# Many-to-few for non-local branching Markov process

Simon C. Harris, Emma Horton, Andreas E. Kyprianou and Ellen Powell University
of Auckland. E-mail<EMAIL_ADDRESS>Research Centre
Bordeaux. E-mail<EMAIL_ADDRESS>of Bath. E-mail:
<EMAIL_ADDRESS>of Durham. E-mail:
<EMAIL_ADDRESS>
###### Abstract

We provide a many-to-few formula in the general setting of non-local branching
Markov processes. This formula allows one to compute expectations of
$\displaystyle k$-fold sums over functions of the population at $\displaystyle
k$ different times. The result generalises [14] to the non-local setting, as
introduced in [11] and [8]. As an application, we consider the case when the
branching process is critical, and conditioned to survive for a large time. In
this setting, we prove a general formula for the limiting law of the death
time of the most recent common ancestor of two particles selected uniformly
from the population at two different times, as $\displaystyle t\to\infty$.
Moreover, we describe the limiting law of the population sizes at two
different times, in the same asymptotic regime.

Key words: non-local branching processes, many-to-few, spines.

MSC 2020: 60J80, 60J25

##  1 Introduction

###  1.1 Main results

Our main result, the so called many-to-few formula, is a way to rewrite the
expectation of a general $\displaystyle k$-fold sum, depending on the entire
configuration of a branching Markov process at $\displaystyle k$ different
times, as an expectation with respect to the behaviour of $\displaystyle k$
distinguished lines of descent under a tilted measure. We generalise the
original and well cited main result of [14], by allowing for non-local
branching, and not requiring the $\displaystyle k$ individuals to be sampled
at the same time.

The many-to-few formula generalises the role of the classical spine
decomposition for spatial branching processes, which converts expectation
identities for additive functionals of spatial branching processes to Feynman-
Kac formulae for a single Markov particle trajectory. The latter has proved to
be an important tool in analysing the growth and spread of a rich variety of
branching Markov processes and related models; see for example the monographs
[1, 6, 18, 15, 2] among a wide base of research literature that is to too
extensive to exhaustively list here. The many-to-few formula has already
played an important and similar role to the classical spine decomposition as a
tool to interrogate various questions pertaining to particle correlation that
arise in e.g. genealogical coalescent structure, [12], martingale convergence,
[10], maximal displacement of extreme particles, [17], the structure of level
sets for branching Brownian motion, [4] and the analysis of certain models
from the theory of stochastic genetics, [7]. We further remark that the many-
to-few formula is related to recent work pertaining to asymptotic moment
convergence in [8].

We refrain from attempting to give a precise statement of the many-to-few
formula here, deferring instead to Lemma 6 below, as we will need to introduce
several objects in order for the formula to be understood in a meaningful way.
We note that, simultaneously to the results we present here a general
branching Markov process setting, similar ideas have been developed in [7].

Our main motivating application is to understand the limiting genealogy of a
so-called _critical_ branching Markov process, when conditioned to survive for
an arbitrarily long time. Our second main result, Proposition 8, is a general
statement in this direction. More precisely, for a critical non-local
branching Markov process conditioned to survive until a large time
$\displaystyle t$, we provide a precise asymptotic for the death time of the
most recent common ancestor of two individuals sampled uniformly from the
population at two different times. In Proposition 9, we also describe the
limiting law of the population sizes at two different times, in the same
asymptotic regime.

###  1.2 Set-up and assumptions

Let $\displaystyle E$ be a Lusin space. Throughout, will write $\displaystyle
B(E)$ for the Banach space of bounded measurable functions on $\displaystyle
E$ with norm $\displaystyle\lVert\cdot\rVert$, $\displaystyle B^{+}(E)$ for
non-negative bounded measurable functions on $\displaystyle E$ and
$\displaystyle B^{+}_{1}(E)$ for the subset of functions in $\displaystyle
B^{+}(E)$ which are uniformly bounded by unity.

We consider a spatial branching process in which, given their point of
creation, particles evolve independently according to a Markov process,
$\displaystyle(\xi,\mathbf{P})$, which can be characterised via the semigroup
$\displaystyle\mathtt{P}_{t}[f](x)=\mathbf{E}_{x}[f(\xi_{t})]$, for
$\displaystyle x\in E$, $\displaystyle t\geq 0$ and $\displaystyle f\in
B^{+}_{1}(E)$. In an event which we refer to as ‘branching’, particles
positioned at $\displaystyle x$ die at rate $\displaystyle\beta(x)$ where
$\displaystyle\beta\in B^{+}(E)$ and instantaneously, new particles are
created in $\displaystyle E$ according to a point process. The configurations
of these offspring are described by the random counting measure

$\mathcal{Z}(A)=\sum_{i=1}^{N}\delta_{x_{i}}(A),$ (1)

for Borel $\displaystyle A$ in $\displaystyle E$. The law of the
aforementioned point process may depend on $\displaystyle x$, the point of
death of the parent, and we denote it by $\displaystyle\mathcal{P}_{x}$,
$\displaystyle x\in E$, with associated expectation operator given by
$\displaystyle\mathcal{E}_{x}$, $\displaystyle x\in E$. This information is
captured in the so-called branching mechanism

$\mathtt{G}[f](x):=\beta(x)\mathcal{E}_{x}\left[\prod_{i=1}^{N}f(x_{i})-f(x)\right],\qquad
x\in E,\,f\in B^{+}_{1}(E).$ (2)

Without loss of generality we can assume that
$\displaystyle\mathcal{P}_{x}(N=1)=0$ for all $\displaystyle x\in E$ by
viewing a branching event with one offspring as an extra jump in the motion.
On the other hand, we do allow for the possibility that
$\displaystyle\mathcal{P}_{x}(N=0)>0$ for some or all $\displaystyle x\in E$.

Moreover, we do not need $\displaystyle\mathtt{P}$ to have the Feller
property, and it is not necessary that $\displaystyle\mathtt{P}$ is
conservative. That said, if so desired, we can append a cemetery state
$\displaystyle\\{\dagger\\}$ to $\displaystyle E$, which is to be treated as
an absorbing state, and regard $\displaystyle\mathtt{P}$ as conservative on
the extended space $\displaystyle E\cup\\{\dagger\\}$, which can also be
treated as a Lusin space. Equally, we can extend $\displaystyle\mathtt{G}$ to
$\displaystyle E\cup\\{\dagger\\}$ by defining it to be zero on
$\displaystyle\\{\dagger\\}$, that is no branching activity on the cemetery
state.

Henceforth we refer to this spatial branching process as a
$\displaystyle(\mathtt{P},\mathtt{G})$-branching Markov process. It is well
known that if we arbitrarily choose an order for the particles at each
branching event then we may denote the configuration of particles at time
$\displaystyle t$ by $\displaystyle\\{x_{1}(t),\ldots,x_{N_{t}}(t)\\}$ (where
$\displaystyle N_{t}$ denotes the number of particles alive at time
$\displaystyle t$) and we can also associate an Ulam-Harris label

$\displaystyle v_{i}(t)\in\Omega:=\\{\emptyset\\}\cup\bigcup_{n\geq
1}\mathbb{N}^{n}$

to the $\displaystyle i$-th particle at time $\displaystyle t$. For
$\displaystyle v,w\in\Omega$, we write $\displaystyle v\preceq w$ to mean that
$\displaystyle v$ is an ancestor of $\displaystyle w$, which means there
exists $\displaystyle u\in\Omega$ such that $\displaystyle vu=w$. Moreover, we
write $\displaystyle v\prec w$ to mean that $\displaystyle v\preceq w$ in the
strict sense, that is, the possibility that $\displaystyle v=w$ is excluded.
We say that $\displaystyle v,w\in\Omega$ are siblings if there exists
$\displaystyle u\in\Omega$ and $\displaystyle i\neq j$ such that
$\displaystyle v=ui$ and $\displaystyle w=uj$.

The branching Markov process can be described via the co-ordinate process
$\displaystyle X=(X_{t},t\geq 0)$ in the space of counting measures on
$\displaystyle E\times\Omega$ with non-negative integer total mass, denoted by
$\displaystyle M(E\times\Omega)$, where

$\displaystyle
X_{t}(\cdot)=\sum_{i=1}^{N_{t}}\delta_{(x_{i}(t),v_{i}(t))}(\cdot),\qquad
t\geq 0.$

In particular, $\displaystyle X$ is Markovian in $\displaystyle
M(E\times\Omega)$. Its probabilities will be denoted
$\displaystyle\mathbb{P}:=(\mathbb{P}_{\delta_{x}},x\in E)$ where for
$\displaystyle x\in E$, $\displaystyle\mathbb{P}_{\delta_{x}}$ denotes the law
of the process starting from $\displaystyle\delta_{(x,\emptyset)}\in
M(E\times\Omega)$.

Under the additional assumption that $\displaystyle\mathrm{sup}_{x\in
E}\,\mathcal{E}_{x}(N)<\infty$, where we recall that $\displaystyle N$ is the
(random) number of offspring produced at a branching event, we define the
linear semigroup

$\displaystyle\mathtt{T}_{t}[f](x):=\mathbb{E}_{\delta_{x}}[X_{t}[f]]:=\mathbb{E}_{\delta_{x}}\left[\sum_{i=1}^{N_{t}}f(x_{i}(t))\right],\quad
f\in B^{+}(E).$

Now let us introduce an assumption, that we will use throughout the article
unless stated otherwise. We will use the notation

$\displaystyle\langle f,\mu\rangle:=\int_{E}f(x)\mu({\rm d}x),\quad f\in
B(E),\mu\in M(E),$

where $\displaystyle M(E)$ is the set of finite measures on $\displaystyle E$.

######  Assumption 1.

For the Markov process $\displaystyle(\xi,\mathbf{P})$, we assume that

  1. (a)

it admits a càdlàg modification;

  2. (b)

there exists an eigenvalue $\displaystyle\lambda\in\mathbb{R}$ and a
corresponding right eigenfunction $\displaystyle\varphi\in B^{+}(E)$ and
finite left eigenmeasure $\displaystyle\tilde{\varphi}$ such that, for
$\displaystyle f\in B^{+}(E)$,

$\displaystyle\langle\mathtt{T}_{t}[\varphi],\mu\rangle={\rm e}^{\lambda
t}\langle{\varphi},{\mu}\rangle\text{ and
}\langle{\mathtt{T}_{t}[f]},{\tilde{\varphi}}\rangle={\rm e}^{\lambda
t}\langle f,{\tilde{\varphi}}\rangle,$

for all $\displaystyle\mu\in M(E)$.

The first part of the above assumption is a regularity assumption on the
Markov process $\displaystyle(\xi,\mathbf{P})$, which ensures that we can use
the theory of martingale changes of measure. The second is a Perron Frobenius
assumption that ensures the existence of the leading eigenvalue and
corresponding eigenfunctions.

###  1.3 Outline

The many-to-few formula, Lemma 6, will ultimately allow us to express
expectations of general $\displaystyle k$-fold sums depending on the entire
branching process under $\displaystyle\mathbb{P}$, in terms of an expectation
with respect to $\displaystyle k$ so-called _spine particles_ under a
different measure $\displaystyle\mathbb{Q}^{k}$. In Section 2, we define this
measure $\displaystyle\mathbb{Q}^{k}$, introduce the notion of spines, and
give an explicit expression for the Radon-Nikodym derivative of
$\displaystyle\mathbb{Q}^{k}$, with respect to the original law of our
branching process plus some uniformly chosen marked lines of descent (we call
this measure $\displaystyle\mathbb{P}^{k}$). We then state and prove our main
result, the many-to-few formula, in Section 3. We also explain some special
cases in which the formula simplifies nicely. Section 4 is devoted to our main
application, which is to derive some two-point asymptotics for the geneologies
of critical branching processes, when they are conditioned to survive for a
large time $\displaystyle t$. More precisely, we provide an asymptotic (as
$\displaystyle t\to\infty$) for the death time of the most recent common
ancestor of two particles sampled uniformly from the population at two
different times.

##  2 Spines, martingales and changes of measure

In this section, we introduce two measures under which our class of branching
Markov process additionally identifies $\displaystyle k$ distinguished
genealogical lines of descent, or spines. The first of these two measures is a
simple adaptation of the original law. The second measure is identified via a
change of measure with respect to a certain multiplicative martingale.

###  2.1 Definition of the measures $\displaystyle\mathbb{P}^{k}$ and
$\displaystyle\mathbb{Q}^{k}$

####  2.1.1 Definition of $\displaystyle\mathbb{P}^{k}$

We first introduce a measure $\displaystyle\mathbb{P}^{k}$ on the set of
processes taking values in $\displaystyle
M(E\times\Omega\times\mathcal{P}(\\{1,\ldots,k\\}))$, the space of counting
measures on $\displaystyle E\times\Omega\times\mathcal{P}(\\{1,\ldots,k\\})$,
where $\displaystyle\mathcal{P}(\\{1,\ldots,k\\})$ is the set of subsets of
$\displaystyle\\{1,\dots,k\\}$. For convenience, let $\displaystyle\tilde{X}$
denote the branching process on the space $\displaystyle
M(E\times\Omega\times\mathcal{P}(\\{1,\ldots,k\\}))$, that is

$\displaystyle\tilde{X}_{t}=\sum_{i=1}^{N_{t}}\delta_{(x_{i}(t),v_{i}(t),\mathbf{b}_{i}(t))},$

where $\displaystyle\mathbf{b}_{i}(t)\in\mathcal{P}(\\{1,\dots,k\\})$ denotes
the set of marks carried by the $\displaystyle i$-th particle alive at time
$\displaystyle t$. Whenever $\displaystyle\mathbf{b}_{i}(t)\neq\emptyset$, we
refer to the individual $\displaystyle i$ as a spine. In that case, we say
that the spine carries $\displaystyle|\mathbf{b}_{i}(t)|$ marks. Given
$\displaystyle\tilde{X}$, define $\displaystyle X$ to be its projection onto
$\displaystyle M(E\times\Omega)$. With this notation in hand, we let
$\displaystyle(\mathcal{F}_{t},t\geq 0)$ denote the natural filtration
generated by $\displaystyle X$ and $\displaystyle(\mathcal{F}_{t}^{k},t\geq
0)$ denote the natural filtration generated by $\displaystyle\tilde{X}$.

Then, we have the following description of the measure
$\displaystyle\mathbb{P}^{k}$.

######  Definition 1.

The construction of $\displaystyle\tilde{X}$ under
$\displaystyle\mathbb{P}^{k}$ goes as follows.

  1. 1.

We start with a single particle at $\displaystyle x\in E$ which carries
$\displaystyle k\geq 1$ marks.

  2. 2.

All particles move according to the semigroup $\displaystyle\mathtt{P}$,
independently of each other given their birth times and configurations.

  3. 3.

Let $\displaystyle\xi_{t}^{i}$ denote the position of the particle that
carries the mark $\displaystyle 1\leq i\leq k$ at time $\displaystyle t\geq
0$.

  4. 4.

A particle at $\displaystyle y\in E$ carrying $\displaystyle j$ marks
$\displaystyle b_{1}<b_{2}<\dots<b_{j}$, dies at rate $\displaystyle\beta(y)$
and simultaneously produces a random number of new particles according to
$\displaystyle(\mathcal{Z},\mathcal{P}_{y})$. The $\displaystyle j$ marks each
choose a particle to follow independently and uniformly from the
$\displaystyle N=\langle 1,\mathcal{Z}\rangle$ available particles.

  5. 5.

In the event that a particle carrying $\displaystyle j>0$ marks dies and is
replaced by $\displaystyle 0$ offspring, it is sent to the cemetery state,
along with its marks.

Note that the above definition of $\displaystyle\mathbb{P}^{k}$ is such that
$\displaystyle X$ has the same law under both $\displaystyle\mathbb{P}^{k}$
and $\displaystyle\mathbb{P}$.

####  2.1.2 The measure $\displaystyle\mathbb{Q}^{k}$

We will now introduce a second measure, $\displaystyle\mathbb{Q}^{k}$, under
which particles not carrying any marks evolve in the same manner as particles
under $\displaystyle\mathbb{P}$, while spines (that is, particles carrying
marks) evolve differently. In the next section we will show that
$\displaystyle\mathbb{Q}^{k}$ can be defined via a change of measure of
$\displaystyle\mathbb{P}^{k}$. Before describing the process, we need to
introduce some more notation.

Suppose that we are given a functional $\displaystyle(\zeta(\cdot,t),t\geq 0)$
such that $\displaystyle\zeta(\xi,t)$ is a non-negative unit-mean martingale
with respect to the natural filtration of the Markov process
$\displaystyle(\xi_{t},t\geq 0)$ with semigroup
$\displaystyle(\mathtt{P}_{t},t\geq 0)$. We will assume that
$\displaystyle\zeta$ takes the value $\displaystyle 0$ whenever
$\displaystyle\xi_{t}=\dagger$. Now for $\displaystyle k,n\in\mathbb{N}$,
define

$\langle\varphi,\mathcal{Z}\rangle_{k,n}={\bf 1}_{(n\leq
N)}\sum_{[k_{1},\dots,k_{N}]_{k}^{n}}{{k}\choose{k_{1},\dots,k_{N}}}\prod_{i:k_{i}>0}\varphi(x_{i}),$
(3)

where $\displaystyle(x_{i},i=1,\dots,N)$ are as in (1) and
$\displaystyle[k_{1},\dots,k_{N}]_{k}^{n}$ is the set of non-negative integer
$\displaystyle N$-tuples $\displaystyle(k_{1},\dots,k_{N})$ such that
$\displaystyle k_{1}+\dots+k_{N}=k$ and exactly $\displaystyle n$ of the
$\displaystyle k_{i}$s are positive. If $\displaystyle\mathcal{Z}$ corresponds
to the offspring of a particle carrying $\displaystyle k$ marks, one can think
of a single term in the sum
$\displaystyle\langle\varphi,\mathcal{Z}\rangle_{k,n}$ as a weight associated
to the event that the $\displaystyle k$ marks are distributed among the
offspring, by giving exactly $\displaystyle k_{i}$ marks to the $\displaystyle
i$th offspring particle, $\displaystyle i=1,\ldots,N$ (see below for a more
precise interpretation). With this notation in hand, now define

$\displaystyle\displaystyle\langle\varphi,\mathcal{Z}\rangle_{k}(x)$
$\displaystyle\displaystyle:=\sum_{1\leq n\leq
k}\varphi(x)^{-n}\langle\varphi,\mathcal{Z}\rangle_{k,n}$ (4)

and set
$\displaystyle{\mathtt{m}}_{k}(x)=\mathcal{E}_{x}(\langle\varphi,\mathcal{Z}\rangle_{k}(x))$.
Note that in the case of local branching,
$\displaystyle\langle\varphi,\mathcal{Z}\rangle_{k}\equiv N^{k}$ for any
$\displaystyle x\in E$.

######  Definition 2.

The construction of $\displaystyle\tilde{X}$ under
$\displaystyle\mathbb{Q}^{k}$ goes as follows.

  1. 1.

Again, we begin with one particle at $\displaystyle x\in E$ carrying all the
marks $\displaystyle\\{1,\dots,k\\}$. In what follows, particles carrying
marks are referred to as spines.

  2. 2.

Any spine (that is, any particle carrying any number of marks) moves according
to the semigroup

$\mathtt{P}_{t}[{g}](x):=\frac{1}{\zeta(\xi,0)}\mathbf{E}_{x}\left[\zeta(\xi,t)g(\xi_{t})\right],\qquad
x\in E,$ (5)

where $\displaystyle(\xi_{s},s\geq 0)$ denotes the motion.

  3. 3.

Suppose a spine carries marks $\displaystyle\mathbf{b}=(b_{1},\dots,b_{j})$.
Then for each $\displaystyle 1\leq n\leq j$, an independent exponential clock
rings at rate $\displaystyle\beta(x){\mathtt{m}}_{j,n}(x),$

$\displaystyle{\mathtt{m}}_{j,n}(x):=\varphi(x)^{-n}\mathcal{E}_{x}(\langle\varphi,\mathcal{Z}\rangle_{j,n}),\qquad
x\in E.$

When the first of these clocks rings, a branching event occurs, and if it is
the $\displaystyle n$th clock, the $\displaystyle j$ marks carried by the
parent will be given to exactly $\displaystyle n$ distinct offspring
particles.

More precisely, if the first clock to ring is the $\displaystyle n$th one, the
positions of the offspring are described by $\displaystyle\mathcal{Z}$ with
law $\displaystyle\mathcal{P}_{x}^{(j,n)}$ defined by

$\frac{{\rm d}\mathcal{P}_{x}^{(j,n)}}{{\rm
d}\mathcal{P}_{x}}:=\frac{\langle\varphi,\mathcal{Z}\rangle_{j,n}}{\mathcal{E}_{x}(\langle\varphi,\mathcal{Z}\rangle_{j,n})}.$
(6)

Then given $\displaystyle\mathcal{Z}$, for each
$\displaystyle(k_{1},\dots,k_{N})\in[k_{1},\dots,k_{N}]^{n}_{j}$, the
probability that the $\displaystyle i$th offspring particle receives exactly
$\displaystyle k_{i}$ marks for each $\displaystyle 1\leq i\leq N$, is given
by

$\displaystyle\frac{{{k}\choose{k_{1},\dots,k_{N}}}\prod_{i:k_{i}>0}\varphi(x_{i})}{\langle\varphi,\mathcal{Z}\rangle_{j,n}}.$

On this event, the way that the marks $\displaystyle b_{1},\dots,b_{j}$ are
distributed among the offspring is that such that any valid configuration
(that is, satisfying the constraint that exactly $\displaystyle k_{i}$ marks
are given to offspring particle $\displaystyle i$ for each $\displaystyle
1\leq i\leq N$) has the same probability:

$\displaystyle\frac{1}{{{k}\choose{k_{1},\dots,k_{N}}}}.$

  4. 4.

Particles that do not carry marks issue independent copies of
$\displaystyle(X,\mathbb{P})$. Marked particles then continue from Step 2.

######  Remark 1.

An alternative description of the third step above, is in terms of the _total_
branching rate. Namely, suppose a spine carries marks
$\displaystyle\mathbf{b}=(b_{1},\dots,b_{j})$. Then it branches at rate
$\displaystyle\beta(x)\mathtt{m}_{j}(x)$, and on such a branching event, the
offspring positions are described by $\displaystyle\mathcal{Z}$ whose law is
weighted (with respect to $\displaystyle\mathcal{P}_{x}$) by
$\displaystyle\langle\varphi,\mathcal{Z}\rangle_{j}/\mathtt{m}_{j}(x)$.
Moreover, given $\displaystyle\mathcal{Z}$ (and the position $\displaystyle x$
of the spine before branching), any particular allocation of the marks
$\displaystyle b_{1},\ldots,b_{j}$ among the $\displaystyle N$ offspring (that
is, a partition $\displaystyle S_{1},\dots,S_{N}$ of
$\displaystyle\\{b_{1},\dots,b_{j}\\}$) has probability equal to
$\displaystyle\langle\varphi,\mathcal{Z}\rangle_{j}^{-1}\textstyle\prod_{i:|S_{i}|>0}(\varphi(x_{i})/\varphi(x))$.
However, we prefer to highlight the decomposition according to $\displaystyle
n$ (as given in the description of the measure above), since this will be the
one we use in practice.

######  Remark 2.

There are many different variants of the measure $\displaystyle\mathbb{Q}^{k}$
that we could have described, that would also be related to
$\displaystyle\mathbb{P}^{k}$ by a martingale change of measure, and would
also lead to a many-to-few type formula. Our specific choice of
$\displaystyle\mathbb{Q}^{k}$ is motivated by our main application: describing
the genealogical structure of the branching process when it is conditioned on
survival. In particular, when we use our many-to-few formula for this purpose,
we get an extremely simple structure - see (3.2). Note that in the case of
local branching, this measure $\displaystyle\mathbb{Q}^{k}$ is identical to
that which appears in [14]. When $\displaystyle k=1$, and we make a particular
choice for $\displaystyle\zeta$, this also agrees with the “spine
decomposition” given in [11].

###  2.2 The martingale change of measure

We will now explain how the measures $\displaystyle\mathbb{P}^{k}$ and
$\displaystyle\mathbb{Q}^{k}$ are connected via a change of measure. Let us
first introduce some further notation.

Given $\displaystyle v\in\Omega$, note that $\displaystyle
X_{t}(E\times\\{v\\})=0$ except on some unique (possibly empty) interval
$\displaystyle[\sigma_{v},\tau_{v})$, on which $\displaystyle
X_{t}(E\times\\{v\\})=1$. We will often use the notation
$\displaystyle\tau_{v}^{-}$ for the left limit of $\displaystyle\tau_{v}$. If
$\displaystyle t\in[\sigma_{v},\tau_{v})$, there exists a unique
$\displaystyle X_{v}(t)\in E$ such that $\displaystyle
X_{t}(X_{v}(t)\times\\{v\\})=1$ and a unique
$\displaystyle\mathbf{b}_{v}\in\mathcal{P}(\\{1,\dots,k\\})$ such that
$\displaystyle\tilde{X}_{t}(E\times\\{v\\}\times\mathbf{b}_{v})=1$ for all
$\displaystyle t\in[\sigma_{v},\tau_{v})$. Heuristically,
$\displaystyle\sigma_{v}$ and $\displaystyle\tau_{v}$ are the birth and death
times of particle $\displaystyle v$, respectively, $\displaystyle X_{v}(t)$
represents its position at time $\displaystyle t$ during its lifetime, and
$\displaystyle{\bf b}_{v}$ represents the set of marks it carries. We further
set $\displaystyle D_{v}=|\mathbf{b}_{v}|$ to be the number of marks carried
by the particle with label $\displaystyle v$. For each $\displaystyle
v\in\Omega$, let $\displaystyle N_{v}$ denote the number of offspring produced
by $\displaystyle X_{v}(\tau_{v})$.

Set $\displaystyle\mathcal{N}_{t}:=\\{v\in\Omega:t\in[\sigma_{v},\tau_{v})\\}$
so that $\displaystyle N_{t}=|\mathcal{N}_{t}|$. For each $\displaystyle t\geq
0$ and $\displaystyle j=1,\dots,k$, let $\displaystyle\psi_{t}^{j}$ and
$\displaystyle\xi_{t}^{j}$ denote the unique elements of $\displaystyle\Omega$
and $\displaystyle E$, respectively, such that there exists
$\displaystyle\mathbf{b}\in\mathcal{P}(\\{1,\dots,k\\})$ with $\displaystyle
j\in\mathbf{b}$ and
$\displaystyle\tilde{X}_{t}(\xi_{t}^{j}\times\\{\psi_{t}^{j}\\}\times\mathbf{b})=1$.
Finally, let us define the skeleton at time $\displaystyle t$ to be
$\displaystyle S_{k}(t):=\\{\psi_{s}^{1},\ldots,\psi_{s}^{k}:s\leq t\\}$, so
that $\displaystyle S_{k}(t)$ is a subset of labels in the tree. For
$\displaystyle v\in S_{k}(t)$, let $\displaystyle M_{v}$ denote the number of
distinct offspring of $\displaystyle v$ that are given a mark (that is, the
number of distinct spine offspring).

######  Definition 3.

Define the $\displaystyle{\mathcal{F}}_{t}^{k}$-adapted process
$\displaystyle({W}_{t}^{k},t\geq 0)$ by

$\displaystyle\displaystyle{W}_{t}^{k}:=$
$\displaystyle\displaystyle\prod_{v\in
S_{k}(t)}\frac{\zeta(X_{v},\tau_{v}^{-}\wedge
t)}{\zeta(X_{v},\sigma_{v})}\prod_{v\in S_{k}(t)}{\rm
e}^{-\int_{\sigma_{v}}^{\tau_{v}\wedge
t}\beta(X_{v}(s))({\mathtt{m}}_{D_{v}}(X_{v}(s))-1){\rm d}s}$
$\displaystyle\displaystyle\times\prod_{v\in
S_{k}(t)\backslash\\{\emptyset\\}}{\varphi(X_{v}(\sigma_{v}))}\prod_{v\in
S_{k}(t)\backslash\mathcal{N}_{t}}\frac{N_{v}^{D_{v}}}{\varphi(X_{v}(\tau_{v}^{-}))^{M_{v}}}.$
(7)

######  Remark 3.

Henceforth, for $\displaystyle s\leq t$ and $\displaystyle x,y\in E$, we set
$\displaystyle\zeta(x,t)/\zeta(y,s)=1$ whenever
$\displaystyle\zeta(x,t)=\zeta(y,s)=0$.

######  Remark 4.

We emphasise that then when a branching event occurs and the spines all choose
the same particle to follow, this is included as an element of the skeleton,
$\displaystyle S_{k}(t)$.

######  Remark 5.

Note that we may equivalently write

$\displaystyle\displaystyle W_{t}^{k}=$ $\displaystyle\displaystyle\prod_{v\in
S_{k}(t)}\frac{\zeta(X_{v},\tau_{v}^{-}\wedge t)}{\zeta(X_{v},\sigma_{v})}$
$\displaystyle\displaystyle\times\prod_{v\in
S_{k}(t)\backslash\mathcal{N}_{t}}{\mathtt{m}_{D_{v}}(X_{v}(\tau_{v}^{-}))}{\rm
e}^{-\int_{\sigma_{v}}^{\tau_{v}}\beta(X_{v}(s))(\mathtt{m}_{D_{v}}(X_{v}(s))-1){\rm
d}s}\prod_{v\in\mathcal{N}_{t}}{\rm
e}^{-\int_{\sigma_{v}}^{t}\beta(X_{v}(s))(\mathtt{m}_{D_{v}}(X_{v}(s))-1){\rm
d}s}$ $\displaystyle\displaystyle\times\prod_{v\in
S_{k}(t)\backslash\mathcal{N}_{t}}\frac{\langle\varphi,\mathcal{Z}_{v}\rangle_{D_{v}}(X_{v}(\tau_{v}^{-}))}{\mathtt{m}_{D_{v}}(X_{v}(\tau_{v}^{-}))}$
$\displaystyle\displaystyle\times\prod_{v\in
S_{k}(t)\backslash\\{\emptyset\\}}\frac{N_{p_{v}}^{D_{p_{v}}}}{\langle\varphi,\mathcal{Z}_{p_{v}}\rangle_{D_{p_{v}}}(X_{v}(\tau_{p_{v}}^{-}))}\frac{\varphi(X_{v}(\sigma_{v}))}{\varphi(X_{v}(\tau_{p_{v}}^{-}))^{M_{p_{v}}}},$

where $\displaystyle p_{v}$ is the label of the parent of particle with label
$\displaystyle v$. The above decomposition holds since

$\displaystyle\displaystyle\prod_{v\in S_{k}(t)\backslash\mathcal{N}_{t}}$
$\displaystyle\displaystyle\frac{N_{v}^{D_{v}}}{\varphi(X_{v}(\tau_{v}^{-}))^{M_{v}}\,\langle\varphi,\mathcal{Z}_{v}\rangle_{D_{v}}(X_{v}(\tau_{v}^{-}))}\prod_{v\in
S_{k}(t)\backslash\\{\emptyset\\}}\varphi(X_{v}(\sigma_{v}))$
$\displaystyle\displaystyle=\prod_{v\in
S_{k}(t)\backslash\\{\emptyset\\}}\frac{N_{p_{v}}^{D_{p_{v}}}}{\langle\varphi,\mathcal{Z}_{p_{v}}\rangle_{D_{p_{v}}}(X_{v}(\tau_{p_{v}}^{-}))}\frac{\varphi(X_{v}(\sigma_{v}))}{\varphi(X_{v}(\tau_{p_{v}}^{-}))^{M_{p_{v}}}},$

by noting that in the product on the left-hand side of the above equality,
each of the terms in the first product is a parent of an element in the second
product. Using Remark 1, one can see that each of the terms above describes a
change of measure with respect to $\displaystyle\mathbb{P}^{k}$ for, in order:
the motion; branch rate; offspring distribution; and selection of the spine
particles immediately after a branching event. These changes of measure
account for the differences between the pathwise constructions of the measures
$\displaystyle\mathbb{P}^{k}$ and $\displaystyle\mathbb{Q}^{k}$. Indeed, we
have the following result.

######  Proposition 4.

For $\displaystyle x\in E$, $\displaystyle({W}_{t}^{k},t\geq 0)$ is a
martingale. Define

$\frac{{\rm d}\widetilde{\mathbb{Q}}_{\delta_{x}}^{k}}{{\rm
d}{\mathbb{P}}_{\delta_{x}}^{k}}\Bigg{|}_{\mathcal{F}_{t}^{k}}={W}_{t}^{k},\qquad
t\geq 0,\,x\in E.$ (8)

Then for all $\displaystyle x\in E$,
$\displaystyle\mathbb{Q}_{\delta_{x}}^{k}=\widetilde{\mathbb{Q}}_{\delta_{x}}^{k}$.

###### Proof.

In the spirit of [3], and the proofs of Proposition 11 and Theorem 12 in [9],
it suffices to demonstrate that the change of measure holds for the behaviour
of the initial particle, up to and including its branching event. Thereafter,
the Markov property ensures that the result is true in general.

To this end, let us suppose that $\displaystyle T_{1}$ is the first branch
time. According to the definition of
$\displaystyle\mathbb{Q}_{\delta_{x}}^{k}$ we have, for $\displaystyle x\in E$
and any bounded measurable $\displaystyle H$,

$\displaystyle\mathbb{Q}_{\delta_{x}}^{k}[H(\xi_{s},0\leq s\leq
t);t<T_{1}]=\mathbf{E}_{x}\left[\frac{\zeta(\xi,t)}{\zeta(\xi,0)}H(\xi_{s},s\leq
t){\rm e}^{-\int_{0}^{t}\beta(\xi_{s}){\mathtt{m}}_{k}(\xi_{s}){\rm
d}s}\right],\qquad t\geq 0,$

where, on $\displaystyle\\{t<T_{1}\\}$,
$\displaystyle\xi=\xi^{1}=\dots=\xi^{k}$. Noting similarly that

$\displaystyle\mathbb{P}_{\delta_{x}}^{k}[H(\xi_{s},0\leq s\leq
t);t<T_{1}]=\mathbf{E}_{x}\left[H(\xi_{s},s\leq t){\rm
e}^{-\int_{0}^{t}\beta(\xi_{s}){\rm d}s}\right],\qquad t\geq 0,$

it follows that

$\displaystyle\mathbb{Q}_{\delta_{x}}^{k}[H(\xi_{s},0\leq s\leq
t);t<T_{1}]=\mathbb{P}^{k}_{x}\left[\frac{\zeta(\xi,t)}{\zeta(\xi,0)}H(\xi_{s},s\leq
t){\rm e}^{-\int_{0}^{t}\beta(\xi_{s})({\mathtt{m}}_{k}(\xi_{s})-1){\rm
d}s}\right],\qquad t\geq 0,$

which agrees with (8) on $\displaystyle\\{t<T_{1}\\}$.

Next, we extend this to include what happens at the first branch event. Again
referring to the definition of $\displaystyle\mathbb{Q}^{k}_{\delta_{x}}$, for
$\displaystyle t>0$, $\displaystyle x\in E$, $\displaystyle n\in\mathbb{N}$
and $\displaystyle H$ and $\displaystyle\xi$ as before, $\displaystyle f\in
B(E)$ and $\displaystyle i_{1},\dots,i_{k}\in\mathbb{N}$ such that
$\displaystyle|\\{i_{1},\dots,i_{k}\\}|=n$, $\displaystyle
L_{\mathbf{i}}:=\\{l\in\mathbb{N}\text{ s.t. }i_{j}=l\text{ for some }1\leq
j\leq k\\}$

$\displaystyle\displaystyle\mathbb{Q}_{\delta_{x}}^{k}[H(\xi_{s},0\leq s\leq
t){\rm e}^{-\langle f,\mathcal{Z}\rangle}\mathbf{1}_{\\{T_{1}\in{\rm
d}t\\}}\mathbf{1}_{\\{i_{1},\dots,i_{k}\in\\{1,\dots
N\\}\\}}\mathbf{1}_{\\{\psi_{t}^{1}=i_{1},\dots,\psi^{k}_{t}=i_{k}\\}}]$
$\displaystyle\displaystyle=\mathbb{P}^{k}_{x}\Bigg{[}\frac{\zeta(\xi,t)}{\zeta(\xi,0)}H(\xi_{s},s\leq
t){\rm e}^{-\langle f,\mathcal{Z}\rangle}{\rm
e}^{-\int_{0}^{t}\beta(\xi_{s})({\mathtt{m}}_{k}(\xi_{s})-1){\rm d}s}$
$\displaystyle\displaystyle\hskip
28.45274pt\times\mathbf{1}_{\\{i_{1},\dots,i_{k}\in\\{1,\dots
N\\}\\}}\mathbf{1}_{\\{\psi_{t}^{1}=i_{1},\dots,\psi^{k}_{t}=i_{k}\\}}\beta(\xi_{t})\mathtt{m}_{k}(\xi_{t})\frac{{\mathtt{m}}_{k,n}(\xi_{t})}{\mathtt{m}_{k}(\xi_{t})}\frac{\langle\varphi,\mathcal{Z}\rangle_{k,n}}{\mathcal{E}_{\xi_{t}}(\langle\varphi,\mathcal{Z}\rangle_{k,n})}\frac{\prod_{l\in
L_{\mathbf{i}}}\varphi(x_{l})}{\langle\varphi,\mathcal{Z}\rangle_{k,n}}\frac{1}{1/N^{k}}\Bigg{]}{\rm
d}t,$
$\displaystyle\displaystyle=\mathbb{P}^{k}_{x}\Bigg{[}\frac{\zeta(\xi,t)}{\zeta(\xi,0)}H(\xi_{s},s\leq
t){\rm e}^{-\langle f,\mathcal{Z}\rangle}{\rm
e}^{-\int_{0}^{t}\beta(\xi_{s})({\mathtt{m}}_{k}(\xi_{s})-1){\rm d}s}$
$\displaystyle\displaystyle\hskip
113.81102pt\times\mathbf{1}_{\\{i_{1},\dots,i_{k}\in\\{1,\dots
N\\}\\}}\mathbf{1}_{\\{\psi_{t}^{1}=i_{1},\dots,\psi^{k}_{t}=i_{k}\\}}\beta(\xi_{t})\varphi(\xi_{t})^{-n}\prod_{l\in
L_{\mathbf{i}}}\varphi(x_{l})\,N^{k}\Bigg{]}{\rm d}t,$

where the factor $\displaystyle 1/N^{k}$ is removing the selection bias of the
$\displaystyle k$-spines under $\displaystyle\mathbb{P}^{k}$. Noting that, in
the above calculation, on $\displaystyle\\{T_{1}\in{\rm d}t\\}$,
$\displaystyle S_{k}(t)\backslash\mathcal{N}_{t}=\\{\emptyset\\}$, the set
$\displaystyle S_{k}(t)\backslash\\{\emptyset\\},$ agrees with the offspring
of $\displaystyle\emptyset$,
$\displaystyle\varphi(X_{\emptyset}(\tau_{\emptyset}^{-}))=\varphi(\xi_{t})$,
$\displaystyle M_{\emptyset}=n$ and $\displaystyle N_{\emptyset}=k$. As such
we note that on $\displaystyle\\{T_{1}\in{\rm d}t\\}$, the change of measure
(8) is valid. ∎

###  2.3 Spines at different times

We would also like to consider the “skeleton at different times”. To this end,
fix $\displaystyle k\geq 1$, suppose $\displaystyle 0\leq s_{k}\leq\dots\leq
s_{1}$ and write $\displaystyle\mathbf{s}=(s_{1},\dots,s_{k})$. Let

$\displaystyle\mathcal{N}_{\mathbf{s}}:=\\{(v_{1},\dots,v_{k}):v_{i}\in\mathcal{N}_{s_{i}}\,1\leq
i\leq k\\}$

and for
$\displaystyle\mathbf{v}=(v_{1},\dots,v_{k})\in\mathcal{N}_{\mathbf{s}}$, let

$\displaystyle S_{k}(\mathbf{v},\mathbf{s}):=\\{w\in\Omega:w\preceq
v_{i}\text{ for some }1\leq i\leq k\\}$

be the “skeleton” formed by the ancestors of $\displaystyle v_{i}$ up to times
$\displaystyle s_{i}$. We also write

$\displaystyle
S_{k}(\mathbf{s}):=S_{k}((\psi_{s_{1}}^{1},\dots,\psi_{s_{k}}^{k}),\mathbf{s})$

for the skeleton generated by the spines and

$\displaystyle\partial S_{k}(\mathbf{s})=\\{v\in S_{k}(\mathbf{s})\text{ such
that }\nexists\,w\in S_{k}(\mathbf{s})\text{ with }v\prec w\\}$

for the “leaves” of this skeleton. If for each $\displaystyle 1\leq i\leq k$
we set

$\displaystyle r_{i}=\sup\\{s>0:\psi_{s}^{i}=\psi_{s}^{k}\text{ for some
}k\text{ with }s_{k}\geq s\\}.$

then for each $\displaystyle i$ with $\displaystyle r_{i}=s_{i}$,
$\displaystyle v=\psi_{s_{i}}^{i}$ is an element of $\displaystyle\partial
S_{k}(\mathbf{s})$. We define a further set of labels associated to those
$\displaystyle i$ for which $\displaystyle r_{i}>s_{i}$:

$\displaystyle\hat{\partial}S_{k}(\mathbf{s})=\\{\psi_{r_{i}}^{i},\,1\leq
i\leq k,r_{i}>s_{i}\\}.$

Figure 1: The spines carrying marks $\displaystyle 1$, $\displaystyle 2$ and
$\displaystyle 3$ are depicted in cyan, purple and orange respectively. The
times $\displaystyle s_{i}$ and $\displaystyle r_{i}$ for $\displaystyle
i=1,2,3$ are shown (the dotted vertical lines correspond to $\displaystyle
r_{1},r_{2}$ and $\displaystyle r_{3}$).

Note that while $\displaystyle\partial S_{k}(\mathbf{s})$ is a subset of
$\displaystyle S_{k}(\mathbf{s})$,
$\displaystyle\hat{\partial}S_{k}(\mathbf{s})$ is not. Finally, for
$\displaystyle
v=\psi_{r_{i}}^{i}\in\hat{\partial}S_{k}(\mathbf{s})\cup\partial
S_{k}(\mathbf{s})$ we define

$r_{v}:=r_{i}$ (9)

(so for $\displaystyle v\in\psi_{s_{i}}^{i}\in\partial S_{k}(\mathbf{s})$, we
equivalently have $\displaystyle r_{v}=s_{i}$). See Figure 1.

Now fix $\displaystyle t\geq s_{1}$ and let
$\displaystyle\mathcal{F}_{t,\mathbf{s}}^{k}$ denote the
$\displaystyle\sigma$-algebra generated by:

  * •

$\displaystyle\\{\xi^{i}_{s}:s\leq s_{i},1\leq i\leq k\\}$ (the motion of the
spine with mark $\displaystyle i$ up to time $\displaystyle s_{i}$ for each
$\displaystyle i$);

  * •

$\displaystyle\\{\psi^{i}_{s}:s\leq s_{i},1\leq i\leq k\\}$ (the Ulam-Harris
labels associated to the the spine with mark $\displaystyle i$ up to time
$\displaystyle s_{i}$ for each $\displaystyle i$);

  * •

$\displaystyle\\{\mathbf{b}_{\psi^{i}_{s}}:s\leq s_{i},1\leq i\leq k\\}$ (the
collection of marks carried by the spine with mark $\displaystyle i$ up to
time $\displaystyle s_{i}$ for each $\displaystyle i$);

  * •

the subtree rooted at each $\displaystyle w\in\Omega$ that does not carry any
marks and is a sibling of _some_ $\displaystyle v\in
S_{k}(\mathbf{s})$111where by subtree we mean the subprocess started at time
$\displaystyle\sigma_{w}$ with root label $\displaystyle w$, considered up
until (global) time $\displaystyle t$.

Note that the collection of random variables in the third bullet point above
will not always be measurable with respect to the collection in the second.
For example, if $\displaystyle k=2$ and the spine carrying mark $\displaystyle
1$ also carries mark $\displaystyle 2$ at time $\displaystyle
s\in(s_{2},s_{1})$, then since $\displaystyle\\{\psi_{s}^{i}:s\leq
s_{i},i=1,2\\}$ does not tell us about the labels associated to the spine with
mark $\displaystyle 2$ after time $\displaystyle s_{2}$,
$\displaystyle\mathbf{b}_{\psi_{s}^{1}}$ is not measurable with respect to it.

We will also use the notation

$\displaystyle\mathcal{F}_{\mathbf{s}}^{k}:=\mathcal{F}_{s_{1},\mathbf{s}}^{k}.$

Figure 2: The left-hand side figure shows the tree up to time $\displaystyle
s_{1}$ with three spines marked in black. The right-hand side shows the
information from the filtration $\displaystyle\mathcal{F}_{s_{1},\bf{s}}^{3}$,
with $\displaystyle{\bf s}=(s_{1},s_{2},s_{3})$, with dashed lines denoting
information that is not included.

For each $\displaystyle\mathbf{s}$, we define an
$\displaystyle\mathcal{F}_{\mathbf{s}}^{k}$-measurable random variable

$\displaystyle\displaystyle{W}^{k}_{\mathbf{s}}:=$
$\displaystyle\displaystyle\prod_{v\in
S_{k}(\mathbf{s})}\frac{\zeta(X_{v},\tau_{v}^{-}\wedge
r_{v})}{\zeta(X_{v},\sigma_{v})}{\rm e}^{-\int_{\sigma_{v}}^{\tau_{v}\wedge
r_{v}}\beta(X_{v}(s))({\mathtt{m}}_{D_{v}}(X_{v}(s))-1){\rm d}u}$
$\displaystyle\displaystyle\times\prod_{v\in
S_{k}(\mathbf{s})\backslash\\{\emptyset\\}}{\varphi(X_{v}(\sigma_{v}))}\prod_{v\in
S_{k}(\mathbf{s})\backslash\partial
S_{k}(\mathbf{s})}\frac{N_{v}^{D_{v}}}{\varphi(X_{v}(\tau_{v}^{-}))^{M_{v}}},$
(10)

where we have used the notation $\displaystyle r_{v}$ defined in (9) for
$\displaystyle v\in\partial S_{k}(\mathbf{s})$ (and set $\displaystyle
r_{v}=\infty$ otherwise).

Recalling Lemma 4, restricting instead to $\displaystyle\mathcal{F}_{\bf
s}^{k}$ yields the following result.

######  Lemma 5.

For each $\displaystyle k\geq 1$, $\displaystyle 0\leq s_{k}\leq\ldots\leq
s_{1}$ and $\displaystyle x\in E$, we have

$\frac{{\rm d}\mathbb{Q}^{k}_{\delta_{x}}}{{\rm
d}\mathbb{P}^{k}_{\delta_{x}}}\Bigg{|}_{\mathcal{F}^{k}_{\mathbf{s}}}={W}^{k}_{\mathbf{s}}.$
(11)

###### Proof.

Fix $\displaystyle k\geq 1$, $\displaystyle 0\leq s_{k}\leq\dots\leq s_{1}\leq
t$ and $\displaystyle x\in E$. Then, due to the structure of
$\displaystyle{W}_{t}^{k}$, we may write

$\displaystyle\displaystyle{W}_{t}^{k}={W}_{\bf s}^{k}\,$
$\displaystyle\displaystyle\times\prod_{v\in S_{k}(t)\backslash
S_{k}(\mathbf{s})}\frac{\zeta(X_{v},\tau_{v}^{-}\wedge
t)}{\zeta(X_{v},\sigma_{v})}{\rm e}^{-\int_{\sigma_{v}}^{\tau_{v}\wedge
t}\beta(X_{v}(s))({\mathtt{m}}_{D_{v}}(X_{v}(s))-1){\rm d}u}$
$\displaystyle\displaystyle\times\prod_{v\in\partial
S_{k}(\mathbf{s})}\frac{\zeta(X_{v},\tau_{v}^{-}\wedge
t)}{\zeta(X_{v},r_{v})}{\rm e}^{-\int_{r_{v}}^{\tau_{v}\wedge
t}\beta(X_{v}(s))({\mathtt{m}}_{D_{v}}(X_{v}(s))-1){\rm d}u}$
$\displaystyle\displaystyle\times\prod_{v\in
V_{\mathbf{s},t}}\frac{N_{v}^{D_{v}}}{\varphi(X_{v}(\tau_{v}^{-}))^{M_{v}}}\prod_{v\in
S_{k}(t)\backslash S_{k}(\mathbf{s})}\varphi(X_{v}(\sigma_{v})),$ (12)

where $\displaystyle V_{\mathbf{s},t}:=(S_{k}(t)\setminus
S_{k}(\mathbf{s})\cup\mathcal{N}_{t})\cup\partial S_{k}(\mathbf{s})$ (in
words, the difference of the skeletons at times $\displaystyle t$ and
$\displaystyle\mathbf{s}$, minus the boundary at time $\displaystyle t$, but
plus the boundary at time $\displaystyle\mathbf{s}$). Hence,

$\displaystyle\displaystyle\mathbb{P}_{\delta_{x}}^{k}[{W}_{t}^{k}|\mathcal{F}_{\mathbf{s}}^{k}]={W}_{\bf
s}^{k}\,$
$\displaystyle\displaystyle\times\mathbb{P}^{k}_{\delta_{x}}\bigg{[}\prod_{v\in
S_{k}(t)\backslash S_{k}(\mathbf{s})}\frac{\zeta(X_{v},\tau_{v}^{-}\wedge
t)}{\zeta(X_{v},\sigma_{v})}{\rm e}^{-\int_{\sigma_{v}}^{\tau_{v}\wedge
t}\beta(X_{v}(s))({\mathtt{m}}_{D_{v}}(X_{v}(s))-1){\rm d}u}$
$\displaystyle\displaystyle\qquad\quad\times\prod_{v\in\partial
S_{k}(\mathbf{s})}\frac{\zeta(X_{v},\tau_{v}^{-}\wedge
t)}{\zeta(X_{v},r_{v})}{\rm e}^{-\int_{r_{v}}^{\tau_{v}\wedge
t}\beta(X_{v}(s))({\mathtt{m}}_{D_{v}}(X_{v}(s))-1){\rm d}u}$
$\displaystyle\displaystyle\qquad\quad\times\prod_{v\in
V_{\mathbf{s},t}}\frac{N_{v}^{D_{v}}}{\varphi(X_{v}(\tau_{v}^{-}))^{M_{v}}}\prod_{v\in
S_{k}(t)\backslash
S_{k}(\mathbf{s})}\varphi(X_{v}(\sigma_{v}))\bigg{|}\mathcal{F}_{\mathbf{s}}^{k}\bigg{]}.$
(13)

Now consider the collection of subprocesses initiated at times $\displaystyle
r_{v}$ for each $\displaystyle v\in\partial
S_{k}(\mathbf{s})\cup\hat{\partial}S_{k}(\mathbf{s})$. The branching Markov
property implies that these are independent of each other and of
$\displaystyle\mathcal{F}_{\mathbf{s}}^{k}$. Moreover, we may rewrite the
right-hand side of (2.3) as

$\displaystyle
W_{\mathbf{s}}^{k}\times\mathbb{P}^{k}_{\delta_{x}}\bigg{[}\prod_{v\in\hat{\partial}S_{k}(\mathbf{s})\cup\partial
S_{k}(\mathbf{s})}W_{t_{v}}^{(v)}\,\bigg{|}\,\mathcal{F}_{\mathbf{s}}^{k}\bigg{]},$

where for each $\displaystyle v\in\partial
S_{k}(\mathbf{s})\cup\hat{\partial}S_{k}(\mathbf{s})$, $\displaystyle W^{(v)}$
is a copy of the martingale $\displaystyle W^{D_{v}}$ associated to the
subprocess rooted at $\displaystyle v$ and $\displaystyle t_{v}=t-r_{v}$. In
particular,

$\displaystyle\mathbb{P}^{k}_{\delta_{x}}\bigg{[}\prod_{v\in\hat{\partial}S_{k}(\mathbf{s})\cup\partial
S_{k}(\mathbf{s})}W_{t_{v}}^{(v)}\,\bigg{|}\,\mathcal{F}_{\mathbf{s}}^{k}\bigg{]}=\prod_{v\in\hat{\partial}S_{k}(\mathbf{s})\cup\partial
S_{k}(\mathbf{s})}\mathbb{P}^{k}_{\delta_{x}}[W_{t_{v}}^{(v)}]=1,$

which gives the result. ∎

##  3 Many-to-few lemma

###  3.1 Statement and proof of the many-to-few lemma

We are now ready to formulate and prove our main result, which is a many-to-
few lemma for general non-local branching Markov processes at a collection of
different times. This generalises the result of [14] in two ways: firstly, it
holds for non-local branching mechanisms; and secondly, it allows us to deal
with sums over the population at different times.

######  Lemma 6 (Many-to-few at different times).

Let $\displaystyle x\in E$, $\displaystyle k\geq 1$ and $\displaystyle 0\leq
s_{k}\leq\dots\leq s_{1}$ be fixed. Suppose that

$\displaystyle
Y=\sum_{v_{i}\in\mathcal{N}_{s_{i}},\,i=1,\dots,k}Y(v_{1},\dots,v_{k})\mathbf{1}_{\\{\psi_{s_{i}}^{i}=v_{i},\,1\leq
i\leq k\\}}$

is non-negative and $\displaystyle\mathcal{F}_{\mathbf{s}}^{k}$-measurable,
where $\displaystyle Y(v_{1},\dots,v_{k})$ is
$\displaystyle\mathcal{F}_{s_{1}}$-measurable222By this we mean that we have a
collection $\displaystyle(Y(v_{1},\dots,v_{k}),v_{1},\dots,v_{k}\in\Omega)$ of
$\displaystyle\mathcal{F}_{s_{1}}$-measurable random variables such that
$\displaystyle Y(v_{1},\dots,v_{k})=0$ unless $\displaystyle
v_{i}\in\mathcal{N}_{s_{i}}$ for all $\displaystyle i=1,\dots,k$. for every
$\displaystyle(v_{1},\dots,v_{k})\in\mathcal{N}_{\mathbf{s}}$. Then it holds
that

$\displaystyle\displaystyle\mathbb{P}_{\delta_{x}}\Bigg{[}\sum_{\begin{subarray}{c}v_{i}\in\mathcal{N}_{s_{i}}\\\
i=1,\dots,k\end{subarray}}Y(v_{1},\dots,v_{k})\Bigg{]}$
$\displaystyle\displaystyle=\mathbb{Q}_{\delta_{x}}^{{k}}\bigg{[}{Y}\prod_{v\in
S_{k}(\mathbf{s})}\frac{\zeta(X_{v},\sigma_{v})}{\zeta(X_{v},\tau_{v}^{-}\wedge
s_{v})}{\rm e}^{\int_{\sigma_{v}}^{\tau_{v}\wedge
s_{v}}\beta(X_{v}(s))({\mathtt{m}}_{D_{v}}(X_{v}(s))-1){\rm d}u}$
$\displaystyle\displaystyle\hskip 22.76228pt\times\frac{\prod_{v\in
S_{k}(\mathbf{s})\backslash\partial
S_{k}(\mathbf{s})}\varphi(X_{v}(\tau_{v}^{-}))^{M_{v}}}{\prod_{v\in
S_{k}(\mathbf{s})\backslash\\{\emptyset\\}}\varphi(X_{v}(\sigma_{v}))}\prod_{v\in
S_{k}(\mathbf{s})\backslash\partial
S_{k}(\mathbf{s})}N_{v}^{{-D_{v}}}\prod_{i=1}^{k}\prod_{\emptyset\preceq
v\prec\psi_{s_{i}}^{i}}N_{v}\bigg{]}.$

######  Remark 6.

Recall that

$\displaystyle S_{k}(\mathbf{s})\backslash\partial
S_{k}(\mathbf{s})=\\{w\in\Omega:\emptyset\preceq w\prec\psi^{i}_{s_{i}}\text{
for some }1\leq i\leq k\\}.$

Also recall that the mark carried by an individual (spine) $\displaystyle v$
is $\displaystyle{\bf b}_{v}$ with cardinality $\displaystyle D_{v}=|{\bf
b}_{v}|$ (that is the number of marks carried by $\displaystyle v$) and
$\displaystyle M_{v}$ is the number of offspring of $\displaystyle v$ that
inherit a mark from $\displaystyle v$. As a consequence,

$|\\{i\in\\{1,\dots,k\\}:\emptyset\preceq v\prec\psi_{s_{i}}^{i}\\}|\leq
D_{v}.$ (14)

The inequality in (14) can be strict when, for example,
$\displaystyle\emptyset\preceq v=\psi_{s_{j}}^{j}\prec\psi_{s_{i}}^{i}$ for
some pair $\displaystyle i\neq j$.

###### Proof of Lemma 6.

Let us start by rewriting the right-hand side of the expression in the lemma.
Noting that under $\displaystyle\mathbb{Q}_{\delta_{x}}$, $\displaystyle
W_{\mathbf{s}}^{k}$ is positive, we have

$\displaystyle\displaystyle\mathbb{Q}_{\delta_{x}}^{k}\bigg{[}\frac{Y}{{W}^{k}_{\mathbf{s}}}\prod_{i=1}^{k}\prod_{\emptyset\preceq
v\prec\psi_{s_{i}}^{i}}N_{v}\bigg{]}$
$\displaystyle\displaystyle=\mathbb{P}_{\delta_{x}}^{k}\bigg{[}Y\prod_{i=1}^{k}\prod_{\emptyset\preceq
v\prec\psi_{s_{i}}^{i}}N_{v}\bigg{]}$
$\displaystyle\displaystyle=\mathbb{P}^{k}_{\delta_{x}}\bigg{[}\sum_{u_{i}\in\mathcal{N}_{s_{i}},\,1\leq
i\leq k}Y(u_{1},\dots,u_{k})\mathbf{1}_{\\{\psi_{s_{i}}^{i}=u_{i}\,1\leq i\leq
k\\}}\prod_{i=1}^{k}\prod_{\emptyset\preceq v\prec u_{i}}N_{v}\bigg{]}$ (15)

where the first equality holds thanks to the change of measure (11).

Conditioning on $\displaystyle\mathcal{F}_{s_{1}}$, we have

$\displaystyle\mathbb{P}^{k}_{\delta_{x}}\left(\psi_{s_{i}}^{i}=u_{i},\,i=1,\dots,k\bigg{|}\mathcal{F}_{s_{1}}\right)=\prod_{i=1}^{k}\prod_{\emptyset\preceq
v\prec u_{i}}N_{v}^{-1},$

and thus, by the properties of conditional expectation, we can rewrite (15) as

$\displaystyle\displaystyle\mathbb{P}^{k}_{\delta_{x}}\bigg{[}\sum_{u_{i}\in\mathcal{N}_{s_{i}},\,1\leq
i\leq k}Y(u_{1},\dots,u_{k})\prod_{i=1}^{k}\prod_{\emptyset\preceq v\prec
u_{i}}N_{v}\prod_{\emptyset\preceq v\prec
u_{i}}N_{v}^{-1}\bigg{]}=\mathbb{P}_{\delta_{x}}\bigg{[}\sum_{v_{i}\in\mathcal{N}_{s_{i}},\,1\leq
i\leq k}Y(v_{1},\dots,v_{k})\bigg{]},$

as required. ∎

Referring to Remark 6, if we take
$\displaystyle(s_{1},\dots,s_{k})=(t,\dots,t)$ then $\displaystyle
v=\psi_{t}^{j}\prec\psi_{t}^{i}$ cannot occur and the inequality in (14) is an
equality. In that case,

$\displaystyle\displaystyle\prod_{i=1}^{k}\prod_{\emptyset\preceq
v\prec\psi_{s_{i}}^{i}}N_{v}$ $\displaystyle\displaystyle=\prod_{v\in
S_{k}(\mathbf{s})\backslash\partial S_{k}(\mathbf{s})}N_{v}^{D_{v}},$

which yields the following corollary (as a special case).

######  Corollary 7 (Many-to-few).

Let $\displaystyle x\in E$, $\displaystyle k\geq 1$ and $\displaystyle t\geq
0$ be fixed. Suppose that

$\displaystyle
Y=\sum_{v_{i}\in\mathcal{N}_{t},\,i=1,\dots,k}Y(v_{1},\dots,v_{k})\mathbf{1}_{\\{\psi_{t}^{i}=v_{i},\,1\leq
i\leq k\\}}$

is $\displaystyle\mathcal{F}_{t}^{k}$-measurable with $\displaystyle
Y(v_{1},\dots,v_{k})$ $\displaystyle\mathcal{F}_{t}$-measurable for
$\displaystyle v_{i}\in\mathcal{N}_{t}$ for $\displaystyle i=1,\dots k$. Then

$\displaystyle\displaystyle\mathbb{P}_{\delta_{x}}\bigg{[}\sum_{\begin{subarray}{c}v_{i}\in\mathcal{N}_{t}\\\
i=1,\dots,k\end{subarray}}Y(v_{1},\dots,v_{k})\bigg{]}=\mathbb{Q}_{\delta_{x}}^{k}$
$\displaystyle\displaystyle\bigg{[}Y\prod_{v\in
S_{k}(t)}\frac{\zeta(X_{v},\sigma_{v})}{\zeta(X_{v},\tau_{v}^{-}\wedge t)}{\rm
e}^{\int_{\sigma_{v}}^{\tau_{v}\wedge
t}\beta(X_{v}(s))({\mathtt{m}}_{D_{v}}(X_{v}(s))-1){\rm d}s}$
$\displaystyle\displaystyle\hskip 56.9055pt\times\frac{\prod_{v\in
S_{k}(t)\backslash\mathcal{N}_{t}}{\varphi(X_{v}(\tau_{v}^{-}))^{M_{v}}}}{\prod_{v\in
S_{k}(t)\backslash\\{\emptyset\\}}\varphi(X_{v}(\sigma_{v}))}\bigg{]}.$

###  3.2 Natural choice of $\displaystyle\zeta$ and separated skeletons

We will now consider a specific example of the many-to-few formula given in
Lemma 6 by choosing a particular form for the martingale $\displaystyle\zeta$.
Using the fact that $\displaystyle\varphi$ is the right eigenfunction for the
MBP, $\displaystyle(X,\mathbb{P})$, with corresponding eigenvalue
$\displaystyle\lambda$, it isn’t too difficult to show that under
$\displaystyle\mathbf{P}_{x}$

$\zeta(\xi,t)=\frac{\varphi(\xi_{t})}{\varphi(x)}\exp\left(\int_{0}^{t}{\beta}(\xi_{s})(\mathtt{m}_{1}(\xi_{s})-1){\rm
d}s\right),\quad t\geq 0,$ (16)

is a martingale.

In this case, we have that

$\displaystyle\displaystyle W^{k}_{t}$
$\displaystyle\displaystyle=\frac{1}{\varphi(x)}\prod_{v\in S_{k}(t)}{\rm
e}^{-\int_{\sigma_{v}}^{\tau_{v}\wedge
t}\beta(X_{v}(s))({\mathtt{m}}_{D_{v}}(X_{v}(s))-{\mathtt{m}}_{1}(X_{v}(s))){\rm
d}s}$ $\displaystyle\displaystyle\qquad\qquad\times\prod_{v\in
S_{k}(t)\cap\mathcal{N}_{t}}\varphi(X_{v}(t))\prod_{v\in
S_{k}(t)\backslash\mathcal{N}_{t}}\frac{N_{v}^{D_{v}}}{\varphi(X_{v}(\tau_{v}^{-}))^{M_{v}-1}}.$
(17)

We now further assume that each of the nodes $\displaystyle
v_{i}\in\mathcal{N}_{s_{i}}$, $\displaystyle i=1,\dots,k$, that make up the
skeleton $\displaystyle S_{k}(\mathbf{v},\mathbf{s})$, are distinct. We refer
to a skeleton with this property as a separated skeleton. This implies that at
time $\displaystyle s_{i}$, node $\displaystyle v_{i}$ only carries one mark
for each $\displaystyle i=1,\dots,k$. Recalling again Remark 6, we thus have
the advantage of writing

$\displaystyle\prod_{v\in
S_{k}(\mathbf{s})\backslash\mathcal{N}_{\mathbf{s}}}N_{v}^{-D_{v}}=\prod_{i=1}^{k}\prod_{\emptyset\leq
v<\psi_{s_{i}}^{i}}N_{v}^{-1}.$

Applying the many-to-few formula with this martingale, for the special choice
of $\displaystyle\zeta$, and in the case of separated skeletons, we get

$\displaystyle\displaystyle\mathbb{P}_{\delta_{x}}^{k}\bigg{[}\sum_{v_{i}\in\mathcal{N}_{t}\text{
distinct }}Y(v_{1},\dots,v_{k})\bigg{]}$
$\displaystyle\displaystyle=\varphi(x)\mathbb{Q}^{k}_{\delta
x}\bigg{[}Y\mathbf{1}_{\\{\\{\psi_{t}^{i}\\}_{1\leq i\leq k}\text{
distinct}\\}}\prod_{j=1}^{k}\varphi(\xi^{j}_{t})^{-1}$
$\displaystyle\displaystyle\hskip 85.35826pt\times\prod_{v\in S_{k}(t)}{\rm
e}^{\int_{\sigma_{v}}^{\tau_{v}\wedge
t}\beta(X_{v}(s))({\mathtt{m}}_{D_{v}}(X_{v}(s))-{\mathtt{m}}_{1}(X_{v}(s))){\rm
d}s}\prod_{v\in
S_{k}(t)\backslash\mathcal{N}_{t}}{\varphi(X_{v}(\tau_{v}^{-}))^{M_{v}-1}}\bigg{]}.$
(18)

##  4 Application to genealogies in the critical case

As an application, we determine the asymptotic law of the death time of the
most recent common ancestor, henceforth referred to as _split time_ , of two
particles sampled uniformly from a critical population at two different times.
The limit is taken as $\displaystyle t\to\infty$, when we have conditioned on
survival of the process up to time $\displaystyle t$.

We assume in this section that the measures $\displaystyle\mathbb{Q}^{k}$ and
$\displaystyle\mathbb{P}^{k}$ are as defined in section 3.2. We remind the
reader of the notation $\displaystyle\xi^{1},\dots,\xi^{k}$ for the motion of
the $\displaystyle k$ spines under $\displaystyle\mathbb{Q}^{k}$ and
$\displaystyle\mathbb{P}^{k}$, and $\displaystyle\psi^{1},\dots,\psi^{k}$ for
the Ulam-Harris labels that they carry.

We assume throughout the section that our branching process satisfies the
following additional criticality requirement.

######  Assumption 2.

The following criticality assumptions hold.

  1. 1.

$\displaystyle\lambda=0$, where $\displaystyle\lambda$ was introduced in
Assumption 1

  2. 2.

Define $\displaystyle\Delta_{t}:=\sup_{x\in E,f\in
B_{1}^{+}(E)}|\varphi(x)^{-1}\mathtt{T}_{t}[f](x)-\langle
f,\tilde{\varphi}\rangle|$. Then

$\displaystyle\sup_{t\geq 0}\Delta_{t}<\infty\text{ and
}\lim_{t\to\infty}\Delta_{t}=0.$

  3. 3.

The number of offspring produced at a branching event is bounded above by
$\displaystyle n_{max}<\infty$.

  4. 4.

There exists a constant $\displaystyle C>0$ such that for all $\displaystyle
g\in B^{+}(E)$,

$\displaystyle{\Sigma:=}\langle\beta\mathbb{V}[g],\tilde{\varphi}\rangle\geq
C\langle g,\tilde{\varphi}\rangle^{2},$

where $\displaystyle\mathbb{V}[g](x):=\mathcal{E}_{x}[\langle
g,\mathcal{Z}\rangle_{2,2}]$ and where the notation $\displaystyle\langle
g,\mathcal{Z}\rangle_{k,n}$ was defined in (3).

  5. 5.

For all $\displaystyle x\in E$, $\displaystyle\mathbb{P}_{\delta_{x}}(\exists
t>0\text{ such that }N_{t}=0)=1$.

######  Remark 7.

We note that the assumptions above are inherited from [11, 8], where
asymptotic results concerning criticality and moment growth were considered.
Whereas the Assumption 2.1 is clearly a standard criticality assumption, the
remaining assumptions 2.2 - 2.5 can be interpreted as follows. Roughly
speaking, Assumption 2.2 describes the uniform stability of the mean
semigroup. In particular, by taking $\displaystyle f\equiv 1$, Assumption 2.2
ensures that the first moment of the process settles down to a stationary
value. Assumption 2.3 rather obviously requires the random number of offspring
to be deterministically bounded. Assumption 2.4 can be thought of as an
irreducibility condition written in terms of the two-point correlation (or
variance) functional $\displaystyle\mathbb{V}[g]$ and ensures a minimal level
of spatial mixing occurring for second order effects associated to the
semigroup assumption in Assumption 2.2. Finally Assumption 2.5 ensures that,
even though at criticality in Assumption 2.1, we can guarantee there is
extinction almost surely. Assumption 2.5 is automatically satisfied e.g. for a
branching Brownian motion in a compact domain with killing on the boundary, or
the category of neutron branching process considered in [16, 10, 5].

Let us now present our main application. Here and in the rest of this section
$\displaystyle\Rightarrow$ means convergence in distribution.

######  Proposition 8.

Let $\displaystyle 0<a<1$ and let $\displaystyle x\in E$ be fixed. Let
$\displaystyle T_{t}$ have the
$\displaystyle\mathbb{P}_{\delta_{x}}(\cdot|N_{t}>0)$ law of the split time of
two particles: one chosen uniformly from those alive at time $\displaystyle t$
and one chosen uniformly from those alive at time $\displaystyle at$. Then

$\displaystyle\frac{T_{t}}{t}\Rightarrow T\quad\text{ as }t\to\infty$

where $\displaystyle T$ has density

$\displaystyle
f_{a}(u):=\frac{2a}{1-a}\frac{2(a-u)\log(1-\tfrac{u}{a})-(2-u-\tfrac{u}{a})\log(1-u)}{u^{3}}$

with respect to Lebesgue measure $\displaystyle{\rm d}u$ on
$\displaystyle[0,a]$.

######  Remark 8.

Note that

$\displaystyle\displaystyle\lim_{a\nearrow 1}\frac{2a}{1-a}$
$\displaystyle\displaystyle\frac{2(a-u)\log(1-\tfrac{u}{a})-(2-u-\tfrac{u}{a})\log(1-u)}{u^{3}}=2(-2u+(u-2)\log(1-u))$

This agrees with the density in the case $\displaystyle a=1$ (for critical GW
processes) calculated in [13].

######  Remark 9.

It is not obvious a priori that $\displaystyle f_{a}$ is the density function
of a random variable. However, this is indeed the case, since we can calculate
that

$\displaystyle\displaystyle\frac{2a}{1-a}$
$\displaystyle\displaystyle\int_{0}^{a}\frac{2(a-u)\log(1-\tfrac{u}{a})-(2-u-\tfrac{u}{a})\log(1-u)}{u^{3}}\,du$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{2}{a(1-a)}\int_{0}^{1}\frac{2a(1-u)\log(1-u)-(2-ua-u)\log(1-au)}{u^{3}}\,du$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{2}{a(1-a)}\left[\frac{(1-u)}{u^{2}}\left(a(u-1)\log(1-u)+(1-au)\log(1-au)\right)\right]_{0}^{1}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{2}{a(1-a)}\lim_{u\to
0}\frac{a(u-1)(-u-\tfrac{u^{2}}{2}+\mathrm{o}(u^{2}))+(1-au)(-au-\tfrac{a^{2}u^{2}}{2}+\mathrm{o}(u^{2}))}{u^{2}}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle 1.$

We also have the following result concerning the joint convergence of the
(normalised) population size at two different times under
$\displaystyle\mathbb{Q}^{1}$. Recall from Assumption 2.3 that
$\displaystyle\Sigma:=\langle\beta\mathbb{V}[\varphi],\tilde{\varphi}\rangle.$

######  Proposition 9.

Under $\displaystyle\mathbb{Q}_{\delta_{x}}^{1}$,

$\displaystyle\bigg{(}\frac{N_{at}}{t},\frac{N_{t}}{t}\bigg{)}\Rightarrow(Z,\hat{Z}),\qquad\text{as
}t\to\infty,$

where $\displaystyle Z$ is equal in law to a
$\displaystyle\mathtt{Gamma}(2,(a\Sigma\langle
1,\tilde{\varphi}\rangle/2)^{-1})$ random variable and, conditionally on
$\displaystyle Z$, the law of $\displaystyle\hat{Z}$ is that of a
$\displaystyle\mathtt{Gamma}(2+K,(\Sigma(1-a)\langle
1,\tilde{\varphi}\rangle/2)^{-1})$ random variable where $\displaystyle
K\sim\mathtt{Poisson}(((1-a)\Sigma\langle 1,\tilde{\varphi}\rangle/2)^{-1}Z)$.

Equivalently, under $\displaystyle\mathbb{P}_{\delta_{x}}(\cdot|N_{t}>0)$, we
have the joint convergence of $\displaystyle(N_{at}/t,N_{t}/t)$ to
$\displaystyle(Y,\hat{Y})$, where the joint law of $\displaystyle(Y,\hat{Y})$
is that of $\displaystyle(Z,\hat{Z})$ _weighted by_ $\displaystyle 1/\hat{Z}$.

Before moving on to the proofs of the above two propositions, we first state a
lemma that will be used throughout the aforementioned proofs.

######  Lemma 10.

In what follows, we suppose that $\displaystyle(g_{t},t\geq 0)$ are a
collection of functions with $\displaystyle g_{t}\in B_{1}^{+}(E)$ for each
$\displaystyle t>0$, and such that $\displaystyle g_{t}\to g\in B^{+}_{1}(E)$
pointwise as $\displaystyle t\to\infty$. For any $\displaystyle x\in E$, the
following hold.

  1. (a)

$\displaystyle t\mathbb{P}_{\delta_{x}}(N_{t}>0)\to\frac{2\varphi(x)}{\Sigma}$
as $\displaystyle t\to\infty$,
$\displaystyle\sup_{t,x}|t\mathbb{P}_{\delta_{x}}(N_{t}>0)|<\infty$ and
$\displaystyle\inf_{t}|t\mathbb{P}_{\delta_{x}}(N_{t}>0)|>0$.

  2. (b)

The joint law of

$\displaystyle\left(\frac{X_{t}[g_{t}]}{t},\frac{N_{t}}{t}\right)\text{ under
}\mathbb{P}_{\delta_{x}}(\cdot|\,N_{t}>0)$

converges to that of $\displaystyle(\langle g,\tilde{\varphi}\rangle Z,\langle
1,\tilde{\varphi}\rangle Z)$ as $\displaystyle t\to\infty$, where
$\displaystyle Z\sim\mathtt{Exponential}({2}/{\Sigma})$.

  3. (c)

The joint law of

$\left(\frac{X_{t}[g_{t}]}{t},\frac{N_{t}}{t},\xi_{t}\right)\text{ under
}\mathbb{Q}^{1}_{\delta_{x}}$ (19)

converges to that of $\displaystyle(\langle g,\tilde{\varphi}\rangle Z,\langle
1,\tilde{\varphi}\rangle Z,\bar{\xi})$ as $\displaystyle t\to\infty$, where
$\displaystyle Z\sim\mathtt{Gamma}(2,2/\Sigma)$ and $\displaystyle\bar{\xi}$
is independent of $\displaystyle Z$, with law given by

$P(\bar{\xi}\in A)=\langle\mathbf{1}_{A}\varphi,\tilde{\varphi}\rangle$ (20)

###### Proof.

(a) follows from [11, Theorem 1.2], [11, Lemma 7.4] and [11, Lemma 7.2]. For
the proof of (b), first note that the joint convergence of
$\displaystyle(X_{t}[g]/t,N_{t}/t)$ under
$\displaystyle\mathbb{P}_{\delta_{x}}(\cdot|\,N_{t}>0)$ follows from [11,
Theorem 1.3], together with the fact that $\displaystyle
t^{-1}[X_{t}[f]-\langle f,\tilde{\varphi}\rangle X_{t}[\varphi]]\to 0$ in
probability as $\displaystyle t\to\infty$ for any bounded $\displaystyle f$
(see the proof of [11, Theorem 1.3]). Writing

$\displaystyle X_{t}[g_{t}]=X_{t}[g]+X_{t}[g_{t}-g],$

it thus suffices to show that $\displaystyle X_{t}[g_{t}-g]/t\to 0$ in
$\displaystyle\mathbb{P}_{\delta_{x}}(\cdot|N_{t}>0)$-probability. For this,
we will show that we actually have $\displaystyle L^{1}$ convergence. Setting
$\displaystyle f_{t}=|g_{t}-g|$ (which is bounded by $\displaystyle 2$), we
have

$\displaystyle\displaystyle\frac{1}{t}\mathbb{E}_{\delta_{x}}[|X_{t}[g_{t}-g]||N_{t}>0]$
$\displaystyle\displaystyle\leq\frac{1}{t\mathbb{P}_{\delta_{x}}(N_{t}>0)}\mathbb{E}_{\delta_{x}}[X_{t}[f_{t}]]$
$\displaystyle\displaystyle\leq\frac{1}{t\mathbb{P}_{\delta_{x}}(N_{t}>0)}(|\mathbb{E}_{\delta_{x}}[X_{t}[f_{t}]]-\varphi(x)\langle
f_{t},\tilde{\varphi}\rangle|+\varphi(x)\langle
f_{t},\tilde{\varphi}\rangle).$

From Lemma 10(a), it follows that
$\displaystyle(t\mathbb{P}_{\delta_{x}}(N_{t}>0))^{-1}$ is uniformly bounded.
By the first part of Assumption 2, the first term in the parentheses on the
right-hand side above converges to $\displaystyle 0$ uniformly. Finally, by
dominated convergence (since $\displaystyle\tilde{\varphi}$ is a finite
measure), the second term in the parentheses also converges to $\displaystyle
0$.

For (c), note that if $\displaystyle W_{t}^{1}$ is the martingale from (11)
(in the classical case of one spine),

$\mathbb{P}_{\delta_{x}}[W_{t}^{1}|\mathcal{F}_{t}]=X_{t}[\varphi],$ (21)

so that for any bounded, continuous function $\displaystyle F$,

$\displaystyle\displaystyle\mathbb{Q}_{\delta_{x}}^{1}\left[F\left(\frac{X_{t}[g_{t}]}{t},\frac{N_{t}}{t}\right)\right]=\frac{t\mathbb{P}_{\delta_{x}}(N_{t}>0)}{\varphi(x)}\mathbb{P}_{\delta_{x}}\left[F\left(\frac{X_{t}[g_{t}]}{t},\frac{N_{t}}{t}\right)\frac{X_{t}[\varphi]}{t}{\,\bigg{|}\,N_{t}>0}\right].$
(22)

Taking $\displaystyle F$ to be of the form $\displaystyle F(x,y)={\rm
e}^{-\theta x-\mu y}$, for $\displaystyle\theta,\mu\geq 0$ and using (a) and
(b) yields the convergence of the first two components of the triple under
$\displaystyle\mathbb{Q}_{\delta_{x}}^{1}$. The marginal convergence in law of
$\displaystyle\xi_{t}$ to $\displaystyle\xi$ under
$\displaystyle\mathbb{Q}_{\delta_{x}}$ follows from [11, section 5].

To see the joint convergence in law of the triple in (19), note that due to
the aforementioned marginal convergence of $\displaystyle\xi$ and the first
two components, we immediately have tightness. Moreover, any subsequential
limit has the form $\displaystyle({\langle g,\tilde{\varphi}\rangle}Z,\langle
1,\tilde{\varphi}\rangle Z,\bar{\xi})$, where the marginals of $\displaystyle
Z$ and $\displaystyle\bar{\xi}$ are as desired. Thus it remains to show that
for any such subsequential limit, $\displaystyle Z$ and
$\displaystyle\bar{\xi}$ (or equivalently $\displaystyle\langle
1,\tilde{\varphi}\rangle Z$ and $\displaystyle\bar{\xi}$) are independent.

For this, define $\displaystyle N_{t}^{*}$ to be the contribution to
$\displaystyle N_{t}$ of all descendants branching off the (single) spine
particle _before_ time $\displaystyle t-t^{1/3}$. Then $\displaystyle
N_{t}-N^{*}_{t}$ behaves like $\displaystyle N_{t^{1/3}}$ under
$\displaystyle\mathbb{Q}^{1}$. Applying the Markov property at time
$\displaystyle t-t^{1/3}$, it follows that $\displaystyle
t^{-1/3}({N_{t}-N_{t}^{*}})$ converges in law to $\displaystyle\langle
1,\tilde{\varphi}\rangle Z$, where $\displaystyle
Z\sim\mathtt{Gamma}(2,2/\Sigma)$, thanks to the discussion following (22).
Thus, for $\displaystyle\varepsilon>0$, we have that
$\displaystyle\mathbb{Q}_{\delta_{x}}^{1}(N_{t}-N_{t}^{*}\geq
t^{1-\varepsilon})\to 0$ as $\displaystyle t\to\infty$. We also have that
$\displaystyle\mathbb{Q}_{\delta_{x}}^{1}(N_{t}\geq t^{1-\varepsilon/2})\to
0$, uniformly in $\displaystyle x$, thanks to the Markov inequality and
Assumption 2.2.

Now note that, on the one hand, $\displaystyle N_{t}^{*}/N_{t}\leq 1$. On the
other hand, $\displaystyle
1-(N^{*}_{t}/N_{t})=(N_{t}-N^{*}_{t})/N_{t}=t^{-1}(N_{t}-N^{*}_{t})/(t^{-1}N_{t})$.
In the final equality, the numerator tends to zero in
$\displaystyle\mathbb{Q}^{1}$-probability, and the denominator converges under
$\displaystyle\mathbb{Q}^{1}$ to a Gamma distributed random variable. It
follows that $\displaystyle N_{t}^{*}/N_{t}\to 1$ under
$\displaystyle\mathbb{Q}_{\delta_{x}}^{1}$ as $\displaystyle t\to\infty$. Next
note that the part of the spatial branching tree consisting of all descendants
branching off the spine particle _before_ time $\displaystyle t-t^{1/3}$ is
conditionally independent (given the position of the spine at time
$\displaystyle t-t^{1/3}$) of descendants branching off the spine particle
_after_ time $\displaystyle t-t^{1/3}$. This implies that any subsequential
distributional limit of $\displaystyle(N_{t}/t,\xi_{t})$ as $\displaystyle
t\to\infty$ under $\displaystyle\mathbb{Q}_{\delta_{x}}^{1}$, say
$\displaystyle(\langle 1,\tilde{\varphi}\rangle Z,\bar{\xi})$, can be extended
to a subsequential limit $\displaystyle(Y^{*},\langle 1,\tilde{\varphi}\rangle
Z,\bar{\xi})$ of $\displaystyle(N_{t}^{*}/t,N_{t}/t,\bar{\xi})$ satisfying
$\displaystyle Y^{*}=\langle 1,\tilde{\varphi}\rangle Z$ almost surely. On the
other hand, ergodicity of the spine motion [11, section 5] implies that
$\displaystyle Y^{*}$ and $\displaystyle\bar{\xi}$ are independent. That is to
say, $\displaystyle Z$ and $\displaystyle\bar{\xi}$ are independent. ∎

######  Remark 10.

Observe that a slight variant of the proof of (c) given above, implies that
for any $\displaystyle c\in(0,1]$, the joint law of
$\displaystyle(X_{t}[g_{t}]/t,N_{t}/t,\xi_{ct},\xi_{t})$ under
$\displaystyle\mathbb{Q}_{\delta_{x}}^{1}$ also converges to that of
$\displaystyle(\langle g,\tilde{\varphi}\rangle Z,\langle
1,\tilde{\varphi}\rangle Z,\bar{\xi},\bar{\xi}^{\prime})$ as $\displaystyle
t\to\infty$, where $\displaystyle(\bar{\xi},\bar{\xi}^{\prime},Z)$ are
mutually independent and $\displaystyle\bar{\xi},\bar{\xi}^{\prime}$ both have
law (20).

We are now ready to prove Proposition 8. However, let us first give a sketch
proof, which (roughly) describes the main steps of the argument (although some
details look slightly different in the final version, in order to deal with
technicalities that arise).

###### Sketch proof of Proposition 8..

For $\displaystyle v,w\in\Omega$, let $\displaystyle\tau_{v,w}$ be the split
time of $\displaystyle v$ and $\displaystyle w$. That is, the death time of
the most recent common ancestor of $\displaystyle v$ and $\displaystyle w$.

  * •

We first show that the asymptotic probability of selecting $\displaystyle
w\in\mathcal{N}_{at}$ and $\displaystyle v\in\mathcal{N}_{t}$ with
$\displaystyle w\preceq v$ is zero. Roughly speaking, this is because the
probability that any specific individual at time $\displaystyle at$ has _any_
descendants at time $\displaystyle t$, tends to $\displaystyle 0$ as
$\displaystyle t\to\infty$. This means that to obtain the asymptotic law of
the split time, it is enough to provide an asymptotic for

$\displaystyle\mathbb{P}_{\delta_{x}}\bigg{[}\sum_{w\in\mathcal{N}_{at},v\in\mathcal{N}_{t},w\npreceq
v}\frac{1}{N_{at}\hat{N}^{w}_{t}}F\big{(}\frac{\tau_{v,w}}{t}\big{)}\,\bigg{|}\,N_{t}>0\bigg{]}$

when $\displaystyle F$ is an arbitrary bounded continuous function, and where
for $\displaystyle w\in\mathcal{N}_{at}$, $\displaystyle\hat{N}^{w}_{t}$ is
the size of the population at time $\displaystyle t$ _without_ counting the
descendants of $\displaystyle w$. _(This roughly corresponds to Step 1 in the
full proof below)._

  * •

The many-to-two lemma at times $\displaystyle at,t$ allows the above
expectation to be written as

$\displaystyle\frac{\varphi(x)}{\mathbb{P}_{\delta_{x}}(N_{t}>0)}\mathbb{Q}_{\delta_{x}}^{2}\left[F(\frac{\tau}{t})\mathbf{1}_{\\{\tau\leq
at\\}}\frac{\varphi(\xi^{1}_{\tau-})}{\varphi(\xi^{1}_{t})\varphi(\xi^{2}_{at})}\frac{1}{N_{at}\hat{N}_{t}}{\rm
e}^{\int_{0}^{\tau}\beta(\xi^{1}_{s})(\mathtt{m}_{2}(\xi^{1}_{s})-\mathtt{m}_{1}(\xi^{1}_{s}))\,{\rm
d}s}\right]$

where $\displaystyle\tau$ is the split time of the two spines under
$\displaystyle\mathbb{Q}_{\delta_{x}}^{2}$ and $\displaystyle\hat{N}_{t}$ is
the population size at time $\displaystyle t$, not counting descendants of the
second spine at time $\displaystyle at$. _(This roughly corresponds to Step 3
in the full proof below)._

Figure 3: Suppose the two spines split from each other at time
$\displaystyle\tau=ut$. The population at time $\displaystyle at$ can be
broken up into those individuals that have branched off the first spine before
time $\displaystyle at$ (depicted in orange) and those individuals that have
branched off the second spine between times $\displaystyle ut$ and
$\displaystyle at$ (depicted in green). Given the population at time
$\displaystyle at$, the size of the population at time $\displaystyle t$
(without the descendants of the second spine, that is,
$\displaystyle\hat{N}_{t}$) can again be broken up into two subpopulations:
those that branch off the first spine between times $\displaystyle at$ and
$\displaystyle t$ (depicted in black) and those that are descendants of non-
spine particles at time $\displaystyle at$ (depicted in blue).

  * •

Next we consider $\displaystyle\hat{\mathbb{Q}}_{\delta_{x}}^{2}$ obtained by
reweighting $\displaystyle\mathbb{Q}_{\delta_{x}}^{2}$ by

$\displaystyle(\beta(\xi_{\tau^{-}}^{1})(\mathtt{m}_{2}(\xi_{\tau^{-}}^{1})-\mathtt{m}_{1}(\xi_{\tau^{-}}^{1})))^{-1}{\rm
e}^{\int_{0}^{\tau}\beta(\xi^{1}_{s})(\mathtt{m}_{2}(\xi^{1}_{s})-\mathtt{m}_{1}(\xi^{1}_{s})){\rm
d}s-\tau}.$

This change of measure alters the rate at which the spine particles split into
two distinct spines (from rate
$\displaystyle\beta(\mathtt{m}_{2}-\mathtt{m}_{1})$ to rate $\displaystyle 1$)
but doesn’t affect the rate at which branching events occur that don’t result
in the spines splitting. Combining this with a change of variables and
conditioning on $\displaystyle\tau$ (which has an exponential $\displaystyle
1$ distribution under $\displaystyle\hat{\mathbb{Q}}^{2}_{\delta_{x}}$), we
rewrite our expectation again, as

$\displaystyle\frac{\varphi(x)}{t\mathbb{P}_{\delta_{x}}(N_{t}>0)}\int_{0}^{a}{\rm
d}uF(u)\hat{\mathbb{Q}}_{\delta_{x}}^{2}\left[\frac{\beta(\xi^{1}_{ut})\varphi(\mathtt{m}_{2}(\xi^{1}_{ut})-\mathtt{m}_{1}(\xi^{1}_{ut}))}{\varphi(\xi^{1}_{t})\varphi(\xi^{2}_{at})}\frac{t^{2}}{N_{at}\hat{N}_{t}}\,\bigg{|}\,\tau=ut\right].$

Here the law
$\displaystyle\hat{\mathbb{Q}}_{\delta_{x}}^{2}\left[\cdot|\tau=ut\right]$
makes rigorous sense: the system has a single spine and in fact evolves as
under $\displaystyle\mathbb{Q}_{\delta_{x}}^{1}$ until time $\displaystyle
ut$, where some (biased) branching event occurs, two spines are selected, and
each of these initiates an independent $\displaystyle\mathbb{Q}^{1}$ process.
_(This roughly corresponds to Step 4 in the full proof below)._

  * •

Now, we know by Lemma 10 that
$\displaystyle\varphi(x)(t\mathbb{P}_{\delta_{x}}(N_{t}>0))^{-1}\to\Sigma/2$
as $\displaystyle t\to\infty$. Moreover, under
$\displaystyle\hat{\mathbb{Q}}^{2}_{\delta_{x}}$, similar arguments to those
in the proof of Lemma 10 (in particular, due to ergodicity of the spine
motion) imply that the positions
$\displaystyle\xi_{ut}^{1},\xi_{t}^{1},\xi_{at}^{2}$ of the spines are
asymptotically independent of each other and of $\displaystyle
N_{at},\hat{N}_{t}$ as $\displaystyle t\to\infty$, with limiting laws
described by $\displaystyle P(\xi\in
A)=\langle\mathbf{1}_{A}\varphi,\tilde{\varphi}\rangle$ for $\displaystyle
A\subset E$.

Furthermore, the limiting law of $\displaystyle N_{at}/t$ as $\displaystyle
t\to\infty$ is described by $\displaystyle aZ+(a-{u})Z^{\prime}$, where
$\displaystyle(Z,Z^{\prime})$ are a pair of independent
$\displaystyle\mathtt{Gamma}(2,\,{2}/{\Sigma\langle
1,\tilde{\varphi}\rangle})$ random variables; this is because of the explicit
description of the process under
$\displaystyle\hat{\mathbb{Q}}_{\delta_{x}}^{2}(\cdot|\tau=ut)$ and item (c)
of Lemma 10. In Figure 3, $\displaystyle aZ$ and
$\displaystyle(a-u)Z^{\prime}$ correspond to the sizes of the orange and green
populations respectively (after rescaling by $\displaystyle t$).

Finally, the conditional limiting law of $\displaystyle\hat{N}_{t}/t$ given
$\displaystyle N_{at}/t$ is that of a Gamma random variable with parameter
$\displaystyle(2+K,\,{2}/{\Sigma(1-a)\langle 1,\tilde{\varphi}\rangle})$,
where $\displaystyle K\sim\mathtt{Poisson}({2N}/{(1-a)\Sigma\langle
1,\tilde{\varphi}\rangle}N)$ is itself random. This is because, given the
collection of particles alive at time $\displaystyle at$, the first spine
particle will have a number of offspring at time $\displaystyle t$ which is
asymptotically like $\displaystyle t$ times a
$\displaystyle\mathtt{Gamma}(2,{2}/{\Sigma(1-a)\langle
1,\tilde{\varphi}\rangle})$ random variable (Lemma 10(c) again; this
corresponds to the population depicted in black in Figure 3). Then,
independently, each of the non-spine particles alive will have some descendant
alive at time $\displaystyle t$ with probability asymptotically proportional
to $\displaystyle t^{-1}$ times $\displaystyle\varphi$ of their positions.
Using (essentially) the Poisson approximation of the binomial distribution,
this results in a total number of non-spine particles with some descendant
alive at time $\displaystyle t$ having asymptotic conditional distribution
given by a $\displaystyle\mathtt{Poisson}({2N}/{(1-a)\Sigma\langle
1,\tilde{\varphi}\rangle}N)$ random variable. By Lemma 10 (b), the number of
offspring of each of these will approximately $\displaystyle t$ times an
independent $\displaystyle\mathtt{Exponential}({2}/{\Sigma(1-a)\langle
1,\tilde{\varphi}\rangle})$, that is, a
$\displaystyle\mathtt{Gamma}(1,{2}/{\Sigma(1-a)\langle
1,\tilde{\varphi}\rangle})$ random variable. This corresponds to the
population depicted in blue in Figure 3. The additivity property of
independent Gamma distributions completes the argument. _(This roughly
corresponds to Step 5 in the full proof below)._

  * •

Plugging these asymptotics into the
$\displaystyle\hat{\mathbb{Q}}_{\delta_{x}}^{2}$ expectation, and performing
some simple explicit computations, we obtain the desired formula. _(This
roughly corresponds to Steps 6 and 7 in the full proof below)._

∎

###### Proof of Proposition 8.

Fix $\displaystyle 0<a<1$ and for $\displaystyle w\in\mathcal{N}_{at}$ and
$\displaystyle v\in\mathcal{N}_{t}$, write $\displaystyle\tau_{v,w}$ for the
split time of $\displaystyle v$ and $\displaystyle w$ (as in the sketch
proof). It suffices to show that, for each continuous $\displaystyle
F:[0,\infty)\to[0,1]$,

$\mathbb{P}_{\delta_{x}}\left[\frac{1}{N_{at}N_{t}}\sum_{v\in\mathcal{N}_{t},w\in\mathcal{N}_{at}}F({\tau_{v,w}}/{t})\,\bigg{|}\,N_{t}>0\right]\to\int_{0}^{a}F(u)f_{a}(u){\rm
d}u$ (23)

as $\displaystyle t\to\infty$.

Step 1 In order to apply the many-to-two lemma, we first need to write the
left-hand side of (23) in a slightly different form (that is asymptotically
equivalent). We claim that

$\mathbb{P}_{\delta_{x}}\bigg{[}\frac{1}{N_{at}N_{t}}\sum_{v\in\mathcal{N}_{t},w\in\mathcal{N}_{at}}F({\tau_{v,w}}/{t})\,\bigg{|}\,N_{t}>0\bigg{]}\sim\mathbb{P}_{\delta_{x}}\bigg{[}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{t},w\in\mathcal{N}_{at}\\\
w\npreceq
v\end{subarray}}\frac{1}{N_{at}\hat{N}_{t}^{w}}F({\tau_{v,w}}/{t})\,\bigg{|}\,N_{t}>0\bigg{]}$
(24)

as $\displaystyle t\to\infty$ where
$\displaystyle\hat{N}_{t}^{w}=N_{t}-N_{t}^{w}$ and $\displaystyle
N_{t}^{w}=|\\{v\in\mathcal{N}_{t}:w\preceq v\\}|$.

To prove this claim, let us consider the difference of the two quantities in
the above asymptotic. Recalling that $\displaystyle F\in[0,1]$, we have

$\displaystyle\displaystyle\bigg{|}\mathbb{P}_{\delta_{x}}\bigg{[}\frac{1}{N_{at}N_{t}}\sum_{v\in\mathcal{N}_{t},w\in\mathcal{N}_{at}}F({\tau_{v,w}}/{t})\,\big{|}\,N_{t}>0\bigg{]}-\mathbb{P}_{\delta_{x}}\bigg{[}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{t},w\in\mathcal{N}_{at}\\\
w\npreceq
v\end{subarray}}\frac{1}{N_{at}\hat{N}_{t}^{w}}F({\tau_{v,w}}/{t})\,\big{|}\,N_{t}>0\bigg{]}\bigg{|}$
$\displaystyle\displaystyle\leq\bigg{|}\mathbb{P}_{\delta_{x}}\bigg{[}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{t},w\in\mathcal{N}_{at}\\\
w\preceq
v\end{subarray}}\frac{F(\tau_{v,w}/t)}{N_{t}N_{at}}\big{|}N_{t}>0\bigg{]}\bigg{|}+\bigg{|}\mathbb{P}_{\delta_{x}}\bigg{[}\sum_{{\begin{subarray}{c}v\in\mathcal{N}_{t},w\in\mathcal{N}_{at}\\\
w\npreceq
v\end{subarray}}}\frac{F(\tau_{v,w}/t)}{N_{at}}(\frac{1}{\hat{N}_{t}^{w}}-\frac{1}{N_{t}})\big{|}N_{t}>0\bigg{]}\big{|}$
$\displaystyle\displaystyle\leq\mathbb{P}_{\delta_{x}}\bigg{[}\sum_{v\in\mathcal{N}_{t}}\frac{1}{N_{t}N_{at}}\big{|}N_{t}>0\bigg{]}+\mathbb{P}_{\delta_{x}}\bigg{[}\frac{1}{N_{at}}\sum_{w\in\mathcal{N}_{at}}\frac{N_{t}^{w}}{N_{t}}\big{|}N_{t}>0\bigg{]}$
$\displaystyle\displaystyle=2\mathbb{P}_{\delta_{x}}\bigg{[}\frac{1}{N_{at}}\bigg{|}N_{t}>0\bigg{]},$

where we have used the simple decomposition
$\displaystyle\textstyle\sum_{w\in\mathcal{N}_{at}}{N_{t}^{w}}={N_{t}}$ in the
final equality. Note that, for arbitrarily large $\displaystyle M>0$,

$\displaystyle\displaystyle\mathbb{P}_{\delta_{x}}\bigg{[}\frac{1}{N_{at}}\bigg{|}N_{t}>0\bigg{]}$
$\displaystyle\displaystyle\leq\frac{1}{M}\mathbb{P}_{\delta_{x}}\bigg{[}N_{at}\geq
M\bigg{|}N_{t}>0\bigg{]}+\mathbb{P}_{\delta_{x}}\bigg{[}N_{at}\leq
M\bigg{|}N_{t}>0\bigg{]}.$ (25)

The first term on the right-hand side of (25) can be made arbitrarily small by
taking $\displaystyle M$ sufficiently large. Moreover, the second term on the
right-hand side of (25) converges to zero by Lemma 10.

It therefore suffices to prove that

$\mathbb{P}_{\delta_{x}}\bigg{[}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{t},w\in\mathcal{N}_{at}\\\
w\npreceq
v\end{subarray}}\frac{1}{N_{at}\hat{N}_{t}^{w}}F({\tau_{v,w}}/{t})\,\bigg{|}\,N_{t}>0\bigg{]}\to\int_{0}^{a}F(u)f_{a}(u){\rm
d}u,\quad\text{as }t\to\infty$ (26)

This will be the new goal for the remainder of the proof.

Step 2 In order to apply some bounded convergence results, it is convenient to
define the following event for $\displaystyle\delta>0$. Namely, we write
$\displaystyle A^{\delta}_{v,w}$ for the event that

$\displaystyle\varphi(X_{v}(t))\geq\delta\text{ ,
}\varphi(X_{w}(at))\geq\delta\text{ ,
}\varphi(X_{v}(\tau_{v,w}^{-}))=\varphi(X_{w}(\tau_{v,w}^{-}))\geq\delta\text{
, }\frac{N_{at}}{t}\geq\delta\text{ and }\frac{\hat{N}_{t}^{w}}{t}\geq\delta.$

We claim that it suffices to show that for each $\displaystyle\delta>0$ and
continuous $\displaystyle F:[0,\infty)\to[0,1]$,

$\mathbb{P}_{\delta_{x}}\bigg{[}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{t},w\in\mathcal{N}_{at}\\\
w\npreceq
v\end{subarray}}\frac{\mathbf{1}_{A^{\delta}_{v,w}}}{N_{at}\hat{N}^{w}_{t}}F({\tau_{v,w}}/{t})\,\bigg{|}\,N_{t}>0\bigg{]}\to\frac{c_{\delta}}{\langle
1,\tilde{\varphi}\rangle^{2}\Sigma}\int_{0}^{a}F(u)f_{a}^{\delta}(u){\rm d}u$
(27)

as $\displaystyle t\to\infty$, where

$c_{\delta}:=\langle{\mathbf{1}_{\\{\varphi\geq\delta\\}}},\tilde{\varphi}\rangle^{2}\langle\beta\varphi^{2}(\mathtt{m}_{2}-\mathtt{m}_{1})\mathbf{1}_{\\{\varphi\geq\delta\\}},\tilde{\varphi}\rangle$
(28)

and for some $\displaystyle f_{a}^{\delta}(u)\nearrow f_{a}(u)$ as
$\displaystyle\delta\searrow 0$, pointwise on $\displaystyle[0,a]$. To see why
(27) suffices, note that using the definitions of $\displaystyle\mathbb{V}$
and $\displaystyle\mathtt{m}_{k}$,

$\displaystyle\langle\beta\varphi^{2}(\mathtt{m}_{2}-\mathtt{m}_{1})\mathbf{1}_{\\{\varphi\geq\delta\\}},\tilde{\varphi}\rangle=\langle\beta\mathbb{V}[\varphi]\mathbf{1}_{\\{\varphi\geq\delta\\}},\tilde{\varphi}\rangle.$

Thus, using the boundedness of $\displaystyle\tilde{\varphi}$,
$\displaystyle\varphi$, $\displaystyle\beta$ and Assumption 2.2, it follows
that $\displaystyle c_{\delta}\uparrow\langle
1,\tilde{\varphi}\rangle^{2}\Sigma$ as $\displaystyle\delta\downarrow 0$.
Moreover, since we know by Remark 8 that $\displaystyle f_{a}(u)$ integrates
to $\displaystyle 1$ over $\displaystyle[0,a]$, we can take $\displaystyle
F\equiv 1$ in (27) to see that

$\displaystyle\lim_{\delta\to
0}\lim_{t\to\infty}\mathbb{P}_{\delta_{x}}\bigg{[}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{t},w\in\mathcal{N}_{at}\\\
w\npreceq
v\end{subarray}}\frac{\mathbf{1}_{(A^{\delta}_{v,w})^{c}}}{N_{at}\hat{N}^{w}_{t}}\,\bigg{|}\,N_{t}>0\bigg{]}=\lim_{\delta\to
0}\lim_{t\to\infty}\bigg{(}1-\mathbb{P}_{\delta_{x}}\bigg{[}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{t},w\in\mathcal{N}_{at}\\\
w\npreceq
v\end{subarray}}\frac{\mathbf{1}_{A^{\delta}_{v,w}}}{N_{at}\hat{N}^{w}_{t}}\,\bigg{|}\,N_{t}>0\bigg{]}\bigg{)}=0.$

Thus, given (27), we can take $\displaystyle t\to\infty$ and then
$\displaystyle\delta\downarrow 0$ to deduce that the right-hand side of (24)
converges to the right-hand side of (23) as $\displaystyle t\to\infty$.

The remaining steps will focus on the proof of (27) for an appropriate choice
of $\displaystyle f_{a}^{\delta}$ and with $\displaystyle c_{\delta}$ defined
in (28).

Step 3 We may apply the many-to-two formula, Lemma 7, with $\displaystyle
s_{1}=t,s_{2}=at$, to write

$\displaystyle\displaystyle\mathbb{P}_{\delta_{x}}$
$\displaystyle\displaystyle\bigg{[}\sum_{v\in\mathcal{N}_{t},w\in\mathcal{N}_{at},w\npreceq
v}\frac{\mathbf{1}_{A^{\delta}_{v,w}}}{N_{at}\hat{N}_{t}^{w}}F({\tau_{v,w}}/{t})\,\bigg{|}\,N_{t}>0\bigg{]}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{\varphi(x)}{\mathbb{P}_{\delta_{x}}(N_{t}>0)}\mathbb{Q}_{\delta_{x}}^{2}\left[F({\tau}/{t})\mathbf{1}_{\\{\tau\leq
at\\}}\frac{\varphi(\xi^{1}_{\tau-})}{\varphi(\xi^{1}_{t})\varphi(\xi^{2}_{at})}\frac{\mathbf{1}_{A^{\delta}_{t}}}{N_{at}\hat{N}_{t}}{\rm
e}^{\int_{0}^{\tau}\beta(\xi^{1}_{s})(\mathtt{m}_{2}(\xi^{1}_{s})-\mathtt{m}_{1}(\xi^{1}_{s}))\,{\rm
d}s}\right]$ (29)

where $\displaystyle\tau=\tau_{\psi_{t}^{1},\psi_{at}^{2}}$,
$\displaystyle\hat{N}_{t}=\hat{N}_{t}^{\psi_{at}^{2}}$, and $\displaystyle
A^{\delta}_{t}=A^{\delta}_{\psi^{1}_{t},\psi^{2}_{at}}$. The application of
Lemma 6 is justified since

$\displaystyle Y=F\big{(}{\tau}/{t}\big{)}\mathbf{1}_{\\{\tau\leq
at\\}}\frac{\mathbf{1}_{A^{\delta}_{t}}}{N_{at}\hat{N}_{t}}$

is $\displaystyle\mathcal{F}_{(at,t)}^{2}$\- measurable.

Step 4 Next we consider $\displaystyle\hat{\mathbb{Q}}_{\delta_{x}}^{2}$
obtained by reweighting $\displaystyle\mathbb{Q}_{\delta_{x}}^{2}$ by

$\displaystyle\frac{1}{\beta(\xi_{\tau^{-}}^{1})(\mathtt{m}_{2}(\xi_{\tau^{-}}^{1})-\mathtt{m}_{1}(\xi_{\tau^{-}}^{1}))}{\rm
e}^{\int_{0}^{\tau}\beta(\xi^{1}_{s})(\mathtt{m}_{2}(\xi^{1}_{s})-\mathtt{m}_{1}(\xi^{1}_{s})){\rm
d}s-\tau}.$

This change of measure alters the rate at which the spine particles split into
two distinct spines (from rate
$\displaystyle\beta(\mathtt{m}_{2}-\mathtt{m}_{1})$ to rate $\displaystyle
1$). Note, however, that it doesn’t affect the rate at which branching events
occur that don’t result in the spines splitting. Combining this with a change
of variables and conditioning on $\displaystyle\tau$, it follows that the
right-hand side of (4) is equal to

$\frac{\varphi(x)}{t\mathbb{P}_{\delta_{x}}(N_{t}>0)}\int_{0}^{a}{\rm
d}uF(u)\hat{\mathbb{Q}}_{\delta_{x}}^{2}\left[\frac{\beta(\xi^{1}_{ut})\varphi(\xi^{1}_{ut})(\mathtt{m}_{2}(\xi^{1}_{ut})-\mathtt{m}_{1}(\xi^{1}_{ut}))\mathbf{1}_{A^{\delta}_{t}}}{\varphi(\xi^{1}_{t})\varphi(\xi^{2}_{at})}\frac{t^{2}}{N_{at}\hat{N}_{t}}\,\bigg{|}\,\tau=ut\right].$
(30)

Note that under
$\displaystyle\hat{\mathbb{Q}}_{\delta_{x}}^{2}(\cdot|\tau=ut)$, the process
behaves as follows.

  * •

Until time $\displaystyle ut$, the process moves with biased motion as in (5)
with $\displaystyle\zeta$ given by (16). At rate
$\displaystyle\beta\mathtt{m}_{1}$ branching events occur, at which point, the
offspring distribution is given by $\displaystyle\mathcal{P}^{2,1}$ (as in
(6)) and the $\displaystyle i$-th particle is chosen to be the spine with
probability proportional to $\displaystyle\varphi(x_{i})$.

  * •

At time $\displaystyle ut$ a branching event occurs, where the law of the
offspring is given by $\displaystyle\mathcal{P}^{2,2}$ and particles
$\displaystyle i,j$ with $\displaystyle i\neq j$ are chosen as the two
(distinct) spines with probability proportional to
$\displaystyle\varphi(x_{i})\varphi(x_{j})$.

  * •

After time $\displaystyle ut$, the processes issued from the two spine
particles evolve under $\displaystyle\mathbb{Q}^{1}$ and those issued from the
non-spine particles evolve under $\displaystyle\mathbb{P}$.

Note also that by Lemma 10,
$\displaystyle\varphi(x)/(t\mathbb{P}_{\delta_{x}}(N_{t}>0))\to\Sigma/2$ as
$\displaystyle t\to\infty$. Thus, if we write
$\displaystyle\hat{\mathbb{Q}}^{2}_{\delta_{x},ut}(\cdot):=\hat{\mathbb{Q}}_{\delta_{x}}^{2}(\cdot\,|\,\tau=ut)$,
we have that

$\displaystyle\displaystyle\mathbb{P}_{\delta_{x}}$
$\displaystyle\displaystyle\Bigg{[}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{t},\,w\in\mathcal{N}_{at}\\\
w\npreceq
v\end{subarray}}\frac{\mathbf{1}_{A^{\delta}_{v,w}}}{N_{at}\hat{N}_{t}^{w}}F({\tau_{v,w}}/{t})\,\bigg{|}\,N_{t}>0\Bigg{]}$
$\displaystyle\displaystyle\sim\frac{\Sigma}{2}\int_{0}^{a}{\rm
d}uF(u)\hat{\mathbb{Q}}_{\delta_{x},ut}^{2}\bigg{[}\frac{\beta(\xi^{1}_{ut})\varphi(\xi^{1}_{ut})(\mathtt{m}_{2}(\xi^{1}_{ut})-\mathtt{m}_{1}(\xi^{1}_{ut}))}{\varphi(\xi^{1}_{t})\varphi(\xi^{2}_{at})}\frac{t^{2}\mathbf{1}_{A_{t}^{\delta}}}{N_{at}\hat{N}_{t}}\bigg{]},$
(31)

as $\displaystyle t\to\infty$.

Step 5 Our next goal is to describe the limit in law of
$\displaystyle(\xi_{ut}^{1},\xi_{t}^{1},\xi_{at}^{2},N_{at}/t,\hat{N}_{t}/t)$
under $\displaystyle\hat{\mathbb{Q}}_{x,ut}^{2}$ as $\displaystyle
t\to\infty$. More precisely, we claim that it converges to
$\displaystyle(\bar{\xi},\bar{\xi}^{\prime},\bar{\xi}^{\prime\prime},N,\hat{N})$
where

  * •

$\displaystyle\bar{\xi},\bar{\xi}^{\prime},\bar{\xi}^{\prime\prime}$ are
independent of eachother and of $\displaystyle(N,\hat{N})$, each with law
given by (20);

  * •

the law of $\displaystyle N$ is that of $\displaystyle aZ+(a-u)Z^{\prime}$,
where $\displaystyle(Z,Z^{\prime})$ are a pair of independent
$\displaystyle\mathtt{Gamma}(2,\,2/\Sigma\langle 1,\tilde{\varphi}\rangle)$
random variables;

  * •

conditionally on $\displaystyle N$, the law of $\displaystyle\hat{N}$ is that
of a $\displaystyle\mathtt{Gamma}(2+K,\,2/\Sigma(1-a)\langle
1,\tilde{\varphi}\rangle)$ random variable with random $\displaystyle
K\sim\mathtt{Poisson}({2N}/{(1-a)\Sigma\langle 1,\tilde{\varphi}\rangle}N)$.

To justify this claim, we identify the limiting Laplace transform of
$\displaystyle(\xi_{ut}^{1},\xi_{t}^{1},\xi_{at}^{2},N_{at}/t,\hat{N}_{t}/t)$.
To this end, for arbitrary $\displaystyle\theta,\mu,\eta,\rho,\chi\geq 0$, let
us consider

$\hat{\mathbb{Q}}_{x,ut}^{2}\big{[}{\rm e}^{-\theta\xi_{ut}^{1}}{\rm
e}^{-\mu\xi_{t}^{1}}{\rm e}^{-\eta\xi_{at}^{2}}{\rm e}^{-\rho N_{at}/t}{\rm
e}^{-\chi{\hat{N}_{t}}/{t}}\big{]}\\\ =\hat{\mathbb{Q}}_{x,ut}^{2}\big{[}{\rm
e}^{-\theta\xi_{ut}^{1}-\eta\xi_{at}^{2}-\rho{N_{at}}/{t}}\,\hat{\mathbb{Q}}_{x,ut}^{2}\big{[}{\rm
e}^{-\mu\xi_{t}^{1}}{\rm
e}^{-\chi{\hat{N}_{t}}/{t}}\,|\,\mathcal{F}_{at}^{2}\big{]}\big{]},$ (32)

where $\displaystyle\mathcal{F}_{at}^{2}$ is the $\displaystyle\sigma$-algebra
containing all the information about the process, including the spines, up to
time $\displaystyle at$.

Recalling the description of the process under
$\displaystyle\hat{\mathbb{Q}}_{x,ut}^{2}$, we see that

$\displaystyle\displaystyle\hat{\mathbb{Q}}_{\delta_{x},ut}^{2}\big{[}{\rm
e}^{-\mu\xi_{t}^{1}}{\rm
e}^{-\chi\hat{N}_{t}/t}\,|\,\mathcal{F}_{at}^{2}\big{]}=\mathbb{Q}^{1}_{\delta_{\xi_{at}^{1}}}\big{[}{\rm
e}^{-\mu\xi_{(1-a)t}}{\rm
e}^{-\chi{N_{(1-a)t}}/{t}}\big{]}\prod_{\begin{subarray}{c}v\in\mathcal{N}_{at}\\\
v\neq\psi_{at}^{1},\psi_{at}^{2}\end{subarray}}\mathbb{P}_{\delta_{X_{v}(at)}}\big{[}{\rm
e}^{-\chi{N_{(1-a)t}}/{t}}\big{]}$
$\displaystyle\displaystyle=\mathbb{Q}^{1}_{\delta_{\xi_{at}^{1}}}\big{[}{\rm
e}^{-\mu\xi_{(1-a)t}}{\rm
e}^{-\chi{N_{(1-a)t}}/{t}}\big{]}\exp\bigg{(}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{at}\\\
v\neq\psi_{at}^{1},\psi_{at}^{2}\end{subarray}}\log\big{(}1-(1-\mathbb{P}_{\delta_{X_{v}(at)}}[{\rm
e}^{-\chi{N_{(1-a)t}}/{t}}])\big{)}\bigg{)}$
$\displaystyle\displaystyle=\mathbb{Q}^{1}_{\delta_{\xi_{at}^{1}}}\big{[}{\rm
e}^{-\mu\xi_{(1-a)t}}{\rm
e}^{-\chi{N_{(1-a)t}}/{t}}\big{]}(1-E_{t})\,\exp\bigg{(}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{at}\\\
v\neq\psi_{at}^{1},\psi_{at}^{2}\end{subarray}}-(1-\mathbb{P}_{\delta_{X_{v}(at)}}[{\rm
e}^{-\chi{N_{(1-a)t}}/{t}}])\bigg{)}$ (33)

where

$\displaystyle
E_{t}:=1-\exp\bigg{(}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{at}\\\
v\neq\psi_{at}^{1},\psi_{at}^{2}\end{subarray}}\log\big{(}1-(1-\mathbb{P}_{\delta_{X_{v}(at)}}[{\rm
e}^{-\chi{N_{(1-a)t}}/{t}}])\big{)}+(1-\mathbb{P}_{\delta_{X_{v}(at)}}[{\rm
e}^{-\chi{N_{(1-a)t}}/{t}}])\bigg{)}.$

We claim that $\displaystyle E_{t}$ belongs to
$\displaystyle[0,({cN_{at}}/{t^{2}})\wedge 1]$ for some absolute deterministic
constant $\displaystyle c$, by Lemma 10(a). The lower bound of zero and the
upper bound of 1 are obvious. To see where the upper bound of
$\displaystyle{cN_{at}}/{t^{2}}$comes from, note that there are at most
$\displaystyle N_{at}$ elements of the sum defining $\displaystyle E_{t}$,
$\displaystyle\log(1-x)+x\leq-x^{2}/2$ and $\displaystyle\textstyle
1-\mathbb{P}_{\delta_{X_{v}(at)}}[{\rm
e}^{-\chi{N_{(1-a)t}}/{t}}]\leq\chi\sup_{x\in
E}\mathbb{P}_{\delta_{x}}[N_{(1-a)t}]/t\leq K/t$, for some appropriate
$\displaystyle K>0$ (where this final inequality follows from criticality and
Assumption 2).

Note also that by Lemma 10(c),

$\displaystyle\mathbb{Q}^{1}_{\delta_{\xi_{at}^{1}}}\big{[}{\rm
e}^{-\mu\xi_{(1-a)t}}{\rm
e}^{-\chi{N_{(1-a)t}}/{t}}\big{]}=s^{2}\langle\varphi{\rm
e}^{-\mu\cdot},\tilde{\varphi}\rangle(1+e_{t}(\xi_{at}^{1}))$

where $\displaystyle s:=(1+\chi{\Sigma(1-a)\langle
1,\tilde{\varphi}\rangle}/{2})^{-1}<1$ and $\displaystyle e_{t}(x)$ is such
that $\displaystyle e_{t}(x)\to 0$ in the pointwise sense on $\displaystyle E$
and

$\displaystyle\sup_{t,x}|e_{t}(x)|<\infty.$

Let us further denote for $\displaystyle x\in E,t\geq 0$,

$\displaystyle g_{t}(x):=t\big{(}1-\mathbb{P}_{\delta_{x}}[{\rm
e}^{-\chi{N_{(1-a)t}}/{t}}]\big{)}>0$

so that by Lemma 10, $\displaystyle\sup\nolimits_{t,x}g_{t}(x)<\infty$ and

$\displaystyle\displaystyle
g_{t}(x)=t\mathbb{P}_{\delta_{x}}(N_{(1-a)t}>0)\big{(}1-\mathbb{P}_{\delta_{x}}({\rm
e}^{-\chi{N_{(1-a)t}}/{t}}\,|\,N_{(1-a)t}>0)\big{)}\to\frac{2(1-s)}{\Sigma(1-a)}\varphi(x)$

pointwise on $\displaystyle E$ as $\displaystyle t\to\infty$. Using this
notation in the right hand side of (33), we see that

$\hat{\mathbb{Q}}_{\delta_{x},ut}^{2}\big{[}{\rm e}^{-\mu\xi_{t}^{1}}{\rm
e}^{-\chi\hat{N}_{t}/t}\,|\,\mathcal{F}_{at}^{2}\big{]}=s^{2}\langle\varphi{\rm
e}^{-\mu\cdot},\tilde{\varphi}\rangle{\rm
e}^{-\frac{X_{at}[g_{t}]}{t}}(1+e_{t}(\xi_{at}^{1}))(1-E_{t})$

so that

$\hat{\mathbb{Q}}_{x,ut}^{2}\big{[}{\rm e}^{-\theta\xi_{ut-}^{1}}{\rm
e}^{-\mu\xi_{t}^{1}}{\rm e}^{-\eta\xi_{at}^{2}}{\rm e}^{-\rho{N_{at}}/{t}}{\rm
e}^{-\chi{\hat{N}_{t}}/{t}}\big{]}\\\ =s^{2}\langle\varphi{\rm
e}^{-\mu\cdot},\tilde{\varphi}\rangle\hat{\mathbb{Q}}_{x,ut}^{2}\big{[}{\rm
e}^{-\theta\xi_{ut-}^{1}-\eta\xi_{at}^{2}-\rho{N_{at}}/{t}}{\rm
e}^{-{X_{at}[g_{t}]}/{t}}(1+e_{t}(\xi_{at}^{1}))(1-E_{t})\big{]}.$ (34)

Now, we claim that under $\displaystyle\hat{\mathbb{Q}}_{x,ut}^{2}$,

$(\xi_{ut}^{1},\xi_{at}^{2},e_{t}(\xi_{at}^{1}),N_{at}/t,X_{t}[g_{t}]/t,E_{t})\Rightarrow(\bar{\xi},\bar{\xi}^{\prime\prime},0,N,\tfrac{2(1-s)}{\Sigma(1-a)\langle
1,\tilde{\varphi}\rangle}N,0)$ (35)

as $\displaystyle t\to\infty$, where
$\displaystyle(\bar{\xi},\bar{\xi}^{\prime\prime},N)$ have joint law as
described in the bullet points at the start of Step 5. Since everything inside
the expectation on right-hand side of (34) is deterministically bounded, (35)
implies that

$\hat{\mathbb{Q}}_{x,ut}^{2}\big{[}{\rm e}^{-\theta\xi_{ut-}^{1}}{\rm
e}^{-\mu\xi_{t}^{1}}{\rm e}^{-\eta\xi_{at}^{2}}{\rm e}^{-\rho{N_{at}}/{t}}{\rm
e}^{-\chi{\hat{N}_{t}}/{t}}\big{]}\to\\\ \langle\varphi{\rm
e}^{-\theta\cdot},\tilde{\varphi}\rangle\langle\varphi{\rm
e}^{-\mu\cdot},\tilde{\varphi}\rangle\langle\varphi{\rm
e}^{-\lambda\cdot},\tilde{\varphi}\rangle s^{2}\int_{0}^{\infty}p_{a,u}(x){\rm
e}^{-\rho x}{\rm e}^{\tfrac{2(s-1)}{(1-a)\Sigma(1,\tilde{\varphi})}x}{\rm
d}x,$

where $\displaystyle p_{a,u}$ is the density of $\displaystyle N$ (with law as
described in the second bullet point). It is easy to check using the explicit
expressions for the Laplace transforms of Poisson and Gamma random variables,
that the right-hand side above is exactly the joint Laplace transform of our
desired limit $\displaystyle(\xi,\xi^{\prime},\xi^{\prime\prime},N,\hat{N})$.
Thus, it only remains to justify (35).

To this end, let us write $\displaystyle\mathcal{N}_{at}^{1}$ for the
collection of particles alive at time $\displaystyle at$ that have branched
off the first spine between times $\displaystyle 0$ and $\displaystyle at$
(depicted in orange in Figure 3). We also write
$\displaystyle\mathcal{N}_{at}^{2}$ for those particles alive at time
$\displaystyle at$ that have branched off the second spine between times
$\displaystyle ut$ and $\displaystyle at$ (depicted in green in Figure 3).
Then, using Lemma 10 (and its extension Remark 10), it follows that

$(\xi_{ut}^{1},e_{t}(\xi_{at}^{1}),\frac{1}{t}{|\mathcal{N}_{at}^{1}|},\frac{1}{t}{\sum_{v\in\mathcal{N}_{at}^{1}}g_{t}(X_{v}(at))})\Rightarrow(\xi,0,N^{\prime},\tfrac{2(1-s)}{\Sigma(1-a)\langle
1,\tilde{\varphi}\rangle}N^{\prime})$ (36)

say, where $\displaystyle(\xi,N^{\prime})$ are independent, $\displaystyle\xi$
has law given in (20) and $\displaystyle N^{\prime}$ has the law of
$\displaystyle a$ times a $\displaystyle\mathtt{Gamma}(2,2/\Sigma\langle
1,\tilde{\varphi}\rangle)$ random variable (recall that $\displaystyle
e_{t}(x)$ is deterministically uniformly bounded over $\displaystyle t,x$ and
converges pointwise to $\displaystyle 0$ on $\displaystyle E$). Now, given all
the information in the quadruple displayed on the left-hand side of (36), the
(conditional) joint law of

$\displaystyle(\xi_{at}^{2},\frac{1}{t}{|\mathcal{N}_{at}^{2}|},\frac{1}{t}{\sum_{v\in\mathcal{N}_{at}^{2}}g_{t}(X_{v}(at))})$

is given by the $\displaystyle\mathbb{Q}_{\delta_{X}}^{1}$ law of
$\displaystyle(\xi_{(a-u)t}^{2},N_{(a-u)t}/t,X_{(a-u)t}[g_{t}]/t)$, where the
(conditional) law of $\displaystyle X$ is explicit but not required here.
Again, by Lemma 10, in particular part (c),

$\displaystyle(\xi_{(a-u)t}^{2},N_{(a-u)t}/t,X_{(a-u)t}[g_{t}]/t)\Rightarrow(\hat{\xi},\hat{N},\tfrac{2(1-s)}{\Sigma(1-a)\langle
1,\tilde{\varphi}\rangle}\hat{N})$

as $\displaystyle t\to\infty$ under
$\displaystyle\mathbb{Q}_{\delta_{X}}^{1}$, where
$\displaystyle\hat{\xi},\hat{N}$ are independent, $\displaystyle\hat{\xi}$ has
law given by (20), and $\displaystyle\hat{N}$ has the law of
$\displaystyle(a-u)$ times a $\displaystyle\mathtt{Gamma}(2,2/\Sigma\langle
1,\tilde{\varphi}\rangle))$ random variable, independently of the value of
$\displaystyle X$. Putting these observations together, plus the fact that
$\displaystyle E_{t}\in[0,cN_{at}/t^{2}]$ for some deterministic
$\displaystyle c$, gives us (35).

Step 6 Using the convergence in law (and associated notation for limiting
variables) from Step 5, plus boundedness of the functionals in question, we
deduce that for each $\displaystyle 0\leq u\leq a$,

$\displaystyle\displaystyle\frac{\Sigma}{2}\int_{0}^{a}{\rm
d}uF(u)\hat{\mathbb{Q}}_{\delta_{x},ut}^{2}\bigg{[}\frac{\beta(\xi^{1}_{ut})\varphi(\xi^{1}_{ut})(\mathtt{m}_{2}(\xi^{1}_{ut})-\mathtt{m}_{1}(\xi^{1}_{ut}))}{\varphi(\xi^{1}_{t})\varphi(\xi^{2}_{at})}\frac{t^{2}\mathbf{1}_{A_{t}^{\delta}}}{N_{at}\hat{N}_{t}}\bigg{]}$
$\displaystyle\displaystyle{\to}\frac{\Sigma}{2}\int_{0}^{a}{\rm
d}uF(u)\mathbb{E}\left[\frac{\beta(\bar{\xi})\varphi(\bar{\xi})(\mathtt{m}_{2}(\bar{\xi})-\mathtt{m}_{1}(\bar{\xi}))}{\varphi(\bar{\xi}^{\prime})\varphi(\bar{\xi}^{\prime\prime})}\mathbf{1}_{\\{\varphi(\bar{\xi})\geq\delta,\varphi(\bar{\xi}^{\prime})\geq\delta,\varphi(\bar{\xi}^{\prime\prime})\geq\delta\\}}\frac{\mathbf{1}_{\\{N\geq\delta,\hat{N}\geq\delta\\}}}{N\hat{N}}\right],$
(37)

as $\displaystyle{t\to\infty}$. The expectation in the integrand on the right-
hand side of (37) is equal to

$\displaystyle\displaystyle
c_{\delta}\mathbb{E}\left[\frac{\mathbf{1}_{\\{N\geq\delta\\}}}{N}\mathbb{E}\left[\frac{\mathbf{1}_{\\{\hat{N}\geq\delta\\}}}{\hat{N}}\,\bigg{|}\,N\right]\right]$
$\displaystyle\displaystyle=c_{\delta}\mathbb{E}\left[\frac{\mathbf{1}_{\\{N\geq\delta\\}}}{N}\left(\mathbb{E}\left[\frac{1}{\hat{N}}\,\bigg{|}\,N\right]-\mathbb{E}\left[\frac{\mathbf{1}_{\\{\hat{N}<\delta\\}}}{\hat{N}}\,\bigg{|}\,N\right]\right)\right]$
$\displaystyle\displaystyle={c_{\delta}}\mathbb{E}\left[\frac{\mathbf{1}_{\\{N\geq\delta\\}}}{N}\frac{1-\exp(-\tfrac{1}{1-a}(\tfrac{2}{\Sigma\langle
1,\tilde{\varphi}\rangle}N))}{N}\right]-c_{\delta}\mathbb{E}\left[\frac{\mathbf{1}_{\\{N\geq\delta\\}}}{N}\mathbb{E}\left[\frac{\mathbf{1}_{\\{\hat{N}<\delta\\}}}{\hat{N}}\,\bigg{|}\,N\right]\right]$
$\displaystyle\displaystyle={c_{\delta}}\mathbb{E}\left[\frac{1-\exp(-\tfrac{1}{1-a}(\tfrac{2}{\Sigma\langle
1,\tilde{\varphi}\rangle}N))}{N^{2}}\right]-c_{\delta}\mathbb{E}\left[\frac{\mathbf{1}_{\\{N<\delta\\}}}{N}\frac{1-\exp(-\tfrac{1}{1-a}(\tfrac{2}{\Sigma\langle
1,\tilde{\varphi}\rangle}N))}{N}\right]$ $\displaystyle\displaystyle\hskip
227.62204pt-
c_{\delta}\mathbb{E}\left[\frac{\mathbf{1}_{\\{N\geq\delta\\}}}{N}\mathbb{E}\left[\frac{\mathbf{1}_{\\{\hat{N}<\delta\\}}}{\hat{N}}\,\bigg{|}\,N\right]\right]$
$\displaystyle\displaystyle=:\frac{4c_{\delta}}{a^{2}\Sigma^{2}\langle
1,\tilde{\varphi}\rangle^{2}}\mathbb{E}\left[\frac{1-\exp(-\tfrac{a}{1-a}(\tfrac{2}{a\Sigma\langle
1,\tilde{\varphi}\rangle}N))}{(\tfrac{2}{a\Sigma\langle
1,\tilde{\varphi}\rangle}N)^{2}}\right]-h(\delta)$ (38)

as $\displaystyle t\to\infty$, where under $\displaystyle\mathbb{E}$,
$\displaystyle(\bar{\xi},\bar{\xi}^{\prime},\bar{\xi}^{\prime\prime},N,\hat{N})$
are described in Step 5 (recalling in particular the law of
$\displaystyle\hat{N}$ given $\displaystyle N$) and $\displaystyle
h(\delta)\geq 0$ and $\displaystyle
c_{\delta}:=\langle{\mathbf{1}_{\\{\varphi\geq\delta\\}}},\tilde{\varphi}\rangle^{2}\langle\beta\varphi^{2}(\mathtt{m}_{2}-\mathtt{m}_{1})\mathbf{1}_{\\{\varphi\geq\delta\\}},\tilde{\varphi}\rangle.$

Step 7 Recall that by (27) and (31), our aim is to prove that

$\frac{\Sigma}{2}\int_{0}^{a}{\rm
d}uF(u)\hat{\mathbb{Q}}_{\delta_{x},ut}^{2}\bigg{[}\frac{\beta(\xi^{1}_{ut})\varphi(\xi^{1}_{ut})(\mathtt{m}_{2}(\xi^{1}_{ut})-\mathtt{m}_{1}(\xi^{1}_{ut}))}{\varphi(\xi^{1}_{t})\varphi(\xi^{2}_{at})}\frac{t^{2}\mathbf{1}_{A_{t}^{\delta}}}{N_{at}\hat{N}_{t}}\bigg{]}\to\frac{c_{\delta}}{\langle
1,\tilde{\varphi}\rangle^{2}\Sigma}\int_{0}^{a}F(u)f_{a}^{\delta}(u){\rm d}u$

as $\displaystyle t\to\infty$, for some $\displaystyle
f_{a}^{\delta}(u)\nearrow f_{a}(u)$ as $\displaystyle\delta\searrow 0$,
pointwise on $\displaystyle[0,a]$.

First notice that $\displaystyle h(\delta)$ in (4) converges to $\displaystyle
0$ as $\displaystyle\delta\searrow 0$, since the law of $\displaystyle
N\hat{N}$ has negative moments of all orders. Then (4), (37) imply the result,
since writing $\displaystyle Y={2N}/{a\Sigma(1,\tilde{\varphi})}$ (so that
$\displaystyle Y\sim Y^{\prime}+(1-\tfrac{u}{a})Y^{\prime\prime}$ for
independent $\displaystyle
Y^{\prime},Y^{\prime\prime}\sim\mathtt{Gamma}(2,1)$) we have:

$\displaystyle\displaystyle\frac{2}{a^{2}}\mathbb{E}\left[\frac{1-\exp(-\tfrac{a}{1-a}(\tfrac{2}{a\Sigma(1,\tilde{\varphi})}N))}{(\tfrac{2}{a\Sigma(1,\tilde{\varphi})}N)^{2}}\right]$
$\displaystyle\displaystyle=\frac{2}{a^{2}}\int_{0}^{\infty}\mathbb{E}\left[\theta\exp(-\theta
Y)-\theta\exp(-(\theta+\tfrac{a}{1-a})Y)\right]\,d\theta$
$\displaystyle\displaystyle=\frac{2}{a^{2}}\int_{0}^{\infty}\left(\frac{\theta}{(1+\theta)^{2}(1+(1-\tfrac{u}{a})\theta)^{2}}-\frac{\theta}{(1+\theta+\tfrac{a}{1-a})^{2}(1+(1-\tfrac{u}{a})(\theta+\tfrac{a}{1-a}))^{2}}\right){\rm
d}\theta$ $\displaystyle\displaystyle=f_{a}(u).$

To calculate the integral in the penultimate line, we have used the change of
variables $\displaystyle x=\theta+\frac{a}{1-a}$ for the second integrand, the
fact that the anti-derivative of $\displaystyle\frac{y}{(1+y)^{2}(1+\gamma
y)^{2}}$ is given by

$\displaystyle\frac{1}{(\gamma-1)^{3}}\left(\frac{(\gamma-1)(\gamma
y+y+2)}{(y+1)(\gamma y+1)}-(\gamma+1)\log\big{(}\frac{y+1}{\gamma
y+1}\big{)}\right),$

and that the anti-derivative of $\displaystyle\frac{1}{(1+y)^{2}(1+\gamma
y)^{2}}$ is given by

$\displaystyle\frac{1}{(\gamma-1)^{3}}\left(\frac{-(\gamma-1)(2\gamma
y+\gamma+1)}{(y+1)(\gamma y+1)}+2\gamma\log\big{(}\frac{y+1}{\gamma
y+1}\big{)}\right).$

The proof is now complete. ∎

###### Proof of Proposition 9.

The proof of this proposition is contained in the proof of Step 5 above,
ignoring the contribution from the second spine. ∎

##### Acknowledgements

This work was partially supported by by EPSRC grant EP/W026899/1, UKRI Future
Leader’s Fellowship MR/W008513/1, and the “PHC Alliance” programme (project
number: 47867UJ), funded by the French Ministry for Europe and Foreign
Affairs, the French Ministry for Higher Education and Research and the UK
Department for Business, Energy and Industrial Strategy.

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