_{0}^{\chi_{\lambda}}=\sum_{u\in\mathcal{I}}\exp(\lambda,-S(u),k)Y_{u}$

converges unconditionally in $L^{2}$ and almost surely along admissible
orderings of $\mathcal{I}$.

###  5.1 Nerman’s martingales with complex parameters

For $u\in\mathcal{I}$ we define
$\mathcal{G}_{u}\vcentcolon=\sigma(\xi_{v}:v\preceq u)$ and, for
$t\in\mathds{R}$,

$\mathcal{F}^{W}_{t}\vcentcolon=\sigma(\\{A\cap\\{S(u)\leq
t\\}:u\in\mathcal{I}\text{ and }A\in\mathcal{G}_{u}\\}).$

######  Lemma 5.2.

Suppose that (A1) holds and that $\mathcal{L}\mu(\vartheta)<\infty$ for some
$0<\vartheta<\alpha$. Let $\lambda\in\mathds{C}$ with
$\mathrm{Re}(\lambda)>\vartheta$ be a root of $z\mapsto\mathcal{L}\mu(z)-1$
with multiplicity $k$. If the characteristic $\phi_{\lambda}$ satisfies (A4),
(A5) and (A6), then following process

$\displaystyle
W_{t}(\lambda,k)\vcentcolon=\exp({\lambda},-t,k)\cdot\mathcal{Z}^{\phi_{\lambda}}_{t},\qquad
t\in\mathds{R}$

is a (matrix-valued) martingale with respect to the filtration
$(\mathcal{F}^{W}_{t})_{t\geq 0}$. Moreover, for any $t\in\mathds{R}$, it
holds

$\displaystyle
W_{t}(\lambda,k)=I_{k}\mathds{1}_{[0,\infty)}(t)+\mathcal{Z}^{\chi_{\lambda,[-t,1)}}_{0}=\sum_{u\in\mathcal{C}_{t}}\exp({\lambda,-S(u)},k)\quad\text{a.\,s.}$
(5.7)

where by definition (see (2.11)) $\mathcal{C}_{t}=\\{uj\in\mathcal{T}:S(u)\leq
t<S(uj)\\}$. In particular,

$\displaystyle
m^{\phi_{\lambda}}_{t}=\mathds{1}_{[0,\infty)}(t)\exp(\lambda,t,k),\quad
t\in\mathds{R}.$

######  Remark 5.3.

It is worth mentioning that if, for some $x,y\in\mathds{R}^{k}$, the
characteristic $x^{\mathsf{T}}\phi_{\lambda}y$ satisfies (A4), (A5) and (A6),
then the proof below carries over and gives that
$x^{\mathsf{T}}W_{t}(\lambda,k)y$ is a martingale and

$\displaystyle
x^{\mathsf{T}}W_{t}(\lambda,k)y=\sum_{u\in\mathcal{C}_{t}}x^{\mathsf{T}}\exp({\lambda,-S(u)},k)y\quad\text{a.\,s.}$

In particular, if the above conditions hold with $x=y=\mathsf{e}_{1}$, then we
obtain that $W_{t}(\lambda,1)$ is a martingale.

###### Proof of Lemma 5.2.

We can apply Proposition 2.2, which yields that $W_{t}(\lambda,k)$ is
integrable for any $t\in\mathds{R}$.

We boldly write

$\displaystyle\exp$
$\displaystyle({\lambda},-t,k)\mathcal{Z}^{\phi_{\lambda}}_{t}=\sum_{u\in\mathcal{I}}\sum_{j=1}^{N_{u}}\mathds{1}_{[0,X_{u,j})}(t-S(u))\exp({\lambda,-S(uj)},k)$
$\displaystyle=\sum_{u\in\mathcal{I}}\sum_{j=1}^{N_{u}}\mathds{1}_{\\{S(u)\leq
t\\}}\mathds{1}_{\\{S(uj)>t\\}}\exp({\lambda,-S(uj)},k)$
$\displaystyle=\sum_{u\in\mathcal{I}}\sum_{j=1}^{N_{u}}\big{(}\mathds{1}_{\\{S(u)\leq
t\\}}-\mathds{1}_{\\{S(uj)\leq t\\}}\big{)}\exp({\lambda,-S(uj)},k)$
$\displaystyle=\sum_{u\in\mathcal{I}}\sum_{j=1}^{N_{u}}\mathds{1}_{\\{S(u)\leq
t\\}}\exp({\lambda,-S(uj)},k)-\sum_{|u|\geq 1}\mathds{1}_{\\{S(u)\leq
t\\}}\exp({\lambda,-S(u)},k)$
$\displaystyle=I_{k}\mathds{1}_{[0,\infty)}(t)+\sum_{u\in\mathcal{I}}\mathds{1}_{\\{S(u)\leq
t\\}}\exp({\lambda,-S(u)},k)\big{(}Z_{1}(\lambda,k)\circ\theta_{u}-I_{k}\big{)},$
(5.8)

where the rearrangements of the infinite series in the last two lines are
justified by the fact that there are only finitely many non-zero terms almost
surely. Next, note that for any $t\in\mathds{R}$,

$\displaystyle\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}}\mathds{1}_{\\{S(u)\leq
t\\}}\big{\|}\exp({\lambda,-S(u)},k)\big{\|}\bigg{(}\sum_{j=1}^{N_{u}}\big{\|}\exp({\lambda,-X_{u,j}},k)\big{\|}+1\bigg{)}\bigg{]}<\infty.$
(5.9)

Indeed, by (4.17), the expectation in (5.9) can be bounded by a finite,
deterministic constant times

$\displaystyle\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}}\mathds{1}_{\\{S(u)\leq
t\\}}e^{-\vartheta
S(u)}\big{(}Z_{1}(\vartheta)\circ\theta_{u}+1\big{)}\bigg{]}\leq\big{(}\mathcal{L}\mu(\vartheta)\\!+\\!1\big{)}\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}}\mathds{1}_{\\{S(u)\leq
t\\}}e^{-\vartheta S(u)}\bigg{]}<\infty,$

where we have used the independence between $S(u)$ and $\xi_{u}$. The
finiteness of the last expectation follows from the many-to-one lemma (Formula
(4.2)), namely,

$\displaystyle\mathds{E}\bigg{[}\sum_{u\in\mathcal{I}}\mathds{1}_{\\{S(u)\leq
t\\}}e^{-\vartheta S(u)}\bigg{]}$ $\displaystyle=\mathsf{E}\bigg{[}\sum_{n\geq
0}\mathds{1}_{\\{S_{n}\leq t\\}}e^{(\alpha-\vartheta)S_{n}}\bigg{]}\leq
e^{(\alpha-\vartheta)t}\mathsf{U}([0,t])<\infty.$

Further, $(W_{t}(\lambda,k))_{t\geq 0}$ is adapted to the filtration
$(\mathcal{F}^{W}_{t})_{t\geq 0}$. In order to show the martingale property
note that, for $0\leq s<t$,

$\displaystyle
W_{t}(\lambda,k)-W_{s}(\lambda,k)=\sum_{u\in\mathcal{I}}\mathds{1}_{\\{s<S(u)\leq
t\\}}\exp({\lambda,-S(u)},k)\big{(}Z_{1}(\lambda,k)\circ\theta_{u}-I_{k}\big{)},$

and by (5.9) it suffices to show that for any $u\in\mathcal{I}$

$\displaystyle\mathds{E}\big{[}\mathds{1}_{\\{s<S(u)\leq
t\\}}\exp({\lambda,-S(u)},k)\big{(}Z_{1}(\lambda,k)\circ\theta_{u}-I_{k}\big{)}\big{|}\mathcal{F}^{W}_{s}\big{]}=0\quad\text{a.\,s.}$
(5.10)

Let $u,v\in\mathcal{I}$ and note that the fact $S(u)>s$, $S(v)\leq s$ implies
$u\not\preceq v$. In particular, for such $u$ and $v$, $\xi_{u}$ is
independent of $\mathcal{G}_{v}$ and hence for any $A\in\mathcal{G}_{v}$

$\displaystyle\mathds{E}\big{[}\mathds{1}_{\\{s<S(u)\leq
t\\}}\exp({\lambda,-S(u)},k)\big{(}Z_{1}(\lambda,k)\circ\theta_{u}-I_{k}\big{)}\mathds{1}_{A\cap\\{S(v)\leq
s\\}}\big{]}=0,$

where $\mathds{E}[Z_{1}(\lambda,k)]=I_{k}$ was used. The argument carries over
if we take a finite intersection of sets of the type $A\cap\\{S(v)\leq s\\}$,
$A\in\mathcal{G}_{v}$ for different $v\in\mathcal{I}$. The
$\pi$-$\lambda$-theorem (or monotone class theorem) gives (5.10) and thus
proves that $(W_{t}(\lambda,k))_{t\geq 0}$ is a martingale.

It remains to prove (5.7). The first identity of this equation is (5.8).
Further, from the calculation leading towards (5.8), we have

$\displaystyle W_{t}(\lambda,k)$
$\displaystyle=\sum_{u\in\mathcal{I}}\sum_{j=1}^{N_{u}}\mathds{1}_{\\{S(u)\leq
t\\}}\mathds{1}_{\\{S(uj)>t\\}}\exp({\lambda,-S(uj)},k)$
$\displaystyle=\sum_{u\in\mathcal{I}}\sum_{j=1}^{N_{u}}\mathds{1}_{\mathcal{C}_{t}}(uj)\exp(\lambda,-S(uj),k)=\sum_{u\in\mathcal{C}_{t}}\exp(\lambda,-S(u),k).$

∎

The random matrix $W_{t}(\lambda,k)$ has the following form

$\displaystyle
W_{t}(\lambda,k)=\begin{pmatrix}W^{(0)}_{t}(\lambda)&0&0&\dots&0\\\
W^{(1)}_{t}(\lambda)&W^{(0)}_{t}(\lambda)&0&\dots&0\\\
W^{(2)}_{t}(\lambda)&2W^{(1)}_{t}(\lambda)&W^{(0)}_{t}(\lambda)&\dots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ W^{(k-1)}_{t}(\lambda)&{k-1\choose
1}W^{(k-2)}_{t}(\lambda)&{k-1\choose
2}W^{(k-3)}_{t}(\lambda)&\dots&W^{(0)}_{t}(\lambda)\end{pmatrix},$ (5.11)

where $W_{t}^{(j)}(\lambda)=\sum_{u\in\mathcal{C}_{t}}(-S(u))^{j}e^{-\lambda
S(u)}$ as in (2.17).

###  5.2 Convergence of Nerman’s martingales

The following lemma implies Theorem 2.6.

######  Lemma 5.4.

Suppose that (A1) through (A2.9) hold and let $\lambda$ be a solution to
(2.16) with multiplicity $k$ and $\mathrm{Re}(\lambda)>\frac{\alpha}{2}$. Then
the process $W_{t}(\lambda,k)$ is an $L^{2}$-bounded martingale with limit
given by

$W(\lambda,k)\vcentcolon=I_{k}+\sum_{u\in\mathcal{I}}\exp(\lambda,-S(u),k)Y_{u}=I_{k}+\mathcal{Z}^{\chi_{\lambda,(-\infty,1)}}_{0},$

where the series above converges unconditionally in $L^{2}$. In particular,
for every $0\leq j\leq k\\!-\\!1$, the martingale
$(W_{r}^{(j)}(\lambda))_{t\geq 0}$ converges a. s. and in $L^{2}$.

###### Proof.

Fix $x,y\in\mathds{R}^{k}$. It suffices to show the corresponding result for
the martingale $x^{\mathsf{T}}W_{t}(\lambda,k)y$. By Lemma 5.1 the centered
characteristics $x^{\mathsf{T}}\chi_{\lambda,(-\infty,1)}y$,
$x^{\mathsf{T}}\chi_{\lambda,(-\infty,-t)}y$ and
$x^{\mathsf{T}}\chi_{\lambda,[-t,1)}y$ satisfy (A5) and (A6). In particular,
by Lemma 4.1, the general branching processes counted with these
characteristics are well-defined as unconditional limits in $L^{2}$, and, for
$t\geq 0$,

$\displaystyle\sum_{u\in\mathcal{I}}$ $\displaystyle
x^{\mathsf{T}}\exp(\lambda,-S(u),k)Y_{u}y=\mathcal{Z}^{x^{\mathsf{T}}\chi_{\lambda,(-\infty,1)}y}_{0}=\mathcal{Z}^{x^{\mathsf{T}}\chi_{\lambda,(-\infty,-t)}y}_{0}+\mathcal{Z}^{x^{\mathsf{T}}\chi_{\lambda,[-t,1)}y}_{0}$
$\displaystyle=\sum_{u\in\mathcal{I}}\mathds{1}_{\\{S(u)>t\\}}x^{\mathsf{T}}\exp(\lambda,-S(u),k)Y_{u}y+\sum_{u\in\mathcal{I}}\mathds{1}_{\\{S(u)\leq
t\\}}x^{\mathsf{T}}\exp(\lambda,-S(u),k)Y_{u}y.$

Taking into account that, for any $s\in\mathds{R}$, $t\geq-1$,

$\displaystyle\operatorname{\mathrm{Var}}[x^{\mathsf{T}}\chi_{\lambda,(-\infty,-t)}y](s)\leq\operatorname{\mathrm{Var}}[x\chi_{\lambda,(-\infty,1)}y](s),$

and applying the identity (4.10) (by splitting the characteristics into real
and imaginary part) the dominated convergence theorem yields

$\displaystyle\lim_{t\to\infty}\mathds{E}\Big{[}\Big{|}\mathcal{Z}^{x^{\mathsf{T}}\chi_{\lambda,(-\infty,-t)}y}_{0}\Big{|}^{2}\Big{]}=0.$

In particular, in view of (5.7), we infer that

$\displaystyle
x^{\mathsf{T}}W_{t}(\lambda,k)y=x^{\mathsf{T}}y+\mathcal{Z}^{x^{\mathsf{T}}\chi_{\lambda,(-\infty,1)}y}_{0}-\mathcal{Z}^{x^{\mathsf{T}}\chi_{\lambda,(-\infty,-t)}y}_{0}$

converges in $L^{2}$ as $t\to\infty$. ∎

###  5.3 Limits of Nerman’s martingales as general branching processes

Suppose now that the martingale $(W_{t}(\lambda,k))_{t\geq 0}$ is uniformly
integrable. Then it converges in $L^{1}$ as $t\to\infty$ to some random matrix
$W(\lambda,k)$ of the form

$\displaystyle W(\lambda,k)=\begin{pmatrix}W^{(0)}(\lambda)&0&0&\dots&0\\\
W^{(1)}(\lambda)&W^{(0)}(\lambda)&0&\dots&0\\\
W^{(2)}(\lambda)&2W^{(1)}(\lambda)&W^{(0)}(\lambda)&\dots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ W^{(k-1)}(\lambda)&{k-1\choose
1}W^{(k-2)}(\lambda)&{k-1\choose
2}W^{(k-3)}(\lambda)&\dots&W^{(0)}(\lambda)\end{pmatrix}.$ (5.12)

By uniform integrability, $\mathds{E}[W(\lambda,k)]=I_{k}$.

######  Lemma 5.5.

Suppose that (A1) through (A2.9) hold and let $\lambda$ be a solution to
(2.16) with multiplicity $k$ and $\mathrm{Re}(\lambda)>\frac{\alpha}{2}$.
Then, for $t\geq 0$,

$\exp(\lambda,-t,k)\mathcal{Z}^{\chi_{\lambda}}_{t}=W(\lambda,k)-W_{t}(\lambda,k).$

In particular, for $t\geq 0$,

$\displaystyle\mathcal{Z}^{\phi_{\lambda}+\chi_{\lambda}}_{t}=\exp(\lambda,t,k)W(\lambda,k).$
(5.13)

###### Proof.

Note that for any $s\in\mathds{R}$, $t\geq 0$, we have

$\displaystyle\exp(\lambda,-t,k)\chi_{\lambda}(s)=\chi_{\lambda,(-\infty,-t)}(s-t).$

In particular,

$\displaystyle\exp(\lambda,-t,k)\mathcal{Z}^{\chi_{\lambda}}_{t}=\mathcal{Z}^{\chi_{\lambda,(-\infty,-t)}}_{0},$

which equals $W(\lambda,k)-W_{t}(\lambda,k)$ by Lemma 5.4. This together with
Lemma 5.2 implies

$\mathcal{Z}^{\phi_{\lambda}+\chi_{\lambda}}_{t}=\exp(\lambda,t,k)W(\lambda,k),$

for any $t\geq 0$. ∎

##  6 Proofs of the main results

In this section we provide a proof of our main Theorem 2.14. We begin by
outlining the main ideas and steps.

The basic step is to decompose a given general branching process
$\mathcal{Z}_{t}^{\varphi}$

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

into $H_{\Lambda}(t)$ and $H_{\partial\Lambda}(t)$, see (2.21) and (2.22), the
leading terms in the expansion, plus two general branching processes
$\mathcal{Z}_{t}^{\varrho-\phi_{\partial\Lambda}}$ and
$\mathcal{Z}_{t}^{\chi}$, the first one with mean of the order
$o(e^{\frac{\alpha}{2}t})$ as $t\to\infty$ and the second one with centered
characteristic, i.e., $\mathds{E}[\chi(t)]=0$ for all $t\in\mathds{R}$.

General branching processes with centered characteristics are studied in
Section 6.1. Theorem 6.3 provides the fluctuations of $\mathcal{Z}_{t}^{\chi}$
for a centered characteristic $\chi$. There are two different cases of
interest. First, when $\chi$ satisfies (A5) and second when
$\int_{0}^{t}\operatorname{\mathrm{Var}}[\chi(x)]e^{-\alpha
x}\,\mathrm{d}\mathit{x}\sim ct^{\theta}$ for some $\theta\geq 0$ and $c>0$.
In both cases, $\mathcal{Z}_{t}^{\chi}$, appropriately rescaled, is
asymptotically normal. The main tools to prove this are the martingale central
limit theorem and Nerman’s strong law of large numbers for supercritical
general branching processes. The second case requires the additional auxiliary
Lemma 6.1.

Characteristics such that the corresponding general branching process has mean
function of the order $o(e^{\frac{\alpha}{2}t})$ as $t\to\infty$ are treated
in Section 6.3. Theorem 6.6 of this section yields asymptotic normality for
such processes.

Section 6.2 provides a connection between the cases studied in Sections 6.1
and 6.3. Roughly speaking, Lemma 6.4 enables us to rewrite the process
$\mathcal{Z}_{t}^{f}$ for a deterministic characteristic $f$ in the form

$\mathcal{Z}^{f}_{t}=\mathcal{Z}^{\chi_{f}}_{t}+m^{f}_{t},$

for an appropriately chosen centered characteristic $\chi_{f}$. This enables
us to reduce the case of general branching processes with mean function of the
order $o(e^{\frac{\alpha}{2}t})$ as $t\to\infty$ to the case of centered
characteristics.

In Section 6.4, we put all the pieces together and prove the main Theorem
2.14.

We investigate the asymptotic behavior of the general branching process
$\mathcal{Z}_{t}^{\varphi}$ counted with characteristic $\varphi$ as
$t\to\infty$ in several steps. In the first step, we prove convergence of