# Genetic Composition of Supercritical Branching Populations under Power Law
Mutation Rates

Vianney Brouard ENS de Lyon, UMPA, CNRS UMR 5669, 46 Allée d’Italie, 69364
Lyon Cedex 07, France; E-mail<EMAIL_ADDRESS>
(September 21, 2023)

###### Abstract

We aim at understanding the evolution of the genetic composition of cancer
cell populations. To this aim, we consider a branching individual based model
representing a cell population where cells divide, die and mutate along the
edges of a finite directed graph $(V,E)$. The process starts with only one
cell of trait $0$. Following typical parameter values in cancer cell
populations we study the model under large population and power law mutation
rates limit, in the sense that the mutation probabilities are parameterized by
negative powers of $n$ and the typical sizes of the population of our interest
are positive powers of $n$. Under non-increasing growth rate condition (namely
the growth rate of any sub-population is smaller than the growth rate of trait
$0$), we describe the time evolution of the first-order asymptotics of the
size of each sub-population on the $log(n)$ time scale, as well as in the
random time scale at which the initial population, resp. the total population,
reaches the size $n^{t}$. In particular, such results allow to characterize
whose mutational paths along the edges of the graph are actually contributing
to the size order of the sub-populations. Without any condition on the growth
rate, we describe the time evolution of the orders of magnitude of each sub-
population. Adapting techniques from [13], we show that these converges to
positive deterministic non-decreasing piecewise linear continuous functions,
whose slopes are given by an algorithm.

Keywords: cancer evolution, multitype branching processes, finite graph, long
time behavior, power law mutation rates, population genetics.

##  1 Introduction and presentation of the model

Consider a population of cells characterized by a phenotypic trait, where the
trait space $V$ is finite and discrete. For all $v\in V$ denote by
$(Z_{v}(t))_{t\in\mathbb{R}^{+}}$ the number of cells of trait $v$ at time $t$
in the population, and $\left(\mathcal{Z}(t):=\left(Z_{v}(t)\right)_{v\in
V}\right)_{t\in\mathbb{R}^{+}}$ the global process. Assume that $0\in V$ and

$\displaystyle\forall v\in V,Z_{v}(0)=\mathbbm{1}_{\\{v=0\\}},\text{ almost
surely.}$ (1)

Cells with trait $0$ are called wild-type cells, and all cells with trait
$v\in V\backslash\\{0\\}$ are called mutant cells.
The population dynamics will follow a continuous time branching process on
$\mathbb{N}_{0}^{V}$. More precisely cells divide to give birth to two
daughter cells and die with rates depending only on their phenotypic trait.
The birth, death and growth rate functions are respectively

$\displaystyle\alpha:V\longrightarrow\mathbb{R}^{+},$ (2)
$\displaystyle\beta:V\longrightarrow\mathbb{R}^{+},$ (3)
$\displaystyle\lambda:=\alpha-\beta.$ (4)

During a division event of a cell of trait $v\in V$, independent mutation over
the two daughter cells are considered. Mutation landscape across traits is
encoded via a graph structure $(V,E)$ on the trait space.
$E\subset\\{(v,u),\forall v,u\in V^{2}\\}$ is a set of ordered pairs over $V$
satisfying for all $v\in V$, $(v,v)\cap E=\emptyset$. It means that $(V,E)$ is
a finite oriented graph without self-loop. Mutation from a trait $v$ to a
trait $u$ is possible if and only if $(v,u)\in E$. Let
$\mu:E\longrightarrow[0,1]$ be a mutation kernel satisfying

$\displaystyle\forall v\in V,\overline{\mu}(v):=\sum_{u\in V:(v,u)\in
E}\mu(v,u)\leq 1.$ (5)

A daughter cell mutates from type $v$ to type $u$ with probability $\mu(v,u)$,
meaning that $\overline{\mu}(v)$ is its total mutation probability. Notice
that backward mutations are contained in this model.
Finally the exact transition rates from a state $z=(z_{v})_{v\in
V}\in\mathbb{N}_{0}^{V}$ of the process $\mathcal{Z}$ are

$\displaystyle z\mapsto\left\\{\begin{array}[]{l l l}&z-\delta_{v},&\mbox{ at
rate }z_{v}\beta(v),\\\ &z-\delta_{v}+\delta_{u}+\delta_{w},&\mbox{ at rate
}2z_{v}\alpha(v)\mu(v,u)\mu(v,w)\mathbbm{1}_{\\{(v,u)\in
E\\}}\mathbbm{1}_{\\{(v,w)\in E\\}}\mathbbm{1}_{\\{u\neq w\\}},\\\
&z-\delta_{v}+2\delta_{u},&\mbox{ at rate
}z_{v}\alpha(v)\mu(v,u)^{2}\mathbbm{1}_{\\{(v,u)\in E\\}},\\\
&z+\delta_{v},&\mbox{ at rate
}z_{v}\alpha(v)\left(1-\overline{\mu}(v)\right)^{2}+2\underset{u\in V:(u,v)\in
E}{\sum}z_{u}\alpha(u)\mu(u,v)\left(1-\overline{\mu}(u)\right),\end{array}\right.$
(10)

where $\forall v\in V,\delta_{v}=\left(\mathbbm{1}_{\\{u=v\\}}\right)_{u\in
V}$. Through the paper, the growth rate of the wild-type sub-population is
assumed to be strictly positive $\lambda(0)>0$, otherwise the wild-type sub-
population won’t survive almost surely.
The biological motivation of this model is to capture the dynamics over time
of the genetic composition of a population of cells during carcinogenesis.
Tumors are detected when they reaches a size of a large amount of cells,
typically $10^{9}$ cells. The mutations rates per base pair per cell division
is generally estimated to be of order $10^{-9}$, see [20, 4]. Then it
naturally invites to consider the framework of large population and power law
mutation rates regime. It means that a parameter $n\in\mathbb{N}$ is used to
quantify both the decrease of the mutation probabilities, as negative powers
of $n$, and also the typical size of the population, depending on $n$ as
positive power of $n$, at which we are interested in understanding the genetic
composition. The aim is to obtain asymptotic results on the sizes of all the
mutant sub-populations when $n$ goes to infinity. It is a classical stochastic
regime studied in particular in [7, 6, 13, 3, 5, 8, 14, 2, 30, 16]. Such
regime is referred in [7, 6] as the large population rare mutations limit, but
we decided to take the precision of power law mutation rates in order to
distinguish such regime with the classical rare mutation regime where the
mutation probabilities $\mu^{(n)}$ scale typically as
$e^{-Cn}\ll\mu^{(n)}\ll\frac{1}{n\log(n)}$. Indeed with the large population
power law mutation rates regime, the mutation probability can be of a higher
order than the rare mutation regime if for instance $\mu^{(n)}\propto
n^{-\alpha}$ with $\alpha\in(0,1]$.
To be more precise, let $L:=\\{\ell(v,u)\in\mathbb{R}_{+}^{*},\forall(v,u)\in
E\\}$ be a set of strictly positive labels on the edges of the graph.
Introduce a sequence of models
$\left(\mathcal{Z}^{(n)}\right)_{n\in\mathbb{N}}$, where for each
$n\in\mathbb{N}$, $\mathcal{Z}^{(n)}$ corresponds to the process described
above with the mutation kernel $\mu^{(n)}:E\longrightarrow[0,1]$ satisfying

$\forall(v,u)\in
E,n^{\ell(v,u)}\mu^{(n)}(v,u)\underset{n\to\infty}{\longrightarrow}\mu(v,u)\in\mathbb{R}^{+}.$
(11)

For each $t\in\mathbb{R}_{+}^{*}$, the stopping times corresponding to the
first time that the wild-type population $Z_{0}^{(n)}$, respectively the total
population $Z^{(n)}_{tot}:=\sum_{v\in V}Z_{v}^{(n)}$, reaches the level
$n^{t}$, are defined as

$\displaystyle\eta_{t}^{(n)}:=\inf\left\\{u\in\mathbb{R}^{+}:Z_{0}^{(n)}(u)\geq
n^{t}\right\\},$ (12)
$\displaystyle\sigma_{t}^{(n)}:=\inf\left\\{u\in\mathbb{R}^{+}:Z^{(n)}_{tot}(u)\geq
n^{t}\right\\}.$ (13)

Two different biological interpretations in different settings can be made in
order to motivate both of them. For instance, when considering metastasis the
wild-type population $Z_{0}^{(n)}$ may represent the primary tumor, and the
mutant sub-populations $Z_{v}^{(n)},$ for $v\in V\backslash\\{0\\}$, may
correspond to secondary tumors. As it is size and not age of a tumor that
clinicians have access to, it is biologically relevant to estimate the genetic
composition of the secondary tumors when the primary one has a given size.
This is mathematically encoded in looking at the first-order asymptotics of
$Z_{v}^{(n)}\left(\eta_{t}^{(n)}\right)$ for any $v\in V\backslash\\{0\\}$.
Another biological setting is when the total population $Z^{(n)}_{tot}$
represents one tumor. It is appropriate to obtain theoretical results about
the size of the mutant cells $Z_{v}^{(n)}$ for any $v\in V\backslash\\{0\\}$
when the tumor has reached a given size. It corresponds exactly to look at the
first-order asymptotics of $Z_{v}^{(n)}\left(\sigma_{t}^{(n)}\right)$. Every
time that results can be stated either with $\eta_{t}^{(n)}$ or
$\sigma_{t}^{(n)}$, the following notation will be used

$\displaystyle\rho_{t}^{(n)}:=\eta_{t}^{(n)}\text{ or }\sigma_{t}^{(n)}.$ (14)

In the present work the population of cells will be studied in different time-
scales: the random time-scale

$\displaystyle\left(\rho_{t}^{(n)}+s\right)_{(t,s)\in\mathbb{R}^{+}\times\mathbb{R}};$
(15)

and the following deterministic approximation

$\displaystyle\left(\mathfrak{t}^{(n)}_{t}+s\right)_{(t,s)\in\mathbb{R}^{+}\times\mathbb{R}},$
(16)

where

$\displaystyle\mathfrak{t}^{(n)}_{t}:=t\frac{\log(n)}{\lambda(0)}.$ (17)

Intuitively, the lineage of wild-type cells generated from the initial cell is
the first sub-population that will allow to create mutations. Then
understanding its growth gives the natural time scale to consider for seeing
mutations. Its birth and death rates are
$\alpha(0)\left(1-\overline{\mu}^{(n)}(0)\right)^{2}$ and
$\beta(0)+\alpha(0)\left(\overline{\mu}^{(n)}(0)\right)^{2}$ respectively.
Because of the power law mutation rates regime of Equation (11) they are
converging to $\alpha(0)$ and $\beta(0)$ when $n$ grows to $\infty$. Meaning
that this lineage should behave asymptotically as a birth and death process
with rates $\alpha(0)$ and $\beta(0)$. Indeed such a result emerges on the
natural martingale associated to a birth and death process, see Lemma 3.1. In
particular the growth rate of this lineage is close to $\lambda(0)$ thus this
population reaches a size of order $n^{t}$ approximately at the deterministic
time $\mathfrak{t}_{t}^{(n)}$, see Lemma 3.2.
For any finite oriented labeled graph $(V,E,L)$ under the following non-
increasing growth rate condition

$\displaystyle\forall v\in V,\lambda(v)\leq\lambda(0),$ (18)

the first-order asymptotics of the sizes of any mutant populations
$Z_{v}^{(n)}$ are obtained both in random and deterministic time-scales (15)
and (16), see Theorem 2.1. Assumption (18) can be biologically motivated.
Historically tumor dynamics has been seen under the prism of clonal expansion
of selective mutations, i.e. $\lambda(v)>\lambda(0)$. Nevertheless the
paradigm of neutral evolution of cancer has been recently considered, see [31,
26, 33, 32, 9], meaning that the selective mutations are already present in
the initial cell and that the occurring mutations are neutral ones (i.e.
$\lambda(v)=\lambda(0)$). With Assumption (18) deleterious mutations (i.e.
$\lambda(v)<\lambda(0)$) are also considered. This paradigm has been
introduced because the genetic heterogeneity inside a tumor could be explained
only considering neutral mutations. Various statistical methods are developed
to infer the evolutionary history of tumors, including test of neutral
evolution, see [34, 1, 17] for details about that.
Without any assumption on the growth rate function $\lambda$, the study is
made on the deterministic time-scale of Equation (16). As in [13, 3, 5, 8, 14,
2, 30] the asymptotic behaviors are obtained on the following stochastic
exponent processes

$\displaystyle\forall v\in
V,X_{v}^{(n)}(t):=\frac{\log^{+}\left(Z_{v}^{(n)}\left(\mathfrak{t}^{(n)}_{t}\right)\right)}{\log(n)/\lambda(0)}.$
(19)

The results are presented in Theorem 2.2. It is the exponent as a power of $n$
that is tracked for any sub-populations, whereas Theorem 2.1 gives directly
the size order on $n$, this is a refined result. Up to our knowledge, it is
the first model considering the power law mutation rates regime (11) capturing
this level of refinement on the asymptotic behaviors. Two new significant
results emerge.
First it shows the remarkable result that under Assumption (18) the randomness
on the first-order asymptotics of any mutant sub-populations is fully given by
the stochasticity of only one random variable $W$ -encoding the randomness on
the long time for the lineage of wild-type cells issued from the initial cell.
It means that the stochasticity for any mutant sub-population is fully driven,
at least at the first-order asymptotics, by the randomness on the growth of
the wild-type population and not from the dynamics of any lineage of a mutant
cell, as well as the stochastic processes generating mutations.
Second it characterizes exactly whether a mutational path on the graph
structure of the trait space asymptotically contributes to the growth of the
mutant sub-populations. Whereas having asymptotic results on the stochastic
exponents only allows to discriminate some paths and not to determine exactly
whose paths are actually contributing to the asymptotic growth of the mutant
sub-populations. More precisely, if the weight of a path is defined as the sum
of the label of its edges, asymptotic results on the stochastic exponent gives
that for every trait $v$, among the paths from $0$ to $v$ only those with the
less weight might contribute to the asymptotic growth of trait $v$. On the
contrary, having results directly on the first-order asymptotics of the mutant
sub-populations allows to discriminate among those paths with the less weight
those which actually contributes to the dynamics of trait $v$. In particular
among those paths with the less weight only those with the maximal number of
neutral mutations on their edges have an asymptotic impact on the growth of
trait $v$. Indeed an additional multiplicative factor of order $\log(n)$ for
each neutral mutation of a path is captured when looking at the first-order
asymptotics and is obviously not captured with asymptotic results only on the
stochastic exponents.
Moreover it is the first time that this power-law mutation rates regime is
studied in the random time-scale of Equation (15) up to our knowledge. From
the biological point of view it is more interesting to get result on such
random time-scale instead of the deterministic one. We obtain that the
randomness on the first-order asymptotics of any mutant sub-populations is
fully given by the stochasticity on the survival of the lineage of wild-type
cells issued from the initial cell.
In [7, 6] Cheek and Antal study a model that can be seen as an application of
the model of the present work via a specific finite oriented labeled graph
$(V,E,L)$. Among their results, they fully characterize in distribution the
asymptotic sizes of all the mutant sub-populations around the random time at
which the wild-type sub-population reaches the typical size allowing mutations
to occur. In their setting it corresponds to
$\left(\eta_{1}^{(n)}+s\right)_{s\in\mathbb{R}}$. In particular they obtain
that the asymptotic sizes of all the mutant sub-populations around this random
time $\eta_{1}^{(n)}$ is finite almost surely, following a generalised Luria-
Delbrück distribution, see Theorem 5.1 in [6]. The initial Luria and Delbrück
model has generated many subsequent works, see in particular [27, 25, 22, 19,
21, 23, 7, 6]. Two major features explain the latter result. The first one is
that asymptotically only a finite number of mutant cells are generated from
the wild-type population until time $\eta_{1}^{(n)}$, following a Poisson
distribution. The second one is that all the lineages of the mutant cells
generated from the wild-type population have only, up to time
$\eta_{1}^{(n)}$, asymptotically a finite random time to grow, which is
exponentially distributed. We extend their results to larger times, typically
when the total mutation rate from the sub-population of a trait $v$ to the
sub-population of a trait $u$ is growing as a positive power of $n$, instead
of remaining finite.
In [13] Durrett and Mayberry study the exponentially growing Moran model. They
consider the same mutation regime, their size of the total population is
growing exponentially fast at a fixed rate, and new individuals in the
population chose their trait via a selective frequency dependent process. In
Theorem 2.2 a similar result is obtained for the case of a multitype branching
population. In particular, for this setting the exponential speed of the total
population (and of the dominant sub-populations) is evolving through time.
More specifically, we show that the speed is a non-decreasing piecewise
constant function going from $\lambda(0)$ to $\underset{v\in
V}{\max}\lambda(v)$, and taking values only on the set
$\left\\{\lambda(v),\forall v\in V\right\\}$.
In [7, 13, 3, 5, 8, 14, 2, 30], the authors are considering the power law
mutation rates regime of Equation (11) in the special case where all different
traits mutate with the same scaling of a fixed order of a negative power of
$n$. Whereas in the present work the power law mutation rates regime is more
general by allowing traits to mutate with different scalings, as in [6, 16].
As in [7, 6], compared to the different models in [13, 3, 5, 8, 14, 2, 16],
the initial population $\mathcal{Z}^{(n)}(0)$ is not assumed to have a
macroscopic size. It introduces a supplementary randomness on how the wild-
type population is stochastically growing to get a macroscopic size. But
contrary to [7, 6], we do not condition on the survival of the wild-type
population or on the stopping times of Equation (15) to be finite.
In [28] Nicholson and Antal study a similar model under a slightly less
general non-increasing growth rate condition. More precisely, in their case
all the growth rates of the mutant populations are strictly smaller than the
growth rate of the wild-type population: $\forall v\in
V\backslash\\{0\\},\lambda(v)<\lambda(0)$. But the main difference remains the
mutation regime. In their case, only the last mutation is in the power law
mutation rates regime, all other mutations have a fixed probability
independent of $n$. In Theorem 2.1 the case where all mutations are in the
power law mutation rates regime is treated. Also Nicholson and Antal are
interested in obtaining the distribution of the first time that a mutant sub-
population gets a mutant cell. Whereas in the present work the first-order
asymptotics of the sizes of the mutant sub-populations over time are studied.
In [29] Nicholson, Cheek and Antal study the case of a mono-directional graph
where the time tends first to infinity with fixed mutation probabilities. In
particular they obtain the almost sure first-order asymptotics of the size of
the different mutant sub-populations. Under the non-increasing growth rate
condition, they are able to characterized the distribution of the limit random
variables they obtained. Without any condition on the growth rates, they study
the distribution of the random limit they obtained under the small mutation
probabilities limit, using the hypothesis of an approximating model with less
stochasticity. Notice that the mutation regime they study is not the large
population power law mutation rates regime of Eq. (11) as considered in the
present work. Under the latter regime both the size of the population goes to
infinity and the mutation probabilities to 0, through the parameter $n$, see
Equation (11).
In [18] Gunnarson, Leder and Zhang study a similar model as the one in the
present work and are also interested in capturing the evolution over time of
the genetic diversity of a population of cells, using in their case the well-
known summary statistic called the site frequency spectrum (SFS). The main
difference is the mutation regime because they are not considering the power
law mutation rates limit. In their case the mutation probabilities are fixed.
Also, they restrict the study to the neutral cancer evolution case. In
particular, as in the present work, they capture the first-order asymptotics
of the SFS at a fixed time and at the random time at which the population
first reaches a certain size. Two noticeable similarities in the results are
that the first-order asymptotics of the SFS converges to a random limit when
evaluated at a fixed time and to a deterministic limit when evaluated at the
stochastic previous time. One could argue that in the present work the correct
convergence for the latter case is actually a stochastic limit. But the
stochasticity is fully given by the survival of the initial lineage of cells
of trait $0$, so conditioned on such an event at the end the limit is a
deterministic one. In particular the results of Gunnarson, Leder and Zhang are
all conditioned on nonextinction of the population.
In [16] Gamblin and Lambert study a model of an exponentially growing asexual
population that undergoes cyclic bottlenecks under the large population power
law mutation rates regime. Their trait space is composed of 4 sub-populations
$00,10,01$ and $11$, where two paths of mutations are possible $00\mapsto
10\mapsto 11$ and $00\mapsto 01\mapsto 11$. They study the special case where
one mutation ($10$) has a high-rate but is a weakly beneficial mutation
whereas the other mutation ($01$) has a low-rate but is a strongly beneficial
mutation. In particular they show the noticeable result that due to cyclic
bottlenecks only a unique evolutionary path unfolds but modifying their
intensity and period implies that all paths can be explored. Their work relies
on a deterministic approximation of the wild-type sub-population $00$ and some
parts of the analysis of the behavior of the model is only obtained due to
heuristics. The present work, and more specifically Theorem 2.2 because they
are considering selective mutations, can be used and adapted to consider the
case of cyclic bottlenecks in order to prove rigorously their results, in the
specific trait space that they consider as well as on a general finite trait
space.
The rest of the paper is organised as follows. In Section 2 the results and
their biological interpretations are given. Sections 3 and 4 are dedicated to
prove Theorem 2.1, which assumes Equation (18). In Section 3 the mathematical
definition of the model is given for an infinite mono-directional graph as
well as the proof in this particular case. The generalisation of the proof
from an infinite mono-directional graph to a general finite graph is given in
Section 4. In Section 5, Theorem 2.2 is proved adapting results from [13].

##  2 Main results and biological interpretation

In Subsection 2.1 the first-order asymptotics of the size of all the mutant
sub-populations in the time-scales (15) and (16) are given under the non-
increasing growth rate condition (18). In Subsection 2.2 the asymptotic result
on the stochastic exponent of all the mutant sub-populations are given without
any assumption on the growth rate function $\lambda$. In each subsection,
biological interpretations of the results are made.

###  2.1 First-order asymptotics of the mutant sub-population sizes under non-
increasing growth rate condition

In this subsection assume that $(V,E,L)$ satisfies the non-increasing growth
rate graph condition of Equation (18).
Heuristics:
The next definitions, notations and results are first motivated using some
heuristics for the simplest graph that one can think of, i.e. a wild-type and
a mutant population where only mutations from wild-type to mutant cells are
considered. More precisely
$(V,E,L)=\left(\\{0,1\\},\\{(0,1)\\},\\{\ell(0,1)\\}\right)$ as in Figure 1.

Figure 1: Two traits model without backward mutation

Under the power law mutation rates regime, the inner birth and death rate of
the wild-type population is so close to $\alpha(0)$ and $\beta(0)$
respectively that its natural martingale asymptotically behaves as the one of
a birth and death process with rates $\alpha(0)$ and $\beta(0)$ (Lemma 3.1).
This fact allows to approximate the growth of the wild-type population as an
exponential growth of parameter $\lambda(0)$. Then if it survives, at time
$\mathfrak{t}^{(n)}_{t}$ (see (16)) its size is of order
$\mathcal{O}\left(n^{t}\right)$ (Lemma 3.2). From this fact, one understands
why it is necessary to wait for time $\mathfrak{t}^{(n)}_{\ell(0,1)}$ before
seeing any mutation. Indeed, with a mutation probability which scales as
$n^{-\ell(0,1)}$, the total mutation probability up to time
$\mathfrak{t}^{(n)}_{t}$ scales as
$\int_{0}^{t}n^{u}n^{-\ell(0,1)}d\left(u\frac{\log(n)}{\lambda(0)}\right)=\frac{n^{-\ell(0,1)}}{\lambda(0)}\left(n^{t}-1\right)$
which starts to be of order 1 for $t\geq\ell(0,1)$. This is made formal by D.
Cheek and T. Antal in [7, 6] and an illustration can be found at Figure 2.

Figure 2: Heuristics for the first occurrence time of mutant cells

It is also possible to get some heuristics for the size of the mutant
population for time $\mathfrak{t}^{(n)}_{t},$ for $t\geq\ell(0,1)$. Let
$\ell(0,1)\leq u\leq t$, the number of new mutations generated at time
$\mathfrak{t}_{u}^{(n)}$ scales as
$\exp\left(\lambda(0)(u-\ell(0,1))\frac{\log(n)}{\lambda(0)}\right)$. The
remaining time for these new mutant cells to grow exponentially fast at rate
$\lambda(1)$ until time $\mathfrak{t}_{t}^{(n)}$ is
$\mathfrak{t}_{t-u}^{(n)}$. This implies that their lineages get at time
$\mathfrak{t}_{t}^{(n)}$ a size of order

$\displaystyle\mathcal{O}\left(\exp\left(\left[\lambda(1)t+(\lambda(0)-\lambda(1))u-\lambda(0)\ell(0,1)\right]\frac{\log(n)}{\lambda(0)}\right)\right).$
(20)

Then two scenari are possible:

  * (i)

If $\lambda(1)<\lambda(0)$: Equation (20) is maximal for $u=t$ and equal to
$n^{t-\ell(0,1)}$. This means that the dynamics of the mutant population is
driven by the mutation from the wild-type population and not from its inner
growth. Mathematically it means that its size order at time
$\mathfrak{t}_{t}^{(n)}$ is fully given by the mutations generated at this
time -and so is of order $n^{t-\ell(0,1)}$\- and not from the lineages issued
from mutations at strictly previous time. Biologically these mutations are
called deleterious.

  * (ii)

If $\lambda(1)=\lambda(0)$: Equation (20) is independent of $u$. This means
that these lineages have the same size order at time $\mathfrak{t}_{t}^{(n)}$
than any other lineages of mutant cells generated from mutational events at
any other time between $\mathfrak{t}_{\ell(0,1)}^{(n)}$ and
$\mathfrak{t}_{t}^{(n)}$. To put it differently, in the dynamics of the mutant
population there is a balance between the contribution of mutations and its
inner growth. This is a consequence of assuming $\lambda(1)=\lambda(0)$. These
mutations are referred as neutral mutation, even if biologically speaking it
would have exactly mean the more restrictive condition $\alpha(1)=\alpha(0)$
and $\beta(1)=\beta(0)$. Hence to capture the total size of the mutant
population at time $\mathfrak{t}^{(n)}_{t}$, it remains to integrate all the
lineages issued from mutational events over time $\mathfrak{t}_{u}^{(n)},$ for
$\ell(0,1)\leq u\leq t$. It gives exactly the order
$\mathcal{O}\left((t-\ell(0,1))\log(n)n^{t-\ell(0,1)}\right)$.

To sum up, for this simple graph, the mutant population scales after time
$\mathfrak{t}_{\ell(0,1)}^{(n)}$ as

$\displaystyle\mathcal{O}\left(n^{t-\ell(0,1)}\left[\mathbbm{1}_{\\{\lambda(0)>\lambda(1)\\}}+\mathbbm{1}_{\\{\lambda(0)=\lambda(1)\\}}(t-\ell(0,1))\log(n)\right]\right).$
(21)

Notice in particular that in any case, the mutant population has an
exponential growth at rate $\lambda(0)$ after time
$\mathfrak{t}^{(n)}_{\ell(0,1)}$. An illustration of this heuristic can be
found in Figure 3.

Figure 3: Heuristics for the size of the mutant population after time
$\mathfrak{t}_{\ell(0,1)}^{(n)}$

These heuristics on this simple graph can be used as an elementary brick for
getting heuristics on a general finite graph. Considering a vertex $v\in
V\backslash\\{0\\}$, there are potentially many mutational paths from the
initial vertex $0$ to $v$. Then it needs to be understood which ones are
involved in the size order of the mutant population of trait $v$. Using both
the previous heuristics on the first time that mutations are generated and
that after this time the mutant population grows exponentially fast at rate
$\lambda(0)$, it seems natural to iteratively apply the reasoning. Thus given
one path from $0$ to $v$, the time $u$ in the time-scale
$\mathfrak{t}^{(n)}_{u}$ to wait for seeing a cell of trait $v$ generated due
to this specific path is the sum of the labels of the edges of this path,
called the weight of this path. Then, after this time, this sub-population of
cells of trait $v$ grows exponentially fast at rate $\lambda(0)$. In
particular two interesting facts for the total mutant population of trait $v$
are brought out. First it starts having cells after a time which is the
minimum of the weights over the paths from $0$ to $v$. Second, after this time
only the paths whose weights are equal to the latter minimum might contribute
to the size order of the mutant cells of trait $v$. This is due to the fact
that having a time delay create an exponential delay in the size order. But as
seen in (21), for any neutral mutation a supplementary multiplicative factor
of order $\log(n)$ is captured on the size order. Consequently, over the paths
from $0$ to $v$ satisfying that their weights are equal to the latter minimum,
only those with the maximal number of neutral mutations are actually
contributing to the size order of the mutant population of trait $v$. More
specifically with a factor of $\log(n)$ at the power this maximal number of
neutral mutations. Moreover for any of these admissible paths, at each neutral
mutation a supplementary time integral is obtained. An illustration with an
example is given in Figure 4.

Figure 4: Heuristics for the contribution of paths in the size order of a
mutant sub-population: in this example the dashed red path has a weight equal
to 5 whereas the dotted blue and the plain green ones have a weight equal to
4. Thus, only the two latter ones may contribute to the size order of the
mutant. But the dotted blue path has only one neutral mutation compared to the
plain green one which has two neutral mutations. Finally, only the plain green
path will contribute to the size order of the purple mutant sub-population.
For $t\geq 4$, at time $\mathfrak{t}^{(n)}_{t}$ it will grow as
$\log^{2}(n)e^{\lambda(0)\mathfrak{t}^{(n)}_{t-4}}$.

Notations:
Now the natural definitions issued from these heuristics are formally made
before giving the results.

######  Definition 2.1.

(Deleterious and neutral vertices)
A vertex $v\in V$ satisfying $\lambda(v)=\lambda(0)$, respectively
$\lambda(v)<\lambda(0)$, is called a neutral, respectively deleterious,
vertex.

######  Remark 2.1.

In the previous definition the neutral or deleterious denomination of a
mutation originates from the comparison of its inner growth rate to the growth
rate of the wild-type population. But one could imagine a mutation from a
vertex $v$ to a vertex $u$ satisfying $\lambda(v)<\lambda(u)\leq\lambda(0)$.
This mutation should theoretically be called selective but in the previous
definition it is actually called neutral or deleterious (depending on the
value of $\lambda(u)$ compared to $\lambda(0)$). This nomenclature emerges
from the fact that under Assumption (18) any mutant population grows
exponentially fast at rate $\lambda(0)$, as seen in the previous heuristics.
Hence it legitimates the previous definition.

######  Definition 2.2.

(Path on the graph)
$\gamma=(v(0),\cdots,v(k))$ is said to be the path on the graph $(V,E)$
linking $v(0)$ to $v(k)$ by using the edges $(v(i),v(i+1))$ if for all $0\leq
i\leq k,v(i)\in V\text{ and }\forall 0\leq i\leq k-1,(v(i),v(i+1))\in E$.
For a path $\gamma=(v(0),v(1),\cdots v(k))$ on $(V,E,L)$ define

$\displaystyle t(\gamma):=\sum_{i=0}^{k-1}\ell(v(i),v(i+1)),$ (22)
$\displaystyle\gamma_{neut}=\\{v(i),1\leq i\leq
k:\lambda(v(i))=\lambda(0)\\},$ (23)
$\displaystyle\theta(\gamma):=|\gamma_{neut}|,$ (24)

as respectively the sum of the labels of the edges of the path $\gamma$, the
subset of vertices at the end of the edges of $\gamma$ that are neutral, and
the cardinal of the previous subset. Introduce also $w_{del}$, $w_{neut}$ as

$\displaystyle w_{del}(\gamma):=\prod_{1\leq i\leq
k,\lambda(v(i))<\lambda(0)}\frac{2\alpha(v(i-1))\mu(v(i-1),v(i))}{\lambda(0)-\lambda(v(i))},$
(25) $\displaystyle w_{neut}(\gamma):=\prod_{1\leq i\leq
k,\lambda(v(i))=\lambda(0)}\frac{2\alpha(v(i-1))\mu(v(i-1),v(i))}{\lambda(0)}.$
(26)

Introduce $\forall i\leq k,t_{\gamma}(i):=\sum_{j=0}^{i-1}\ell(v(j),v(j+1))$
as the sum of the labels over the i-th first edges of $\gamma$. Finally
introduce $\sigma$ an increasing function from $\llbracket
1,\theta(\gamma)\rrbracket$ to $\llbracket 1,k\rrbracket$, such that
$v(\sigma_{i})$ is the i-th neutral vertex of the path $\gamma$. Then let

$\displaystyle
I_{\gamma}(t):=\int_{t_{\gamma}(\sigma_{\theta(\gamma)})}^{t\vee
t_{\gamma}(\sigma_{\theta(\gamma)})}\int_{t_{\gamma}(\sigma_{\theta(\gamma)-1})}^{u_{1}}\cdots\int_{t_{\gamma}(\sigma_{\theta(\gamma)-k})}^{u_{k}}\cdots\int_{t_{\gamma}(\sigma_{1})}^{u_{\theta(\gamma)-1}}du_{\theta(\gamma)}\cdots
du_{1}.$ (27)

Finally, the weight on the path $\gamma$ at time $t$ is defined as

$\displaystyle w_{\gamma}(t):=w_{del}(\gamma)w_{neut}(\gamma)I_{\gamma}(t).$
(28)

######  Remark 2.2.

Notice that if $\overline{\gamma}:=(v(0),\cdots,v(k-1))$ we have

$\displaystyle w_{\gamma}(t)=$ $\displaystyle 2\alpha(v(k))\mu(v(k-1),v(k))$
(29) $\displaystyle\hskip
56.9055pt\cdot\left(\mathbbm{1}_{\\{\lambda(k)<\lambda(0)\\}}\frac{1}{\lambda(0)-\lambda(v(k))}w_{\overline{\gamma}}(t)+\mathbbm{1}_{\\{\lambda(k)=\lambda(0)\\}}\frac{1}{\lambda(0)}\int_{t(\overline{\gamma})}^{t\vee
t(\overline{\gamma})}w_{\overline{\gamma}}(s)ds\right).$ (30)

######  Definition 2.3.

(Admissible paths)
For all $v\in V$ denote by $P(v)$ the set of all paths $\gamma$ on $(V,E)$
linking the vertex 0 to the vertex $v$. Define also

$\displaystyle t(v):=\min\\{t(\gamma),\forall\gamma\in P(v)\\},$ (31)
$\displaystyle\theta(v):=\max\left\\{\theta(\gamma),\forall\gamma\in
P(v),t(\gamma)=t(v)\right\\},$ (32) $\displaystyle A(v):=\\{\gamma\in
P(v):t(\gamma)=t(v)\text{ and }\theta(\gamma)=\theta(v)\\}.$ (33)

######  Remark 2.3.

In the previous definition $A(v)$ is called the set of admissible paths
because as seen in the heuristics, only paths belonging to $A(v)$ are
contributing to the growth dynamics of the mutant cells of trait $v$. This is
made formal in Theorem 2.1.

######  Definition 2.4.

(Weight of a vertex)
The weight of a vertex $v\in V\backslash\\{0\\}$ at time $t$ is defined as

$\displaystyle w_{v}(t)$ $\displaystyle:=\sum_{\gamma\in A(v)}w_{\gamma}(t).$
(34)

######  Remark 2.4.

$w_{v}(t)$ is called the weight of the vertex $v$ at time $t$ because, as
shown in Theorem 2.1, it is exactly the deterministic part of the first-order
asymptotics of $Z_{v}^{(n)}\left(\mathfrak{t}^{(n)}_{t}\right)$ or
$Z_{v}^{(n)}\left(\rho_{t}^{(n)}\right)$. As mentioned in the heuristics, only
the paths $\gamma$ from $0$ to $v$ in the set $A(v)$ are actually contributing
on the size order of the mutant population of trait $v$. In each of these
paths $\gamma$, each edge has a constant contribution-depending on the
parameters of the edge- that depends on whether it points to a deleterious or
a neutral vertex. It is summed up in the respective weights $w_{del}(\gamma)$
and $w_{neut}(\gamma)$. The time dependency of $w_{v}$ is defined via the time
intensity $I_{\gamma}(t)$ for every $\gamma\in A(v)$. In particular, it is a
succession of integral terms depending only on the neutral mutations of
$\gamma$.

Results:
Now the more refined result under Assumption (18) can formally be stated.

######  Theorem 2.1.

Assume that $(V,E,L)$ satisfies both the mutation regime described in (11) and
the non-increasing growth rate graph condition of (18). Let
$\gamma_{n}=\frac{\log(n)}{\log(\log(n))\theta_{\max}+\varphi_{n}}$, where
$\varphi_{n}\underset{n\to\infty}{\to}\infty$ such that
$\gamma_{n}\underset{n\to\infty}{\to}\infty$ and where
$\theta_{max}:=\max_{v\in V\backslash\\{0\\}}\theta(v)$. Let also $\psi_{n}$
such that $e^{\varphi_{n}}\log(n)=o(\psi^{2}(n))$. Define for all
$(t,s)\in\mathbb{R}^{+}\times\mathbb{R}$

$\displaystyle d_{v}^{(n)}(t,s):=$
$\displaystyle\mathbbm{1}_{\left\\{t\in[0,t(v)-\gamma_{n}^{-1})\right\\}}+\mathbbm{1}_{\left\\{t\in[t(v)-\gamma_{n}^{-1},t(v))\right\\}}\psi_{n}\log^{\theta(v)-1}(n)$
(35) $\displaystyle\hskip
113.81102pt+\mathbbm{1}_{\left\\{t\in[t(v),\infty)\right\\}}n^{t-t(v)}\log^{\theta(v)}(n)e^{\lambda(0)s}.$
(36)

Let $(T,M)\in\left(\mathbb{R}_{+}^{*}\right)^{2}$ and $0<T_{1}<T_{2}$. There
exists a random variable $W$ satisfying

$\displaystyle
W\overset{law}{:=}Ber\left(\frac{\lambda(0)}{\alpha(0)}\right)\otimes
Exp\left(\frac{\lambda(0)}{\alpha(0)}\right),$ (37)

such that for all $v\in V\backslash\\{0\\}$ we obtain the following results
for the different time-scales:

  * (i)

Deterministic time scale (16):
If $\lambda(v)=\lambda(0)$ then

$\displaystyle\left(\frac{Z_{v}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}{d^{(n)}_{v}(t,s)}\right)_{(t,s)\in[0,T]\times[-M,M]}\underset{n\to\infty}{\longrightarrow}\left(Ww_{v}(t)\right)_{(t,s)\in[0,T]\times[-M,M]}.$
(38)

If $\lambda(v)<\lambda(0)$ then

$\displaystyle\left(\frac{Z_{v}^{(n)}\left(\mathfrak{t}^{(n)}_{t(v)+t}+s\right)}{n^{t}\log^{\theta(v)}(n)e^{\lambda(0)s}}\right)_{(t,s)\in[T_{1},T_{2}]\times[-M,M]}\underset{n\to\infty}{\longrightarrow}\left(Ww_{v}(t(v)+t)\right)_{(t,s)\in[T_{1},T_{2}]\times[-M,M]}.$
(39)

  * (ii)

Random time scale (15): Take
$\left(\rho_{t}^{(n)}\right)_{t\in\mathbb{R}^{+}}$ as defined in (14).
If $\lambda(v)=\lambda(0)$ then

$\displaystyle\left(\frac{Z_{v}^{(n)}\left(\rho_{t}^{(n)}+s\right)}{d_{v}^{(n)}(t,s)}\right)_{(t,s)\in[T_{1},T_{2}]\times[-M,M]}\underset{n\to\infty}{\longrightarrow}\left(\mathbbm{1}_{\\{W>0\\}}w_{v}(t)\right)_{(t,s)\in[T_{1},T_{2}]\times[-M,M]}.$
(40)

If $\lambda(v)<\lambda(0)$ then

$\displaystyle\left(\frac{Z_{v}^{(n)}\left(\rho_{t(v)+t}^{(n)}+s\right)}{n^{t}\log^{\theta(v)}(n)e^{\lambda(0)s}}\right)_{(t,s)\in[T_{1},T_{2}]\times[-M,M]}\underset{n\to\infty}{\longrightarrow}\left(\mathbbm{1}_{\\{W>0\\}}w_{v}(t(v)+t)\right)_{(t,s)\in[T_{1},T_{2}]\times[-M,M]}.$
(41)

Using the mathematical definition of the model given in Section 4, see (359)
and (361), the above convergences are obtained in probability in the adequate
$L^{\infty}$-spaces. For any other mathematical description, the convergences
are at least in distribution in $\mathbb{D}\left([0,T]\times[-M,M]\right)$ for
Equation (38) and in $\mathbb{D}\left([T_{1},T_{2}]\times[-M,M]\right)$ for
Equations (39), (40) and (41).

The proof of Theorem 2.1 is based on a martingale approach using Doob’s and
Maximal Inequalities. The first step involves the control of the growth of the
lineage of wild-type cells issued from the initial cell both for the
deterministic and random time-scales (16) and (15) (Lemma 3.1 and 3.2). Then
for any vertex $v\in V\backslash\\{0\\}$, potentially many mutational paths on
the graph $(V,E)$ can start from $0$ and lead to $v$. The contribution on the
first-order asymptotics of the mutant sub-population of trait $v$ for any of
these paths needs to be understood. The proof is then done in 2 steps. The
first one consists in considering an infinite mono-directional graph under
Assumption (18) and in obtaining the result for this particular graph, see
Section 3. Doing the first step for an infinite graph allows in particular to
deal with the cycles (backward mutations for instance) for a general finite
graph. The second step consists in discriminating among all the paths from the
initial vertex $0$ to $v$ the ones that do not contribute to the first-order
asymptotics of the mutant sub-population of trait $v$, see Section 4.

######  Remark 2.5.

(i) Notice that a multiplicative factor of $\log^{\theta(v)}(n)$ is captured,
see Equations (35), (38), (39), (40) and (41). Getting result on the
stochastic exponents (see (19)) does not capture such a factor. For instance
with the model of Figure 1 if $\lambda(1)=\lambda(0)$ Theorem 2.1 gives that
after time $\ell(0,1)$, $Z_{1}^{(n)}\left(\mathfrak{t}^{(n)}_{t}\right)$
behaves approximately as
$\log(n)e^{\lambda(0)\mathfrak{t}^{(n)}_{t-\ell(0,1)}}$. But what is captured
with $X_{1}^{(n)}(t)$ is asymptotically $\lambda(0)(t-\ell(0,1))$ after time
$\ell(0,1)$.
(ii) The random variable $W$ is explicitly defined as the almost sure limit of
the natural positive martingale associated to a specific birth and death
branching process with rates $\alpha(0)$ and $\beta(0)$, see (371). The
martingale associated to the lineage of wild-type cells issued from the
initial cell is shown to behave as the one associated to the latter birth and
death branching process (Lemma 4.1). Thus $W$ quantifies the randomness of
this lineage over the long time. Due to the power law mutation rates regime
mutations arise after a long time such that the stochasticity of this lineage
is already given by $W$. Notice that under Assumption (18) the randomness in
the first-order asymptotics of any mutant sub-population is summed up in $W$.
Meaning that the stochasticity of these sub-populations are driven more by the
stochasticity in the growth of the wild-type sub-population than by both the
randomness in the mutational process and the randomness of any lineages of
mutant cells.
(iii) It seems more than natural not to obtain such a result when considering
selective mutation ($\lambda(v)>\lambda(0)$). Indeed, a selective mutation
would mean that any time advantage is an advantage into growth. Thus the
stochasticity of the mutational process can not be ignored as well as the one
of the lineages of mutant cells. Hence hoping to control the stochasticity of
the mutant population controlling only the randomness of the wild-type
population and not the randomness of the mutational process as well as the one
of the lineages of the mutant cells is vain. Meaning that using a martingale
approach to get the first-order asymptotics can not be successful for a
selective mutation. Nevertheless looking at the stochastic exponent (19) the
martingale approach allows to get convergence results given in Theorem 2.2.
(iv) In view of Theorem 2.1, the mathematical definition of neutral mutation
$\lambda(v)=\lambda(0)$ is well understood instead of the more restrictive but
biologically more meaningful condition $\alpha(v)=\alpha(0)$ and
$\beta(v)=\beta(0)$. Indeed, taking the growth rate $\lambda(v)$ equal to
$\lambda(0)$ when changing birth and death rates $\alpha(v)$ and $\beta(v)$
modify the distribution of any lineage of mutant cells. Consequently one could
naturally believe that it should impact the stochasticity of the size order of
the mutant population. This is not the case, the randomness on the first
asymptotic order is fully summed up by $W$. Hence it is fully consistent with
getting for the neutral assumption only a condition on the growth rate
function instead of on the birth and death rates.
(v) Considering the time-scale $\mathfrak{t}^{(n)}_{t}$ notice that the result
slightly differs depending on whether the vertex is neutral or deleterious.
Indeed, when looking at the asymptotic behavior for a deleterious vertex $v$
our result is true strictly after time $t(v)$, whereas in the case of a
neutral vertex all the trajectory from the initial time can be dealt with.
Mathematically, this difference originates from the supplementary
multiplicative $\log(n)$ factor in the first asymptotic order when considering
a neutral mutation. It allows to control the quadratic variation at time
$t(v)$ for the martingale associated to the mutant population. Then exactly
three different regimes are obtained, see (35) and (38) :

  * •

Up to $t(v)-\gamma_{n}^{-1}$: with high probability no mutational paths from 0
to $v$ have generated a mutant cell of trait $v$. Since $\gamma_{n}\to\infty$
and satisfies $\gamma_{n}=o(\log(n))$, $t(v)$ can be interpreted as the first
time -when considering the time-scale accelerated in $\log(n)$\- it becomes
asymptotically possible to see the first appearance of a mutant cell of trait
$v$. This result is actually also true for deleterious mutation, see Lemma
3.4.

  * •

For $t\in\left[t(v)-\gamma_{n}^{-1},t(v)\right)$: in this time interval, some
mutants cells of trait $v$ are produced, but the growth of the mutant sub-
population of trait $v$ does not start to be an exponential growth. We succeed
to dominate this growth by $\psi_{n}\log^{\theta(v)-1}(n)$, with a well chosen
function $\psi_{n}$. Heuristically what happens is that the total number of
mutant cells of trait $v$ issued from a mutational event up to time $t$ is of
order $\mathcal{O}\left(\log^{\theta(v)-1}(n)\right)$. Moreover with the
remaining time for the lineages of these mutant cells to grow, we succeed to
control the size of the mutant sub-population of trait $v$ by at most
$e^{\frac{\varphi_{n}}{2}}\sqrt{\log(n)}\log^{\theta(v)-1}(n)$. Consequently
dividing by any function $\psi_{n}$ satisfying
$e^{\varphi_{n}}\log(n)=o(\psi_{n}^{2})$ the asymptotic limits is $0$.

  * •

For $t\in[t(v),\infty)$: with high probability the number of mutant cells of
trait $v$ grows exponentially fast at rate $\lambda(0)$. A supplementary
multiplicative $log^{\theta(v)}(n)$ factor is present due to the neutral
mutations on the paths of $A(v)$. Then it globally scales as
$n^{(t-t(v))}\log^{\theta(v)}(n)w_{v}(t)$.

(vi) When comparing point (i) and (ii) of Theorem 2.1 notice that the result
is transferred from the deterministic time-scale $\mathfrak{t}^{(n)}_{t}$ into
the random time-scale $\rho_{t}^{(n)}$ by switching only $W$ to
$\mathbbm{1}_{\\{W>0\\}}$. This a priory surprising fact can be explained by
the essential role of $W$. As mentioned in Remark 2.5 (ii), $W$ encodes the
stochasticity on the long time for the lineage of wild-type cells issued from
the initial cell. By showing that the time-scale $\mathfrak{t}^{(n)}_{t}$ is
the right deterministic approximation of $\rho_{t}^{(n)}$ (Lemma 4.2), one
shows that having an asymptotic result on time-scale $\mathfrak{t}^{(n)}_{t}$
allows to get it for the time scale $\rho_{t}^{(n)}$. This idea is made formal
using a similar technique as in [15] Lemma 3. Then the switch from $W$ to
$\mathbbm{1}_{\\{W>0\\}}$ in the result is due to the fact that the time-scale
$\rho_{t}^{(n)}$ already bears by definition the stochasticity of the random
variable $W$. Consequently the only randomness that needs to be kept is the
survival of the lineage issued from the initial cell, which is asymptotically
given by $\mathbbm{1}_{\\{W>0\\}}$.

###  2.2 Result for a general finite oriented labeled graph

This subsection is free from the non-increasing growth rate condition of
Equation (18). Without this condition, the martingale approach fails in order
to get the first-order asymptotics off all the mutant sub-populations. But,
the stochastic exponent, as defined in (19), off all the mutant sub-
populations can be uniformly tracked over time. In particular, we show that
the limits are positive deterministic non-decreasing piecewise linear
continuous functions. Such limits are defined via a recursive algorithm
tracking their slopes over time. More precisely, we show that the slopes can
only increase and take values on the set of the growth rates. Two different
kinds of updates can be made. The first one is when a non-already born trait
becomes alive and take the slope which is the maximum between its inner growth
rate and the slope of the sub-population that is giving birth to it. The
second one is when an already born trait changes it slope to increase it
because another born trait among its upcoming neighbors with a higher slope
has reached the typical size allowing it to now drive the trait in question,
and consequently giving it its slope. This heuristic is made formal in the
following theorem. The complexity of such an algorithm comes from the trait
structure which is a general finite trait space. On a mono-directional one,
this algorithm would be much easier. In particular, at the same time, the two
kinds of event can happen.

######  Theorem 2.2.

For all $v\in V$ define

$\displaystyle
X_{v}^{(n)}(t):=\frac{\log_{+}\left(Z_{v}^{(n)}\left(\mathfrak{t}^{(n)}_{t}\right)\right)}{\log(n)/\lambda(0)}.$
(42)

Then we have for all $0<T_{1}<T_{2}$,

$\displaystyle\left(\left(X^{(n)}_{v}(t)\right)_{v\in
V}\right)_{t\in[T_{1},T_{2}]}\underset{n\to\infty}{\longrightarrow}\mathbbm{1}_{\\{W>0\\}}\left(\left(x_{v}(t)\right)_{v\in
V}\right)_{t\in[T_{1},T_{2}]},$ (43)

in probability in $L^{\infty}[T_{1},T_{2}]$. Where for all $v\in V,x_{v}$ is a
positive deterministic non-decreasing piecewise linear continuous function
that is obtained via a recursive approach tracking its slope over time. In
particular it exists $k^{*}\in\mathbb{N}$ and
$0=\Delta_{0}<\Delta_{1}<\cdots<\Delta_{k^{*}}<\infty$ such that the slopes of
$(x_{v})_{v\in V}$ only change at these times. For $j\in\\{0,\cdots,k^{*}\\}$
at time $\Delta_{j}$ two kind of changes in the slope can happen: either a new
trait starts to grow or an already growing trait increases its slope due to a
growth driven by another more selective trait. Along the algorithm the
following different quantities are going to be tracked for all
$j\in\\{0,\cdots,k^{*}\\}$ at time $\Delta_{j}$:

  * •

the set of alive traits, $A_{j}$,

  * •

the set of not already born traits, $U_{j}$,

  * •

the slope of $x_{v}$, $\lambda_{j}(v)$,

  * •

and the set of traits whose growth are driven by trait $v$, $C_{j}(v)$.

Initialisation: Set $A_{0}=\\{0\\}$, $U_{0}=V\ \backslash\\{0\\}$ and for all
$v\in V$

$\displaystyle
x_{v}(0)=0,\lambda_{0}(v)=\lambda(0)\mathbbm{1}_{\\{v=0\\}},\text{ and
}C_{0}(v)=\emptyset.$ (44)

Induction: Let $j\in\\{0,\cdots,k^{*}-1\\}$. Assume that it exists times
$0=\Delta_{0}<\Delta_{1}<\cdots<\Delta_{j}<\infty$ such that
$\left(x_{v}\right)_{v\in V}$ are positive deterministic non-decreasing
piecewise linear continuous functions defined on $[0,\Delta_{j}]$, where their
change of slopes happened only on the discrete set
$\\{\Delta_{1},\cdots,\Delta_{j}\\}$. Also assume that it exists
$\lambda_{j}(v)$, $A_{j}$, $U_{j}$, and $C_{j}(v)$, respectively the slope of
$x_{v}$, the set of alive vertices and not already born vertices, and the set
of vertices whose growth are driven by $v$, everything at time $\Delta_{j}$.
Then it exists $\Delta_{j+1}\in(\Delta_{j},\infty)$ such that
$\left(x_{v}\right)_{v\in V}$ are constructed during the time period
$[\Delta_{j},\Delta_{j+1}]$ according to the following schedule. For all $v\in
V$ and for all $t\geq\Delta_{j}$ let the following function

$\displaystyle y_{v}(t)=(t-\Delta_{j})\lambda_{j}(v)+x_{v}(\Delta_{j}).$ (45)

For all $v\in U_{j}$ define

$\displaystyle\forall u\in A_{j},\text{ such that }(u,v)\in
E,\delta_{u,v}:=\inf\\{t\geq\Delta_{j}:y_{u}(t)\geq\lambda(0)\ell(u,v)\\},$
(46) $\displaystyle\delta_{v}:=\inf_{u\in A_{j}:(u,v)\in E}\delta_{u,v},$ (47)
$\displaystyle\nu(v):=\\{u\in A_{j}:(u,v)\in E\text{ and
}\delta_{u,v}=\delta_{v}\\},$ (48)

For all $v\in A_{j}$ define

$\displaystyle B_{j}(v):=\\{u\in A_{j}:(v,u)\in E\text{ and
}\lambda_{j}(v)>\lambda_{j}(u)\\},$ (50) $\displaystyle\forall u\in
B_{j}(v),\delta_{v,u}:=\inf\\{t\geq\Delta_{j}:y_{v}(t)\geq
y_{u}(t)+\lambda(0)\ell(v,u)\\},$ (51) $\displaystyle\delta_{v}:=\inf_{u\in
B_{j}(v)}\delta_{v,u},$ (52) $\displaystyle\nu(v):=\\{u\in
B_{j}(v):\delta_{v,u}=\delta_{v}\\}.$ (53)

Then define $\Delta_{j+1}:=\inf_{v\in V}\delta_{v}$ and $\nu_{j+1}:=\\{v\in
V:\delta_{v}=\Delta_{j+1}\\}$. Then proceed to the following updates:

  * (i)

Let $A_{j+1}:=A_{j}\cup\left(\nu_{j+1}\cap U_{j}\right)\text{ and
}U_{j+1}=U_{j}\backslash\left(\nu_{j+1}\cap U_{j}\right).$ Also let $\forall
v\in U_{j+1},$ $\lambda_{j+1}(v)=\lambda_{j}(v)=0$,
$C_{j+1}(v)=C_{j}(v)=\emptyset$.

  * (ii)

For all $v\in\nu_{j+1}\cap A_{j}$, introduce the set
$\nu^{(-)}(v):=\\{u\in\nu(v):\exists w\in\nu_{j+1}\cap
A_{j},\lambda_{j}(w)>\lambda_{j}(v),\text{ and }u\in\nu(w)\\}$. Then let
$C_{j+1}(v):=C_{j}(v)\bigcup_{u\in\nu(v)\backslash\nu^{(-)}(v)}\left(\\{u\\}\cup
C_{j}(u)\right).$ For all $u\in\nu(v)\backslash\nu^{(-)}(v)$ and $w\in
C_{j}(u)$, $\lambda_{j+1}(u)=\lambda_{j+1}(w)=\lambda_{j}(v)$.

  * (iii)

For all $v\in A_{j}$ whose slope has not been updated yet, let
$\lambda_{j+1}(v)=\lambda_{j}(v)$. And for all $v\in A_{j}$ whose set
$C_{j}(v)$ has not been updated yet, let $C_{j+1}(v):=C_{j}(v)$.

  * (iv)

For all $v\in\nu_{j+1}\cap U_{j}$, let
$\lambda_{j+1}(v):=\max\left(\lambda(v),\max_{u\in\nu(v)}\lambda_{j+1}(u)\right)$,
and $C_{j+1}(v)=C_{j}(v)=\emptyset$. If $\lambda_{j+1}(v)\geq\lambda(v)$,
introduce the set
$\nu^{+}(v):=\\{u\in\nu(v):\lambda_{j+1}(u)=\max_{w\in\nu(v)}\lambda_{j+1}(w)\\}$,
and for all $u\in\nu^{+}(v)$, $C_{j+1}(u):=C_{j+1}(u)\cup\\{v\\}$.

For any other mathematical description as the one given in Section 4, see
(359) and (361), the convergences are at least in distribution in
$\mathbb{D}\left([T_{1},T_{2}]\right)$ .

The proof of this theorem is given in Section 5. It is heavily based on the
proofs of [13], where we exploit the stochastic construction of such a model,
given in the beginning of Section 4, to adapt the proofs of the previous
article to the situation of the present work. For that reason, we introduce
lemmas and explain in the proofs how the adaptations from the proofs of [13]
are made, without reproving them. This theorem is the counterpart of the study
made in [8] in the case of branching sub-populations, instead of having
competition between sub-populations. One difference is that the power law
mutation rates regime is a bit more general in the present work, allowing each
mutation probabilities to scale differently. But, the result in [8] can be
adapted with this more general regime, as mentioned by the authors.
Nevertheless, Theorem 2.2 is a less refined result compared to Theorem 2.1. We
are going to explicit, using the example of Figure 4, the contribution of
Theorem 2.1 compared to Theorem 2.2, under Assumption (18). For this example,
the asymptotic function $x$ obtained due to Theorem 2.2 for the purple trait
is $x(t)=\mathbbm{1}_{\\{t\geq 4\\}}\lambda(0)(t-4).$ In the caption of Figure
4 it is already made explicit that only the plain green path will contribute
to the size order of the purple mutant sub-population. If one denotes by 1, 2
and 3 respectively the vertices on the plain green path such that it is
exactly $(0,1,2,3)$, where 3 is the purple vertex, it follows that Theorem 2.1
gives that the asymptotic limits for the purple vertex is for all $t\geq 4,$
$\frac{2\alpha(0)\mu(0,1)}{\lambda(0)}\cdot\frac{2\alpha(1)\mu(1,2)}{\lambda(0)}\cdot\frac{2\alpha(2)\mu(2,3)}{\lambda(0)-\lambda(3)}W\int_{3}^{t}\left(\int_{1}^{u}ds\right)du\cdot
n^{t-4}\log^{2}(n)=\left(\frac{t^{2}}{2}-t-\frac{3}{2}\right)\frac{16\alpha(0)\alpha(1)\alpha(2)}{\lambda^{2}(0)\left(\lambda(0)-\lambda(3)\right)}Wn^{t-4}\log^{2}(n)$,
because $\mu(0,1)=1,\mu(1,2)=2$ and $\mu(2,3)=1$. In particular Theorem 2.2
captures only the power of $n$ which is $t-4$ and with Theorem 2.1 we capture
the stochasticity $W$, a supplementary scaling factor $\log^{2}(n)$, a time
polynomial $\frac{t^{2}}{2}-t-\frac{3}{2}$ and also a constant depending only
on the parameters of the visited vertices
$\frac{16\alpha(0)\alpha(1)\alpha(2)}{\lambda^{2}(0)\left(\lambda(0)-\lambda(3)\right)}$.
Capturing this level of complexity under the large population power law
mutation rates limit is, up to our knowledge, the first time that it is done.
It opens potential works of identification of the graph structure using data,
as well as designing statistical tools.

##  3 First-order asymptotics of the mutant sub-populations for an infinite
mono-directional graph

In this section consider the model described in Section 1 in the particular
following infinite mono-directional graph

$\displaystyle(V,E)=\left(\mathbb{N}_{0},\\{(i,i+1),i\in\mathbb{N}_{0}\\}\right).$
(54)

Considering this special case will allow to deal with cycles (in particular
cycles generated due to backward mutations) in the general finite graph case.
Assume the non-increasing growth rate condition (18). For simplicity of
notation consider for all $i\in\mathbb{N}_{0}$ the new notations
$\mu_{i}^{(n)}:=\mu^{(n)}(i,i+1)$ and $\ell(i):=\ell(i,i+1)$. It means that
the following mutation regime is considered

$\displaystyle\forall
i\in\mathbb{N}_{0},n^{\ell(i)}\mu_{i}^{(n)}\underset{n\to\infty}{\longrightarrow}\mu_{i}.$
(55)

Assume in the case of the infinite mono-directional graph that

$\displaystyle\ell^{*}:=\inf\\{\ell(i):i\in\mathbb{N}_{0}\\}>0.$ (56)

Again for $i\in\mathbb{N}_{0}$ denote by $\alpha_{i}$, $\beta_{i}$ and
$\lambda_{i}$ the division, death and growth rates associated to trait $i$
instead of $\alpha(i),\beta(i)$ and $\lambda(i)$.
In this particular case three different scenari can happen during a division
event of a cell of trait $i\in\mathbb{N}_{0}$:

  * •

with probability $\left(1-\mu_{i}^{(n)}\right)^{2}$ each daughter cell keep
the trait $i$ of its mother cell,

  * •

with probability $2\mu_{i}^{(n)}\left(1-\mu_{i}^{(n)}\right)$ exactly one of
the daughter cell mutates to the next trait $i+1$ when the second daughter
cell keeps the trait $i$ of its mother cell,

  * •

with probability $\left(\mu_{i}^{(n)}\right)^{2}$ both of the daughter cells
mutate to the next trait $i+1$.

A graphical representation of the model can be found in Figure 5.

Figure 5: Dynamical representation of the infinite mono-directional graph

In particular it means that any lineage of a cell of trait $i$ follows a
birth-death branching process with rates
$\alpha_{i}\left(1-\mu_{i}^{(n)}\right)^{2}$ and
$\beta_{i}+\alpha_{i}^{(n)}\left(\mu_{i}^{(n)}\right)^{2}$ respectively. Thus
introduce the birth, death and growth rate of any lineage of a cell of trait
$i$ as

$\displaystyle\alpha_{i}^{(n)}:=\alpha_{i}\left(1-\mu_{i}^{(n)}\right)^{2},$
(57)
$\displaystyle\beta_{i}^{(n)}:=\beta_{i}+\alpha_{i}^{(n)}\left(\mu_{i}^{(n)}\right)^{2},$
(58)
$\displaystyle\lambda_{i}^{(n)}:=\alpha_{i}^{(n)}-\beta_{i}^{(n)}=\lambda_{i}-2\alpha_{i}\mu_{i}^{(n)}.$
(59)

Compared to the general finite graph, for any trait $i\in\mathbb{N}$ there is
only one path from trait $0$ to $i$ for this mono-directional graph. In
particular it implies that

$\displaystyle t(i)=\sum_{i=0}^{i-1}\ell(i),$ (60)
$\displaystyle\theta(i)=|\\{j\in\llbracket
1,i\rrbracket:\lambda_{j}=\lambda_{0}\\}|.$ (61)

The sequence
$\left(\left(Z^{(n)}_{i}\right)_{i\in\mathbb{N}_{0}}\right)_{n\in\mathbb{N}}$
is mathematically constructed using independent Poisson Point Measures (PPMs).
Let $Q_{0}^{b}(ds,d\theta)$, $Q_{0}^{d}(ds,d\theta)$,
$\left(Q_{i}(ds,d\theta)\right)_{i\in\mathbb{N}}$,
$\left(N_{i}(ds,d\theta)\right)_{i\in\mathbb{N}_{0}}$, and
$\left(Q_{i}^{m}(ds,d\theta)\right)_{i\in\mathbb{N}_{0}}$ be independent PPMs
with intensity $dsd\theta$. The sub-population of wild-type cells is

$\displaystyle Z_{0}^{(n)}(t):=$ $\displaystyle
1+\int_{0}^{t}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq\alpha_{0}^{(n)}Z_{0}^{(n)}(s^{-})\right\\}}Q_{0}^{b}(ds,d\theta)-\int_{0}^{t}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq\beta_{0}Z_{0}^{(n)}(s^{-})\right\\}}Q_{0}^{d}(ds,d\theta)-H_{0}^{(n)}(t),$
(62)

and for all $i\in\mathbb{N}$

$\displaystyle Z_{i}^{(n)}(t)$
$\displaystyle:=\int_{0}^{t}\int_{\mathbb{R}^{+}}\Bigg{(}\mathbbm{1}_{\left\\{\theta\leq\alpha_{i}^{(n)}Z_{i}^{(n)}(s^{-})\right\\}}-\mathbbm{1}_{\left\\{\alpha_{i}^{(n)}Z_{i}^{(n)}(s^{-})\leq\theta\leq\left(\alpha_{i}^{(n)}+\beta_{i}\right)Z_{i}^{(n)}(s^{-})\right\\}}\Bigg{)}Q_{i}(ds,d\theta)$
(63) $\displaystyle+K_{i-1}^{(n)}(t)+2H_{i-1}^{(n)}(t)-H_{i}^{(n)}(t),$ (64)

where for all $i\in\mathbb{N}_{0}$

$\displaystyle
K_{i}^{(n)}(t):=\int_{0}^{t}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq
2\alpha_{i}\mu_{i}^{(n)}\left(1-\mu_{i}^{(n)}\right)Z_{i}^{(n)}(s^{-})\right\\}}N_{i}(ds,d\theta),$
(66) $\displaystyle
H_{i}^{(n)}(t):=\int_{0}^{t}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq\alpha_{i}\left(\mu_{i}^{(n)}\right)^{2}Z_{i}^{(n)}(s^{-})\right\\}}Q_{i}^{m}(ds,d\theta).$
(67)

The processes $\left(K_{i}^{(n)}(t)\right)_{t\in\mathbb{R}^{+}}$ and
$\left(H_{i}^{(n)}(t)\right)_{t\in\mathbb{R}^{+}}$ count the number of
mutations up to time $t$ from the sub-population of trait $i$ leading to
exactly one, respectively two mutated daughter cells of trait $i+1$.
Let $(Z_{0}(t))_{t\in\mathbb{R}^{+}}$ be the birth-death branching process
with rates $\alpha_{0}$ and $\beta_{0}$ respectively, constructed in the
following way

$\displaystyle
Z_{0}(t)=1+\int_{0}^{t}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq\alpha_{0}Z_{0}(s^{-})\right\\}}Q_{0}^{b}(ds,d\theta)-\int_{0}^{t}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq\beta_{0}Z_{0}(s^{-})\right\\}}Q_{0}^{d}(ds,d\theta).$
(68)

Notice in particular that with such a construction it immediately follows the
monotone coupling

$\displaystyle\forall t\geq 0,Z_{0}^{(n)}(t)\leq Z_{0}(t)\ a.s.$ (69)

Denote by

$\displaystyle W:=\lim_{t\to\infty}e^{-\lambda_{0}t}Z_{0}(t),$ (70)

the almost sure limit of the positive martingale
$\left(e^{-\lambda_{0}t}Z_{0}(t)\right)_{t\in\mathbb{R}^{+}}$, whose law is

$\displaystyle
W\overset{law}{=}Ber\left(\frac{\lambda_{0}}{\alpha_{0}}\right)\otimes
Exp\left(\frac{\lambda_{0}}{\alpha_{0}}\right),$ (71)

see [12] Section 1.1, or [11] Theorem 1.

###  3.1 The wild-type dynamics

Using the same PPMs $Q_{0}^{b}$ and $Q_{0}^{d}$ in the construction of
$\left(Z_{0}^{(n)}\right)_{n\in\mathbb{N}}$ and $Z_{0}$, see Equations (62)
and (68), allows to control the size dynamics over time of the sequence by
comparing it with the size of $Z_{0}$. More precisely, we show that the
natural martingale associated to $Z_{0}^{(n)}$ can be compared to the natural
one of $Z_{0}$. It comes from the fact that
$\left(\alpha_{0}^{(n)},\beta^{(n)}_{0}\right)\underset{n\to\infty}{\to}(\alpha_{0},\beta_{0}).$
The control is obtained along the whole trajectory and in probability. The
rate of convergence is quantifies to be at most of order
$\mathcal{O}\left(\mu^{(n)}_{0}\right)$.

######  Lemma 3.1.

It exits $C(\alpha_{0},\lambda_{0})>0$ and $N\in\mathbb{N}$ such that for all
$\varepsilon>0$ and $n\geq N$

$\displaystyle\mathbb{P}\left(\sup_{t\in\mathbb{R}^{+}}\Big{|}e^{-\lambda_{0}t}Z_{0}(t)-e^{-\lambda_{0}^{(n)}t}Z_{0}^{(n)}(t)\Big{|}\geq\varepsilon\right)\leq\frac{C(\alpha_{0},\lambda_{0})}{\varepsilon^{2}}\mu_{0}^{(n)}\underset{n\to\infty}{\longrightarrow}0.$
(72)

###### Proof of Lemma 3.1.

Notice that
$\left(e^{-\lambda_{0}t}Z_{0}(t)-e^{-\lambda_{0}^{(n)}t}Z_{0}^{(n)}(t)\right)_{t\in\mathbb{R}^{+}}$
is a martingale as the difference between the martingales
$\left(e^{-\lambda_{0}t}Z_{0}(t)\right)_{t\in\mathbb{R}^{+}}$ and
$\left(e^{-\lambda_{0}^{(n)}t}Z_{0}^{(n)}(t)\right)_{t\in\mathbb{R}^{+}}$. Let
$(f(m))_{m\in\mathbb{N}}$ be a non decreasing sequence satisfying
$f(m)\underset{m\to\infty}{\to}\infty$. Using Doob’s Inequality in $L^{2}$
(see [24] Proposition 3.15) we get

$\displaystyle\mathbb{P}\left(\sup_{t\in[0,f(m)]}\Big{|}e^{-\lambda_{0}t}Z_{0}(t)-e^{-\lambda_{0}^{(n)}t}Z_{0}^{(n)}(t)\Big{|}\geq\varepsilon\right)$
(73)
$\displaystyle\leq\frac{4}{\varepsilon^{2}}\mathbb{E}\left[\left(e^{-\lambda_{0}f(m)}Z_{0}(f(m))-e^{-\lambda_{0}^{(n)}f(m)}Z_{0}^{(n)}(f(m))\right)^{2}\right]$
(74)
$\displaystyle=\frac{4}{\varepsilon^{2}}\mathbb{E}\left[e^{-2\lambda_{0}f(m)}Z_{0}(f(m))^{2}+e^{-2\lambda_{0}^{(n)}f(m)}Z_{0}^{(n)}(f(m))^{2}-2e^{-(\lambda_{0}+\lambda_{0}^{(n)})f(m)}Z_{0}(f(m))Z_{0}^{(n)}(f(m))\right].$
(75)

Using Ito’s formula and (69) it follows

$\displaystyle\mathbb{E}\left[Z_{0}(t)Z_{0}^{(n)}(t)\right]=1+\int_{0}^{t}\left(\lambda_{0}+\lambda_{0}^{(n)}\right)\mathbb{E}\left[Z_{0}(s)Z_{0}^{(n)}(s)\right]ds+\int_{0}^{t}\left(\alpha_{0}^{(n)}+\beta_{0}\right)\mathbb{E}\left[Z_{0}^{(n)}(s)\right]ds.$
(76)

Solving this equation we obtain for all $t\geq 0$

$\displaystyle\mathbb{E}\left[Z_{0}(t)Z_{0}^{(n)}(t)\right]=\frac{\alpha_{0}+\alpha_{0}^{(n)}}{\lambda_{0}}e^{\left(\lambda_{0}+\lambda_{0}^{(n)}\right)t}-\frac{\alpha_{0}^{(n)}+\beta_{0}}{\lambda_{0}}e^{\lambda_{0}^{(n)}t}.$
(77)

Similarly we have

$\displaystyle\mathbb{E}\left[\left(Z_{0}(t)\right)^{2}\right]=\frac{2\alpha_{0}}{\lambda_{0}}e^{2\lambda_{0}t}-\frac{\alpha_{0}+\beta_{0}}{\lambda_{0}}e^{\lambda_{0}t}\leq\frac{2\alpha_{0}}{\lambda_{0}}e^{2\lambda_{0}t},$
(78)
$\displaystyle\mathbb{E}\left[\left(Z_{0}^{(n)}(t)\right)^{2}\right]=\frac{2\alpha_{0}^{(n)}}{\lambda_{0}^{(n)}}e^{2\lambda_{0}^{(n)}t}-\frac{\alpha_{0}^{(n)}+\beta_{0}^{(n)}}{\lambda_{0}^{(n)}}e^{\lambda_{0}^{(n)}t}\leq\dfrac{2\alpha_{0}^{(n)}}{\lambda_{0}^{(n)}}e^{2\lambda_{0}^{(n)}t}.$
(79)

Consequently combining (73), (77) and (78) gives that

$\displaystyle\mathbb{P}\left(\sup_{t\in[0,f(m)]}\Big{|}e^{-\lambda_{0}t}Z_{0}(t)-e^{-\lambda_{0}^{(n)}t}Z_{0}^{(n)}(t)\Big{|}\geq\varepsilon\right)$
(80) $\displaystyle\hskip
85.35826pt\leq\frac{4}{\varepsilon^{2}}\left(\frac{2\alpha_{0}}{\lambda_{0}}+\frac{2\alpha_{0}^{(n)}}{\lambda_{0}^{(n)}}-2\frac{\alpha_{0}+\alpha_{0}^{(n)}}{\lambda_{0}}+2\frac{\alpha_{0}^{(n)}+\beta_{0}}{\lambda_{0}}e^{-\lambda_{0}f(m)}\right).$
(81)

The event
$\left\\{\sup_{t\in[0,f(m)]}\Big{|}e^{-\lambda_{0}t}Z_{0}(t)-e^{-\lambda_{0}^{(n)}t}Z_{0}^{(n)}(t)\Big{|}\geq\varepsilon\right\\}$
is increasing in the parameter $m$. Then taking the limit $m\to\infty$ and by
monotonicity of the measure it follows

$\displaystyle\mathbb{P}\left(\sup_{t\in\mathbb{R}^{+}}\Big{|}e^{-\lambda_{0}t}Z_{0}(t)-e^{-\lambda_{0}^{(n)}t}Z_{0}^{(n)}(t)\Big{|}\geq\varepsilon\right)\leq\frac{4}{\varepsilon^{2}}\left(\frac{2\alpha_{0}}{\lambda_{0}}+\frac{2\alpha_{0}^{(n)}}{\lambda_{0}^{(n)}}-2\frac{\alpha_{0}+\alpha_{0}^{(n)}}{\lambda_{0}}\right).$
(82)

Recalling that $\lambda_{0}^{(n)}=\lambda_{0}-2\alpha_{0}\mu_{0}^{(n)}$ it
easily follows that

$\displaystyle\frac{2\alpha_{0}^{(n)}}{\lambda_{0}^{(n)}}=\frac{2\alpha_{0}}{\lambda_{0}}+\frac{4\beta_{0}\alpha_{0}}{\lambda_{0}^{2}}\mu_{0}^{(n)}+\mathcal{O}\left(\left(\mu_{0}^{(n)}\right)^{2}\right),$
(83) $\displaystyle
2\frac{\alpha_{0}+\alpha_{0}^{(n)}}{\lambda_{0}}=\frac{4\alpha_{0}}{\lambda_{0}}-\frac{4\alpha_{0}}{\lambda_{0}}\mu_{0}^{(n)}+\mathcal{O}\left(\left(\mu_{0}^{(n)}\right)^{2}\right).$
(84)

Finally we have

$\displaystyle\mathbb{P}\left(\sup_{t\in\mathbb{R}^{+}}\Big{|}e^{-\lambda_{0}t}Z_{0}(t)-e^{-\lambda_{0}^{(n)}t}Z_{0}^{(n)}(t)\Big{|}\geq\varepsilon\right)$
$\displaystyle\leq\frac{4}{\varepsilon^{2}}\left(\frac{4\beta_{0}\alpha_{0}}{\lambda_{0}^{2}}+\frac{4\alpha_{0}}{\lambda_{0}}\right)\mu_{0}^{(n)}+\mathcal{O}\left(\left(\mu_{0}^{(n)}\right)^{2}\right)$
(85)
$\displaystyle=\frac{16\alpha_{0}^{2}}{\varepsilon^{2}\lambda_{0}^{2}}\mu_{0}^{(n)}+\mathcal{O}\left(\left(\mu_{0}^{(n)}\right)^{2}\right),$
(86)

which concludes the proof. ∎

The next Lemma gives an asymptotic comparison between the random stopping
times $\eta_{t}^{(n)}$ at which the wild-type population reaches the size
$n^{t}$, and the deterministic times $\mathfrak{t}_{t}^{(n)}$. This asymptotic
is given in probability and is conditioned on $\\{W>0\\}$. It explains why
these deterministic times are the natural deterministic candidates for
studying the asymptotic behavior of the mutant sub-populations at the random
stopping times. In particular it shows that the stochastic correction between
the random time-scale and the deterministic one is asymptotically
$\frac{\log(W)}{\lambda_{0}}$. The result is obtained uniformly in time on
intervals whose lengths tend to infinity not too quickly.

######  Lemma 3.2.

For all $\varepsilon>0$, $(T_{1},T_{2})\in\mathbb{R}^{+}$ and $\varphi_{n}$
such that $\log(n)=o(\varphi_{n})$ and
$\varphi_{n}=o\left(n^{\ell(0)}\right)$, we have

$\mathbb{P}\left(\sup_{t\in\left[T_{1},T_{2}\frac{\varphi_{n}}{\log(n)}\right]}\left|\eta_{t}^{(n)}-\left(\mathfrak{t}^{(n)}_{t}-\frac{\log(W)}{\lambda_{0}}\right)\right|\geq\varepsilon\Bigg{|}W>0\right)\underset{n\to\infty}{\longrightarrow}0.$
(87)

###### Proof of Lemma 3.2.

Let $\varepsilon>0$ and for all $n\in\mathbb{N}$ introduce the event

$\displaystyle
A^{(n)}:=\Bigg{\\{}\sup_{t\in\left[T_{1},T_{2}\frac{\varphi_{n}}{\log(n)}\right]}\left|\eta_{t}^{(n)}-\left(\mathfrak{t}^{(n)}_{t}-\frac{\log(W)}{\lambda_{0}}\right)\right|\geq\varepsilon\Bigg{\\}}.$
(88)

Step 1: We start by showing that for all $0<\delta_{1}<\delta_{2}$

$\displaystyle\mathbb{P}\left(A^{(n)}\cap\\{\delta_{1}<W<\delta_{2}\\}\right)\underset{n\to\infty}{\longrightarrow}0.$
(89)

Let $\nu>0$ and $\tilde{\varepsilon}<\frac{\delta_{1}}{2}$. Firstly, since
$e^{-\lambda_{0}t}Z_{0}(t)\underset{t\to\infty}{\to}W$ almost surely,
$Y(t):=\sup_{s\in[t,\infty]}\big{|}e^{-\lambda_{0}s}Z_{0}(s)-W\big{|}\underset{t\to\infty}{\longrightarrow}0$
almost surely and as a consequence in probability. Thus introducing for all
$t>0$ the event $B_{t}:=\\{Y(t)\leq\tilde{\varepsilon}\\}$, it exists
$t_{1}>0$ such that for all $t\geq t_{1}$, $\mathbb{P}\left(B_{t}\right)\geq
1-\frac{\nu}{3}.$
Secondly using Lemma 3.1, we have that it exists $n_{1}\in\mathbb{N}$ such
that for all $n\geq n_{1}$

$\mathbb{P}\left(C^{(n)}\right)\geq 1-\frac{\nu}{3}\text{ with
}C^{(n)}:=\left\\{\sup_{t\in\mathbb{R}^{+}}\Big{|}e^{-\lambda_{0}t}Z_{0}(t)-e^{-\lambda_{0}^{(n)}t}Z_{0}^{(n)}(t)\Big{|}\leq\tilde{\varepsilon}\right\\}.$

Combining these two facts, we obtain the following inequality for all $n\geq
n_{1}$

$\mathbb{P}\left(A^{(n)}\cap\\{\delta_{1}<W<\delta_{2}\\}\right)\leq\mathbb{P}\left(A^{(n)}\cap\\{\delta_{1}<W<\delta_{2}\\}\cap
B_{t_{1}}\cap C^{(n)}\right)+\frac{2\nu}{3}.$ (90)

It remains to show that
$\mathbb{P}\left(A^{(n)}\cap\\{\delta_{1}<W<\delta_{2}\\}\cap B_{t_{1}}\cap
C^{(n)}\right)\leq\frac{\nu}{3}$ for n large enough.
Under the event $B_{t_{1}}$ we have

$\forall s\geq
t_{1},\left(W-\widetilde{\varepsilon}\right)e^{\lambda_{0}s}\leq
Z_{0}(s)\leq\left(W+\widetilde{\varepsilon}\right)e^{\lambda_{0}s}.$ (91)

Using that $\lambda_{0}^{(n)}\leq\lambda_{0}$, we get that under the event
$C^{(n)}$

$\forall
s\in\mathbb{R}^{+},\left(e^{-\lambda_{0}s}Z_{0}(s)-\widetilde{\varepsilon}\right)e^{\lambda_{0}^{(n)}s}\leq
Z_{0}^{(n)}(s)\leq\left(e^{-\lambda_{0}s}Z_{0}(s)+\widetilde{\varepsilon}\right)e^{\lambda_{0}^{(n)}s}\leq
Z_{0}(s)+\widetilde{\varepsilon}e^{\lambda_{0}s}.$ (92)

Combining the two previous inequalities it follows that under
$\\{\delta_{1}<W<\delta_{2}\\}\cap B_{t_{1}}\cap C^{(n)}$ we have

$\forall s\geq
t_{1},\left(W-2\widetilde{\varepsilon}\right)e^{\lambda_{0}^{(n)}s}\leq
Z_{0}^{(n)}(s)\leq\left(W+2\widetilde{\varepsilon}\right)e^{\lambda_{0}s}\leq\left(\delta_{2}+2\widetilde{\varepsilon}\right)e^{\lambda_{0}s}.$
(93)

Notice that by definition of $\widetilde{\varepsilon}$, we have that
$W-2\widetilde{\varepsilon}>0$ under the event $\\{\delta_{1}<W\\}$. Now
introduce the following quantities, which almost surely increase with time

$\displaystyle\underline{T}^{(n)}_{\delta_{2},t}:=\inf\\{s>0:(\delta_{2}+2\widetilde{\varepsilon})e^{\lambda_{0}s}\geq
n^{t}\\},$ (94)
$\displaystyle\underline{T}^{(n)}_{t}:=\inf\\{s>0:(W+2\widetilde{\varepsilon})e^{\lambda_{0}s}\geq
n^{t}\\},$ (95)
$\displaystyle\overline{T}^{(n)}_{t}:=\inf\\{s>0:(W-2\widetilde{\varepsilon})e^{\lambda_{0}^{(n)}s}\geq
n^{t}\\}.$ (96)

We have that it exists $n_{2}\in\mathbb{N}$ such that for all $n\geq n_{2}$

$\displaystyle t_{1}\leq\underline{T}^{(n)}_{\delta_{2},T_{1}}.$ (97)

Moreover under the event $\\{\delta_{1}<W<\delta_{2}\\}\cap B_{t_{1}}\cap
C^{(n)}$ we have for all $n\geq\max(n_{1},n_{2})$ and for all
$t\in\left[T_{1},T_{2}\frac{\varphi_{n}}{\log(n)}\right]$

$\displaystyle\underline{T}^{(n)}_{\delta_{2},T_{1}}\leq\underline{T}^{(n)}_{\delta_{2},t}\leq\underline{T}^{(n)}_{t}\leq\eta^{(n)}_{t}\leq\overline{T}^{(n)}_{t}.$
(98)

Using that
$\frac{\lambda_{0}}{\lambda_{0}^{(n)}}=\left(1-\frac{2\alpha_{0}\mu_{0}^{(n)}}{\lambda_{0}}\right)^{-1}$,
and the previous equation we derive that $\forall
t\in\left[T_{1},T_{2}\frac{\varphi_{n}}{\log(n)}\right]$

$\frac{t\log(n)}{\lambda_{0}}-\frac{\log(W)}{\lambda_{0}}-\frac{\log\left(1+2\tilde{\varepsilon}/W\right)}{\lambda_{0}}\leq\eta_{t}^{(n)}\leq\left(\frac{t\log(n)}{\lambda_{0}}-\frac{\log(W)}{\lambda_{0}}-\frac{\log(1-2\tilde{\varepsilon}/W)}{\lambda_{0}}\right)\cdot\frac{1}{1-2\alpha_{0}\mu_{0}^{(n)}/\lambda_{0}},$
(99)

from which we obtain

$\displaystyle-\frac{\log(1+2\tilde{\varepsilon}/W)}{\lambda_{0}}\leq\eta_{t}^{(n)}-$
$\displaystyle\left(\frac{t\log(n)}{\lambda_{0}}-\frac{\log(W)}{\lambda_{0}}\right)$
(100)
$\displaystyle\leq\frac{1}{1-2\alpha_{0}\mu_{0}^{(n)}/\lambda_{0}}\cdot\left(\left(\frac{t\log(n)}{\lambda_{0}}-\frac{\log(W)}{\lambda_{0}}\right)\frac{2\alpha_{0}\mu_{0}^{(n)}}{\lambda_{0}}-\frac{\log(1-2\tilde{\varepsilon}/W)}{\lambda_{0}}\right).$
(101)

In particular it implies that for all $n\geq\max(n_{1},n_{2})$

$\displaystyle\sup_{t\in\left[T_{1},T_{2}\frac{\varphi_{n}}{\log(n)}\right]}\Big{|}\eta_{t}^{(n)}-\left(\frac{t\log(n)}{\lambda_{0}}-\frac{\log(W)}{\lambda_{0}}\right)\Big{|}$
(102)
$\displaystyle\leq\max\left(\frac{\log(1+2\widetilde{\varepsilon}/W)}{\lambda_{0}};\frac{1}{1-2\alpha_{0}\mu_{0}^{(n)}/\lambda_{0}}\cdot\left(\left(\frac{T_{2}\varphi_{n}}{\lambda_{0}}-\frac{\log(W)}{\lambda_{0}}\right)\frac{2\alpha_{0}\mu_{0}^{(n)}}{\lambda_{0}}-\frac{\log(1-2\tilde{\varepsilon}/W)}{\lambda_{0}}\right)\right).$
(103)

Denote by $D^{(n)}$ the right hand side of the last inequality. Then it
directly follows that

$\mathbb{P}\left(A^{(n)}\cap\\{\delta_{1}<W<\delta_{2}\\}\cap B_{t_{1}}\cap
C^{(n)}\right)\leq\mathbb{P}\left(\\{D^{(n)}\geq\varepsilon\\}\cap\\{\delta_{1}<W<\delta_{2}\\}\right).$
(104)

Because $\varphi_{n}$ was defined such that
$\varphi_{n}\mu_{0}^{(n)}\underset{n\to\infty}{\to}0$ it is possible to find
an adequate $\widetilde{\varepsilon}>0$ and $n_{3}\in\mathbb{N}$ such that for
all $n\geq n_{3}$,
$\mathbb{P}\left(\\{D^{(n)}\geq\varepsilon\\}\cap\\{\delta_{1}<W<\delta_{2}\\}\right)\leq\frac{\nu}{3}$.
In addition with (90) and (104) we deduce (89).
Step 2: To complete the proof we are going to prove that
$\mathbb{P}\left(A^{(n)}\cap\\{W>0\\}\right)\underset{n\to\infty}{\longrightarrow}0.$
We have

$\displaystyle\mathbb{P}\left(A^{(n)}\cap\\{W>0\\}\right)\leq\mathbb{P}\left(A^{(n)}\cap\\{\delta_{1}<W<\delta_{2}\\}\right)+\mathbb{P}\left(0<W<\delta_{1}\right)+\mathbb{P}\left(W>\delta_{2}\right).$
(105)

Using Equation (89) we obtain that

$\limsup_{n\to\infty}\mathbb{P}\left(A^{(n)}\cap\\{W>0\\}\right)\leq\mathbb{P}\left(0<W<\delta_{1}\right)+\mathbb{P}\left(\delta_{2}<W\right).$
(106)

Taking the limit when
$(\delta_{1},\delta_{2})\underset{n\to\infty}{\to}(0,\infty)$ and because $W$
is finite almost surely (see (71)) we conclude. ∎

######  Remark 3.1.

From Lemma 3.2, it follows the useful results

$\displaystyle\mathbb{P}\left(\sup_{t\in\left[T_{1},T_{2}\frac{\varphi_{n}}{\log(n)}\right]}\Big{|}\frac{\eta_{t}^{(n)}}{\log(n)}\lambda_{0}-t\Big{|}\geq\varepsilon\Bigg{|}W>0\right)\underset{n\to\infty}{\longrightarrow}0,$
(107)

and

$\displaystyle\mathbb{P}\left(\sup_{t\in\left[T_{1},T_{2}\frac{\varphi_{n}}{\log(n)}\right]}\Big{|}e^{-\lambda_{0}\left(\eta_{t}^{(n)}-\mathfrak{t}^{(n)}_{t}\right)}-W\Big{|}\geq\varepsilon\Bigg{|}W>0\right)\underset{n\to\infty}{\longrightarrow}0.$
(108)

###  3.2 The mutant sub-populations dynamics in the deterministic time-scale
(Theorem 2.1 (i))

In this subsection, Equations (38) and (39) are proven for the mono-
directional graph. It will be done in two steps. The first one will consist in
showing the result for a fixed $s\in\mathbb{R}$ and uniformly in the parameter
$t$. Then in the second step, the result will be proved uniformly in the
parameter $s$.

####  3.2.1 Uniform control on the time parameter t

In this subsection we are going to prove the following proposition, which is a
less refine result of (38) and (39), because the result is not uniform on the
parameter $s$.

######  Proposition 3.1.

Let $i\in\mathbb{N}$,
$(\psi_{n}(i),\gamma_{n}(i))\underset{n\to\infty}{\rightarrow}\infty$ such
that it exists $\varphi_{n}(i)\underset{n\to\infty}{\to}\infty$ such that
$\gamma_{n}(i)=\frac{\log(n)}{\log(\log(n))\theta(i-1)+\varphi_{n}(i)}$ and
$e^{\varphi_{n}(i)}\log(n)=o(\psi_{n}^{2}(i))$. For all
$(t,s)\in\mathbb{R}^{+}\times\mathbb{R}$ define

$\displaystyle d_{i}^{(n)}(t,s):=$
$\displaystyle\mathbbm{1}_{\left\\{t\in[0,t(i)-\gamma_{n}^{-1}(i))\right\\}}+\mathbbm{1}_{\left\\{t\in[t(i)-\gamma_{n}^{-1}(i),t(i))\right\\}}\psi_{n}\log^{\theta(i)-1}(n)$
(109) $\displaystyle\hskip
85.35826pt+\mathbbm{1}_{\left\\{t\in[t(i),\infty)\right\\}}n^{t-t(i)}\log^{\theta(i)}(n)e^{\lambda(0)s}.$
(110)

Let $T>0$, $0<T_{1}<T_{2}$, and $s\in\mathbb{R}$. Then

  * (i)

If $\lambda_{i}=\lambda_{0}$

$\displaystyle\left(\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}{d_{i}^{(n)}(t,s)}\right)_{t\in[0,T]}\underset{n\to\infty}{\longrightarrow}\left(Ww_{i}(t)\right)_{t\in[0,T]},$
(111)

in probability in $L^{\infty}\left([0,T]\right)$.

  * (ii)

If $\lambda_{i}<\lambda_{0}$

$\displaystyle\left(\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)+t}+s\right)}{n^{t}\log^{\theta(i)}(n)e^{\lambda_{0}s}}\right)_{t\in[T_{1},T_{2}]}\underset{n\to\infty}{\longrightarrow}\left(Ww_{i}(t(i)+t)\right)_{t\in[T_{1},T_{2}]},$
(112)

in probability in $L^{\infty}([T_{1},T_{2}])$.

The proof is done by induction on $i\in\mathbb{N}$. As long as the proof is
similar for the initialization and the inductive part the step considered will
not be specified. To make the proof the clearer possible it is cut using
several lemmas.
All the results are obtained using a martingale approach. In the next Lemma
the martingales that are considered for all the mutant sub-populations are
introduced, and their quadratic variations are computed.

######  Lemma 3.3.

For all $i\in\mathbb{N}$ define

$\displaystyle
M_{i}^{(n)}(t):=Z_{i}^{(n)}(t)e^{-\lambda_{i}^{(n)}t}-\int_{0}^{t}2\alpha_{i-1}\mu_{i-1}^{(n)}e^{-\lambda_{i}^{(n)}s}Z_{i-1}^{(n)}(s)ds.$
(113)

$\left(M_{i}^{(n)}(t)\right)_{t\geq 0}$ is a martingale, with quadratic
variation

$\displaystyle\left\langle
M_{i}^{(n)}\right\rangle_{t}=\int_{0}^{t}2\alpha_{i-1}\mu_{i-1}^{(n)}e^{-2\lambda_{i}^{(n)}s}Z_{i-1}^{(n)}(s)ds+\left(\alpha_{i}^{(n)}+\beta_{i}^{(n)}\right)\int_{0}^{t}e^{-2\lambda_{i}^{(n)}s}Z_{i}^{(n)}(s)ds.$
(114)

###### Proof of Lemma 3.3.

For all $t\geq 0$ let $\mathcal{F}_{i,t}^{(n)}$ the $\sigma$-field generated
by $Z^{(n)}_{j}(s)$ for $0\leq j\leq i$ and $0\leq s\leq t$. For all $h\geq 0$
we have

$\displaystyle\mathbb{E}\left[M_{i}^{(n)}(t+h)-M_{i}^{(n)}(t)|\mathcal{F}_{i,t}^{(n)}\right]$
$\displaystyle=\mathbb{E}\left[Z^{(n)}_{i}(t+h)\Big{\lvert}\mathcal{F}_{i,t}^{(n)}\right]e^{-\lambda_{i}^{(n)}(t+h)}-Z^{(n)}_{i}(t)e^{-\lambda_{i}^{(n)}t}$
(115)
$\displaystyle-\int_{t}^{t+h}2\alpha_{i-1}\mu_{i-1}^{(n)}e^{-\lambda_{i}^{(n)}s}\mathbb{E}\left[Z^{(n)}_{i-1}(s)\Big{\lvert}\mathcal{F}_{i,t}^{(n)}\right]ds.$
(116)

The forward Chapman-Kolmogorov equation gives the time-differential equation
for $\mathbb{E}\left[Z^{(n)}_{i}(t)\right]$

$\begin{split}\dfrac{d\mathbb{E}\left[Z^{(n)}_{i}(t)\right]}{dt}&=\mathbb{E}\left[\alpha_{i}^{(n)}Z^{(n)}_{i}(t)-\beta_{i}^{(n)}Z^{(n)}_{i}(t)\right]+\mathbb{E}\left[2\alpha_{i-1}\mu_{i-1}^{(n)}(1-\mu_{i-1}^{(n)})Z^{(n)}_{i-1}(t)\right]\\\
&+2\mathbb{E}\left[\alpha_{i-1}\left(\mu_{i-1}^{(n)}\right)^{2}Z^{(n)}_{i-1}(t)\right]\\\
&=\lambda_{i}^{(n)}\mathbb{E}\left[Z^{(n)}_{i}(t)\right]+2\alpha_{i-1}\mu_{i-1}^{(n)}\mathbb{E}\left[Z^{(n)}_{i-1}(t)\right],\end{split}$
(117)

which leads to

$\mathbb{E}_{Z_{i}^{(n)}(0)}\left[Z^{(n)}_{i}(t)\right]=Z^{(n)}_{i}(0)e^{\lambda_{i}^{(n)}t}+\int_{0}^{t}2\alpha_{i-1}\mu_{i-1}^{(n)}\mathbb{E}_{Z_{i}^{(n)}(0)}\left[Z^{(n)}_{i-1}(s)\right]e^{\lambda_{i}^{(n)}(t-s)}ds.$
(118)

In particular using the Markov property we obtain that

$\displaystyle\mathbb{E}\left[Z^{(n)}_{i}(t+h)\Big{\lvert}\mathcal{F}_{i,t}^{(n)}\right]=Z^{(n)}_{i}(t)e^{\lambda_{i}^{(n)}h}+\int_{t}^{t+h}2\alpha_{i-1}\mu_{i-1}^{(n)}\mathbb{E}\left[Z^{(n)}_{i-1}(s)\lvert\mathcal{F}_{i,t}^{(n)}\right]e^{\lambda_{i}^{(n)}(t+h-s)}ds.$
(119)

Combining (115) and (119) it follows that
$\left(M_{i}^{(n)}(t)\right)_{t\in\mathbb{R}^{+}}$ is a martingale.
Let $F^{(n)}(t,x,y):=(e^{-\lambda_{i}^{(n)}t}x-y)^{2}$, it follows that

$\displaystyle\frac{\partial F^{(n)}}{\partial
t}(t,x,y)=-2\lambda_{i}^{(n)}xe^{-\lambda_{i}^{(n)}t}F^{(n)}(t,x,y),$ (120)
$\displaystyle\frac{\partial F^{(n)}}{\partial y}(t,x,y)=-2F^{(n)}.$ (121)

Applying Itô’s formula with $x=Z_{i}^{(n)}(t)$ and
$y=\int_{0}^{t}2\alpha_{i-1}\mu_{i-1}^{(n)}e^{-\lambda_{i}^{(n)}s}Z^{(n)}_{i-1}(s)ds$
we obtain

$\begin{split}&\left(M^{(n)}_{i}(t)\right)^{2}=F\left(t,Z_{i}^{(n)}(t),\int_{0}^{t}2\alpha_{i-1}\mu_{i-1}^{(n)}e^{-\lambda_{i}^{(n)}s}Z^{(n)}_{i-1}(s)ds\right)\\\
&=F^{(n)}(0,0,0)-2\int_{0}^{t}2\alpha_{i-1}\mu_{i-1}^{(n)}e^{-\lambda_{i}^{(n)}s}Z^{(n)}_{i-1}(s)M^{(n)}_{i}(s)ds-2\lambda_{i}^{(n)}\int_{0}^{t}e^{-\lambda_{i}^{(n)}s}Z^{(n)}_{i}(s)M^{(n)}_{i}(s)ds\\\
&+\int_{0}^{t}\int_{\mathbb{R}^{+}}\Bigg{[}\Bigg{(}M_{i}^{(n)}+e^{-\lambda_{i}^{(n)}s}\Bigg{\\{}\mathbbm{1}_{\\{\theta\leq\alpha_{i}^{(n)}Z^{(n)}_{i}(s^{-})\\}}-\mathbbm{1}_{\left\\{\alpha_{i}^{(n)}Z_{i}^{(n)}(s^{-})\leq\theta\leq\left(\alpha_{i}^{(n)}+\beta_{i}\right)Z_{i}^{(n)}(s^{-})\right\\}}\Bigg{\\}}\Bigg{)}^{2}\\\
&\hskip 227.62204pt-\left(M_{i}^{(n)}\right)^{2}\Bigg{]}Q_{i}(ds,d\theta)\\\
&+\int_{0}^{t}\int_{\mathbb{R}^{+}}\Bigg{[}\Bigg{(}M_{i}^{(n)}+e^{-\lambda_{i}^{(n)}s}\mathbbm{1}_{\left\\{\theta\leq
2\alpha_{i-1}\mu_{i-1}^{(n)}\left(1-\mu_{i-1}^{(n)}\right)Z_{i-1}^{(n)}(s^{-})\right\\}}\Bigg{)}^{2}-\left(M_{i}^{(n)}\right)^{2}\Bigg{]}N_{i-1}(ds,d\theta)\\\
&+\int_{0}^{t}\int_{\mathbb{R}^{+}}\Bigg{[}\Bigg{(}M_{i}^{(n)}+e^{-\lambda_{i}^{(n)}s}2\mathbbm{1}_{\left\\{\theta\leq\alpha_{i-1}\left(\mu_{i-1}^{(n)}\right)^{2}Z_{i-1}^{(n)}(s^{-})\right\\}}\Bigg{)}^{2}-\left(M_{i}^{(n)}\right)^{2}\Bigg{]}Q_{i-1}^{m}(ds,d\theta)\\\
&+\int_{0}^{t}\int_{\mathbb{R}^{+}}\Bigg{[}\Bigg{(}M_{i}^{(n)}-e^{-\lambda_{i}^{(n)}s}\mathbbm{1}_{\left\\{\theta\leq\alpha_{i}\left(\mu_{i}^{(n)}\right)^{2}Z_{i}^{(n)}(s^{-})\right\\}}\Bigg{)}^{2}-\left(M_{i}^{(n)}\right)^{2}\Bigg{]}Q_{i}^{m}(ds,d\theta)\\\
&=-2\int_{0}^{t}\left(2\alpha_{i-1}\mu_{i-1}^{(n)}Z^{(n)}_{i-1}(s)+\lambda_{i}^{(n)}Z^{(n)}_{i}(s)\right)e^{-\lambda_{i}^{(n)}s}M^{(n)}_{i}(s)ds\\\
&+2\int_{0}^{t}\left(2\alpha_{i-1}\mu_{i-1}^{(n)}Z^{(n)}_{i-1}(s)+\lambda_{i}^{(n)}Z^{(n)}_{i}(s)\right)e^{-\lambda_{i}^{(n)}s}M^{(n)}_{i}ds\\\
&+\int_{0}^{t}\left[2\alpha_{i-1}\mu_{i-1}^{(n)}Z^{(n)}_{i-1}(s)+\left(\alpha_{i}^{(n)}+\beta_{i}^{(n)}\right)Z^{(n)}_{i}(s)\right]e^{-2\lambda_{i}^{(n)}s}ds+\widetilde{M}^{(n)}_{i}(t)\\\
&=\widetilde{M}^{(n)}_{i}(t)+\int_{0}^{t}2\alpha_{i-1}\mu_{i-1}^{(n)}e^{-2\lambda_{i}^{(n)}s}Z_{i-1}^{(n)}(s)ds+\left(\alpha_{i}^{(n)}+\beta_{i}^{(n)}\right)\int_{0}^{t}e^{-2\lambda_{i}^{(n)}s}Z_{i}^{(n)}(s)ds,\end{split}$
(122)

where $\widetilde{M}^{(n)}_{i}$ is a martingale. Finally, we get

$\left\langle
M^{(n)}_{i}\right\rangle_{t}=\int_{0}^{t}2\alpha_{i-1}\mu_{i-1}^{(n)}e^{-2\lambda_{i}^{(n)}s}Z_{i}^{(n)}(s)ds+\left(\alpha_{i}^{(n)}+\beta_{i}^{(n)}\right)\int_{0}^{t}e^{-2\lambda_{i}^{(n)}s}Z_{i}^{(n)}(s)ds.$
(123)

∎

Now we can deal with the proof of Proposition 3.1.

###### Proof of Proposition 3.1.

Let $i\in\mathbb{N}^{*}$. For $i\geq 2$ assume that Proposition 3.1 is true
for $i-1$.
We start by showing the result when $i$ is a neutral trait, that is to say we
are going to prove (111). All the lemmas that we are mentioning in the proof
are free from such neutral assumption, and work also for deleterious mutant
traits.
(i) Neutral case: Assume that $\lambda_{i}=\lambda_{0}$. Let
$(\psi_{n}(i),\gamma_{n}(i))$ as in Proposition 3.1. Notice that

$\displaystyle\mathbb{P}\Bigg{(}\sup_{t\in\left[0,T\right]}\Bigg{\lvert}\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}{d_{i}^{(n)}(t,s)}-Ww_{i}(t)\Bigg{\lvert}\geq
3\varepsilon\Bigg{)}$ (124) $\displaystyle\hskip
56.9055pt\leq\mathbb{P}\Bigg{(}\sup_{t\in\left[0,t(i)-\gamma_{n}^{-1}(i)\right)}Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)\geq\varepsilon\Bigg{)}$
(125) $\displaystyle\hskip
56.9055pt+\mathbb{P}\Bigg{(}\sup_{t\in\left[t(i)-\gamma_{n}^{-1}(i),t(i)\right)}\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}{\psi_{n}(i)\log^{\theta(i-1)}(n)}\geq\varepsilon\Bigg{)}$
(126) $\displaystyle\hskip
56.9055pt+\mathbb{P}\Bigg{(}\sup_{t\in\left[t(i),T\right]}\Bigg{\lvert}\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}{n^{t-t(i)}\log^{\theta(i)}(n)e^{\lambda_{0}s}}-Ww_{i}(t)\Bigg{\lvert}\geq\varepsilon\Bigg{)}.$
(127)

We are going to show that (125), (126) and (127) converges to 0 when $n$ goes
to infinity.
Step 1: Convergence to 0 of (125):
The characterisation of $t(i)$ as the first time to see mutant cell of trait
$i$ in the time-scale $t\mapsto\mathfrak{t}^{(n)}_{t}$ is made explicit in the
next Lemma. More precisely, we exactly show that up until time
$t(i)-\gamma_{n}^{-1}(i)$, asymptotically no mutant cells of trait $i$ are
generated. In particular the convergence to 0 of (125) is deduced from the
next lemma.

######  Lemma 3.4.

Let $i\in\mathbb{N}$, and
$\gamma_{n}(i)=\frac{\log(n)}{\log(\log(n))\theta(i-1)+\varphi_{n}(i)}$ where
$\varphi_{n}(i)\underset{n\to\infty}{\to}\infty$ such that
$\gamma_{n}(i)\underset{n\to\infty}{\to}\infty$, and $s\in\mathbb{R}$. For
$i\geq 2$ we prove that if Proposition 3.1 is true for $i-1$ then

$\mathbb{P}\Bigg{(}\sup_{t\in\left[0,t(i)-\gamma_{n}^{-1}(i)\right]}Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)=0\Bigg{)}\underset{n\to\infty}{\longrightarrow}1.$
(128)

For $i=1$, we prove (128) without condition.

###### Proof of Lemma 3.4.

Notice first that

$\displaystyle\Bigg{\\{}\sup_{t\in\left[0,t(i)-\gamma_{n}^{-1}(i)\right]}Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)=0\Bigg{\\}}=A_{n}\cap
B_{n},$ (129)

where

$\displaystyle
A_{n}:=\Bigg{\\{}K_{i-1}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s\right)=0\Bigg{\\}},$
(130) $\displaystyle
B_{n}:=\Bigg{\\{}H_{i-1}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s\right)=0\Bigg{\\}},$
(131)

because the event in the left hand side of Eq. (129) is satisfied if and only
if there is no mutant cell of the sub-population $Z_{i}^{(n)}$ generated from
the sub-population $Z_{i-1}^{(n)}$ up until time
$\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s$. It corresponds almost surely
to $A_{n}\cap B_{n}$. In what follows, we will detail the proof of
$\mathbb{P}(A_{n})\underset{n\to\infty}{\to}1$. Proving
$\mathbb{P}(B_{n})\underset{n\to\infty}{\to}1$ can be done using a similar
method, so the proof will not be detailed. This will conclude the proof of
Lemma 3.4.
So we deal with the proof of $\mathbb{P}(A_{n})\underset{n\to\infty}{\to}1$
which will be slightly different depending on whether $i=1$ or $i\geq 2$. We
begin with $i=1$.
(i) Case $i=1$:
Introduce the following event for all $\widetilde{t}\in\mathbb{R}^{+}$ and
$\varepsilon\in\mathbb{R}^{+}$

$\displaystyle
C_{\varepsilon,\widetilde{t}}:=\Bigg{\\{}\sup_{s\in[\widetilde{t},\infty]}\Big{|}e^{-\lambda_{0}s}Z_{0}(s)-W\Big{|}\leq\varepsilon\Bigg{\\}}.$
(132)

Using the a.s. inequality of (69), under the event
$C_{\varepsilon,\widetilde{t}}$ we have

$\displaystyle
K_{0}^{(n)}\left(\mathfrak{t}^{(n)}_{t(1)-\gamma_{n}^{-1}(1)}+s\right)$
$\displaystyle\leq\int_{0}^{\widetilde{t}}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq
2\alpha_{0}\mu_{0}^{(n)}\sup_{v\in[0,t_{1}]}Z_{0}(v)\right\\}}N_{0}(du,d\theta)$
(133)
$\displaystyle+\int_{\widetilde{t}}^{\mathfrak{t}^{(n)}_{t(1)-\gamma_{n}^{-1}(1)}+s}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq
2\alpha_{0}\mu_{0}^{(n)}e^{\lambda_{0}u}(\varepsilon+W)\right\\}}N_{0}(du,d\theta).$
(134)

Let set the following notations

$\displaystyle
D_{\widetilde{t}}^{(n)}:=\Bigg{\\{}\int_{0}^{\widetilde{t}}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq
2\alpha_{0}\mu_{0}^{(n)}\sup_{v\in[0,t_{1}]}Z_{0}(v)\right\\}}N_{0}(du,d\theta)=0\Bigg{\\}},$
(135) $\displaystyle
E_{\varepsilon,\widetilde{t}}^{(n)}:=\Bigg{\\{}\int_{\widetilde{t}}^{\mathfrak{t}^{(n)}_{t(1)-\gamma_{n}^{-1}(1)}+s}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq
2\alpha_{0}\mu_{0}^{(n)}e^{\lambda_{0}u}(\varepsilon+W)\right\\}}N_{0}(du,d\theta)=0\Bigg{\\}}.$
(136)

Using Equation (133) we have that

$\displaystyle\mathbb{P}(A_{n})\geq\mathbb{P}(A_{n}\cap
C_{\varepsilon,\widetilde{t}})\geq\mathbb{P}\left(C_{\varepsilon,\widetilde{t}}\cap
D_{\widetilde{t}}^{(n)}\cap E_{\varepsilon,\widetilde{t}}^{(n)}\right).$ (137)

It remains to show that the right hand side converges to $1$. By definition of
$W$ as the almost sure time limit of the positive martingale
$e^{-\lambda_{0}t}Z_{0}(t)$ it follows that
$\mathbb{P}\left(C_{\varepsilon,\widetilde{t}}\right)\underset{\widetilde{t}\to\infty}{\to}1.$
We have that $\sup_{v\in[0,\widetilde{t}]}Z_{0}(v)$ is finite almost surely.
In addition with the fact that $Z_{0}$ and $N_{0}$ are independent we deduce
that
$\mathbb{P}\left(D_{\widetilde{t}}^{(n)}\right)\underset{n\to\infty}{\to}1,$
because $\mu_{0}^{(n)}\underset{n\to\infty}{\to}0$.
Recall the distribution of $W$, which is given in (71). Since $W$ and $N_{0}$
are independent, we have

$\displaystyle\mathbb{P}\left(E_{\varepsilon,\widetilde{t}}^{(n)}\right)$
$\displaystyle=\frac{\beta_{0}}{\alpha_{0}}\mathbb{P}\left(\int_{\widetilde{t}}^{\mathfrak{t}^{(n)}_{t(1)-\gamma_{n}^{-1}(1)}+s}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq
2\alpha_{0}\mu_{0}^{(n)}e^{\lambda_{0}u}\varepsilon\right\\}}N_{0}(du,d\theta)=0\right)$
(138)
$\displaystyle+\frac{\lambda_{0}}{\alpha_{0}}\int_{0}^{\infty}\frac{\lambda_{0}}{\alpha_{0}}e^{-\frac{\lambda_{0}}{\alpha_{0}}w}\mathbb{P}\left(\int_{\widetilde{t}}^{\mathfrak{t}^{(n)}_{t(1)-\gamma_{n}^{-1}(1)}+s}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq
2\alpha_{0}\mu_{0}^{(n)}e^{\lambda_{0}u}(\varepsilon+w)\right\\}}N_{0}(du,d\theta)=0\right)dw$
(139)
$\displaystyle=\frac{\beta_{0}}{\alpha_{0}}\exp\left(-\int_{\widetilde{t}}^{\mathfrak{t}^{(n)}_{t(1)-\gamma_{n}^{-1}(1)}+s}2\alpha_{0}\mu_{0}^{(n)}e^{\lambda_{0}u}\varepsilon
du\right)$ (140) $\displaystyle\hskip
28.45274pt+\frac{\lambda_{0}}{\alpha_{0}}\int_{0}^{\infty}\frac{\lambda_{0}}{\alpha_{0}}e^{-\frac{\lambda_{0}}{\alpha_{0}}w}\exp\left(-\int_{\widetilde{t}}^{\mathfrak{t}^{(n)}_{t(1)-\gamma_{n}^{-1}(1)}+s}2\alpha_{0}\mu_{0}^{(n)}e^{\lambda_{0}u}(\varepsilon+w)du\right)dw$
(141)
$\displaystyle\geq\frac{\beta_{0}}{\alpha_{0}}\exp\left(-\frac{2\alpha_{0}(n^{t(1)}\mu_{0}^{(n)})}{\lambda_{0}}\varepsilon
e^{\lambda_{0}s}e^{-\gamma_{n}^{-1}(1)\log(n)}\right)$ (142)
$\displaystyle\hskip
28.45274pt+\frac{\lambda_{0}}{\alpha_{0}}\int_{0}^{\infty}\frac{\lambda_{0}}{\alpha_{0}}e^{-\frac{\lambda_{0}}{\alpha_{0}}w}\exp\left(-\frac{2\alpha_{0}(n^{t(1)}\mu_{0}^{(n)})}{\lambda_{0}}(\varepsilon+w)e^{\lambda_{0}s}e^{-\gamma_{n}^{-1}(1)\log(n)}\right)dw$
(143) $\displaystyle\underset{n\to\infty}{\longrightarrow}1,$ (144)

where we use first that for all $w\geq 0$,
$\frac{2\alpha_{0}\left(n^{t(1)}\mu_{0}^{(n)}\right)}{\lambda_{0}}(\varepsilon+w)e^{\lambda_{0}s}e^{-\gamma_{n}^{-1}(1)\log(n)}\underset{n\to\infty}{\rightarrow}0$.
This fact comes from the choice of $\gamma_{n}$ which implies that
$\gamma_{n}^{-1}(1)\log(n)\underset{n\to\infty}{\rightarrow}\infty$. Then we
apply the dominated convergence theorem to get

$\displaystyle\int_{0}^{\infty}\frac{\lambda_{0}}{\alpha_{0}}e^{-\frac{\lambda_{0}}{\alpha_{0}}w}\exp\left(-\frac{2\alpha_{0}\left(n^{t(1)}\mu_{0}^{(n)}\right)}{\lambda_{0}}(\varepsilon+w)e^{\lambda_{0}s}e^{-\gamma_{n}^{-1}(1)\log(n)}\right)dw\underset{n\to\infty}{\longrightarrow}\int_{0}^{\infty}\frac{\lambda_{0}}{\alpha_{0}}e^{-\frac{\lambda_{0}}{\alpha_{0}}w}dw=1.$
(145)

Finally we have proven that
$\mathbb{P}(A_{n})\underset{n\to\infty}{\rightarrow}1,$ which concludes the
proof in the case $i=1$.
(ii) Case $i\geq 2$: Let $\widetilde{t}(i):=\frac{t(i)+t(i-1)}{2}$ and
$\Psi_{n}\underset{n\to\infty}{\to}\infty$. Introduce the event

$\displaystyle
C_{\varepsilon}^{(n)}:=\left\\{\sup_{t\in[0,t(i)]}\Bigg{|}\frac{Z_{i-1}^{(n)}\left(\mathfrak{t}_{t}^{(n)}\right)}{d^{(n)}(t)}-W\mathbbm{1}_{\\{t\geq\tilde{t}(i)\\}}w_{i-1}(t)\Bigg{|}\leq\varepsilon\right\\},$
(146)

where
$d^{(n)}(t)=\mathbbm{1}_{\left\\{t\in[0,\widetilde{t}(i))\right\\}}n^{\widetilde{t}(i)-t(i-1)}\log^{\theta(i-1)}(n)\Psi_{n}+\mathbbm{1}_{\left\\{t\in\left[\widetilde{t}(i),t(i)\right]\right\\}}n^{t-t(i-1)}\log^{\theta(i-1)}(n)$.
Under $C_{\varepsilon}^{(n)}$ we have

$\displaystyle
K_{i-1}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s\right)\leq\int_{0}^{\mathfrak{t}^{(n)}_{\widetilde{t}(i)}}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq
2\alpha_{i-1}\mu_{i-1}^{(n)}\varepsilon
n^{\tilde{t}(i)-t(i-1)}\log^{\theta(i-1)}(n)\Psi_{n}\right\\}}N_{i-1}(du,d\theta)$
(147)
$\displaystyle+\int_{\mathfrak{t}^{(n)}_{\widetilde{t}(i)}}^{\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq
2\alpha_{i-1}\mu_{i-1}^{(n)}\left(\varepsilon+Ww_{i-1}(t(i))\right)e^{\lambda_{0}u}n^{-t(i-1)}\log^{\theta(i-1)}(n)\right\\}}N_{i-1}(du,d\theta).$
(148)

Let introduce the events

$\displaystyle
D_{\varepsilon}^{(n)}:=\left\\{\int_{0}^{\mathfrak{t}_{\widetilde{t}(i)}^{(n)}}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq
2\alpha_{i-1}\mu_{i-1}^{(n)}\varepsilon
n^{\tilde{t}(i)-t(i-1)}\log^{\theta(i-1)}(n)\Psi_{n}\right\\}}N_{i-1}(du,d\theta)=0\right\\},$
(149) $\displaystyle
E_{\varepsilon}^{(n)}:=\left\\{\int_{\mathfrak{t}_{\widetilde{t}(i)}^{(n)}}^{\mathfrak{t}_{t(i)-\gamma_{n}^{-1}(i)}^{(n)}+s)}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq
2\alpha_{i-1}\mu_{i-1}^{(n)}\left(\varepsilon+Ww_{i-1}(t(i))\right)e^{\lambda_{0}u}n^{-t(i-1)}\log^{\theta(i-1)}(n)\right\\}}N_{i-1}(du,d\theta)=0\right\\}.$
(150)

From (147) we obtain

$\displaystyle\mathbb{P}\left(A_{n}\right)\geq\mathbb{P}\left(A_{n}\cap
C_{\varepsilon}^{(n)}\right)\geq\mathbb{P}\left(C_{\varepsilon}^{(n)}\cap
D_{\varepsilon}^{(n)}\cap E_{\varepsilon}^{(n)}\right).$ (151)

It remains to show that the r.h.s. converge to $1$.
From assuming Proposition 3.1 for trait $i-1$ it follows that
$\mathbb{P}\left(C_{\varepsilon}^{(n)}\right)\underset{n\to\infty}{\longrightarrow}1$.
Secondly we have

$\displaystyle\mathbb{P}\left(D_{\varepsilon}^{(n)}\right)=\exp\left(-\widetilde{t}(i)\frac{\log^{\theta(i-1)+1}(n)}{\lambda_{0}}2\alpha_{i-1}\mu_{i-1}^{(n)}\varepsilon\sqrt{n^{\ell(i-1)}}\Psi_{n}\right)\underset{n\to\infty}{\longrightarrow}1,$
(152)

because $\widetilde{t}(i)-t(i-1)=\frac{\ell(i-1)}{2}$ and also because
$\Psi_{n}$ can be chosen such that it satisfies both
$\Psi_{n}\underset{n\to\infty}{\to}\infty$ and
$\log^{\theta(i-1)+1}(n)\Psi_{n}\sqrt{n^{\ell(i-1)}}\mu_{i-1}^{(n)}\underset{n\to\infty}{\to}0$.
Recall the distribution of $W$ given in Equation (71). Because $W$ and
$N_{i-1}$ are independent, we have

$\displaystyle\mathbb{P}\left(E_{\varepsilon}^{(n)}\right)=\frac{\beta_{0}}{\alpha_{0}}\mathbb{P}\left(\int_{\mathfrak{t}_{\widetilde{t}(i)}^{(n)}}^{\mathfrak{t}_{t(i)-\gamma_{n}^{-1}(i)}^{(n)}+s)}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq
2\alpha_{i-1}\mu_{i-1}^{(n)}\varepsilon
e^{\lambda_{0}u}n^{-t(i-1)}\log^{\theta(i-1)}(n)\right\\}}N_{i-1}(du,d\theta)=0\right)$
(153)
$\displaystyle+\frac{\lambda_{0}}{\alpha_{0}}\int_{0}^{\infty}\frac{\lambda_{0}}{\alpha_{0}}e^{-\frac{\lambda_{0}}{\alpha_{0}}w}$
(154) $\displaystyle\hskip
14.22636pt\cdot\mathbb{P}\left(\int_{\mathfrak{t}_{\widetilde{t}(i)}^{(n)}}^{\mathfrak{t}_{t(i)-\gamma_{n}^{-1}(i)}^{(n)}+s)}\int_{\mathbb{R}^{+}}\mathbbm{1}_{\left\\{\theta\leq
2\alpha_{i-1}\mu_{i-1}^{(n)}\left(\varepsilon+ww_{i-1}(t(i))\right)e^{\lambda_{0}u}n^{-t(i-1)}\log^{\theta(i-1)}(n)\right\\}}N_{i-1}(du,d\theta)=0\right)dw$
(155)
$\displaystyle=\frac{\beta_{0}}{\alpha_{0}}\exp\left(-\int_{\mathfrak{t}_{\widetilde{t}(i)}^{(n)}}^{\mathfrak{t}_{t(i)-\gamma_{n}^{-1}(i)}^{(n)}+s)}2\alpha_{i-1}\mu_{i-1}^{(n)}\varepsilon
e^{\lambda_{0}u}n^{-(t(i-1))}\log^{\theta(i-1)}(n)du\right)$ (156)
$\displaystyle+\frac{\lambda_{0}}{\alpha_{0}}\int_{0}^{\infty}\frac{\lambda_{0}}{\alpha_{0}}e^{-\frac{\lambda_{0}}{\alpha_{0}}w}$
(157) $\displaystyle\hskip
56.9055pt\cdot\exp\left(-\int_{\mathfrak{t}_{\widetilde{t}(i)}^{(n)}}^{\mathfrak{t}_{t(i)-\gamma_{n}^{-1}(i)}^{(n)}+s}2\alpha_{i-1}\mu_{i-1}^{(n)}\left(\varepsilon+ww_{i-1}(t(i))\right)e^{\lambda_{0}u}n^{-t(i-1)}\log^{\theta(i-1)}(n)du\right)dw$
(158)
$\displaystyle\geq\frac{\beta_{0}}{\alpha_{0}}\exp\left[-\frac{2\alpha_{i-1}\left(n^{\ell(i-1)}\mu_{i-1}^{(n)}\right)}{\lambda_{0}}\varepsilon\log^{\theta(i-1)}(n)e^{\lambda_{0}s}e^{-\gamma_{n}^{-1}(i)\log(n)}\right]$
(159)
$\displaystyle+\frac{\lambda_{0}}{\alpha_{0}}\int_{0}^{\infty}\frac{\lambda_{0}}{\alpha_{0}}e^{-\frac{\lambda_{0}}{\alpha_{0}}w}\exp\left(-\frac{2\alpha_{i-1}\left(n^{\ell(i-1)}\mu_{i-1}^{(n)}\right)}{\lambda_{0}}(\varepsilon+ww_{i-1}(t(i)))\log^{\theta(i-1)}(n)e^{\lambda_{0}s}e^{-\gamma_{n}^{-1}(i)\log(n)}\right)dw$
(160) $\displaystyle\underset{n\to\infty}{\longrightarrow}1,$ (161)

where we use that for all $w\geq 0$,
$\frac{2\alpha_{i-1}\left(n^{\ell(i-1)}\mu_{i-1}^{(n)}\right)}{\lambda_{0}}(\varepsilon+ww_{i-1}(t(i)))\log^{\theta(i-1)}(n)e^{\lambda_{0}s}e^{-\gamma_{n}^{-1}(i)\log(n)}\underset{n\to\infty}{\rightarrow}0$,
because
$\log^{\theta(i-1)}(n)e^{-\gamma_{n}^{-1}(i)\log(n)}=\exp\left(\theta(i-1)\log(\log(n))-\log(n)\gamma_{n}^{-1}(i)\right)\underset{n\to\infty}{\rightarrow}0$
by hypothesis on $\gamma_{n}(i)$, and we apply the dominated convergence
theorem to get

$\displaystyle\int_{0}^{\infty}\frac{\lambda_{0}}{\alpha_{0}}e^{-\frac{\lambda_{0}}{\alpha_{0}}w}\exp\left(-\frac{2\alpha_{i-1}\left(n^{\ell(i-1)}\mu_{i-1}^{(n)}\right)}{\lambda_{0}}(\varepsilon+ww_{i-1}(t(i)))\log^{\theta(i-1)}(n)e^{\lambda_{0}s}e^{-\gamma_{n}^{-1}(i)\log(n)}\right)dw$
(162) $\displaystyle\hskip
56.9055pt\underset{n\to\infty}{\longrightarrow}\int_{0}^{\infty}\frac{\lambda_{0}}{\alpha_{0}}e^{-\frac{\lambda_{0}}{\alpha_{0}}w}dw=1.$
(163)

Finally we have proven that
$\mathbb{P}(A_{n})\underset{n\to\infty}{\rightarrow}1,$ which concludes the
proof in the case $i\geq 2$. ∎

Step 2: Convergence to 0 of (126):
In the next Lemma we show that in the time interval
$[t(i)-\gamma_{n}^{-1}(i),t(i))$, the mutant sub-population of trait $i$ does
not growth exponentially fast yet. In particular we control its growth by the
factor $\psi_{n}(i)\log^{\theta(i-1)}(n)$, with a well chosen function
$\psi_{n}(i)$. Heuristically what happens is that the total number of mutant
cells of trait $i$ issued from a mutational event up to time $t(i)$ is of
order $\mathcal{O}\left(\log^{\theta(i-1)}(n)\right)$. Moreover with the
remaining time for the lineages of these mutant cells to grow, we succeed to
control the size of the mutant sub-population of trait $i$ by at most
$e^{\frac{\varphi_{n}(i)}{2}}\sqrt{\log(n)}\log^{\theta(i-1)}(n)$.
Consequently dividing by any function $\psi_{n}(i)$ satisfying
$e^{\varphi_{n}(i)}\log(n)=o(\psi^{2}_{n}(i))$ the asymptotic limits is $0$.

######  Lemma 3.5.

Let $i\in\mathbb{N}$,
$\gamma_{n}(i)=\frac{\log(n)}{\log(\log(n))\theta(i-1)+\varphi_{n}(i)}$ where
$\varphi_{n}(i)\underset{n\to\infty}{\to}\infty$ such that
$\gamma_{n}(i)\underset{n\to\infty}{\to}\infty$, $\psi_{n}(i)$ satisfying
$e^{\varphi_{n}(i)}\log(n)=o(\psi_{n}^{2}(i))$, and $s\in\mathbb{R}$. For
$i\geq 2$ we prove that if Proposition 3.1 is true for $i-1$ then

$\displaystyle\mathbb{P}\left(\sup_{t\in\left[t(i)-\gamma_{n}^{-1}(i),t(i)\right]}\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}{\psi_{n}(i)\log^{\theta(i-1)}(n)}\geq\varepsilon\right)\underset{n\to\infty}{\longrightarrow}0.$
(164)

For $i=1$ we prove (164) without condition.

###### Proof.

For all $t\in\left[t(i)-\gamma_{n}^{-1}(i),t(i)\right]$ we have

$\displaystyle\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}{\psi_{n}(i)\log^{\theta(i-1)}(n)}=\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)e^{-\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}-Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s\right)e^{-\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s\right)}}{\psi_{n}(i)\log^{\theta(i-1)}(n)e^{-\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}}$
(165) $\displaystyle\hskip
56.9055pt+\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s\right)e^{-\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s\right)}}{\psi_{n}(i)\log^{\theta(i-1)}(n)e^{-\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}}$
(166)
$\displaystyle=\frac{M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)-M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s\right)}{\psi_{n}(i)\log^{\theta(i-1)}(n)e^{-\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}}+\frac{\int_{\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s}^{\mathfrak{t}^{(n)}_{t}+s}2\alpha_{i-1}\mu_{i-1}^{(n)}e^{-\lambda_{i}^{(n)}u}Z_{i-1}^{(n)}(u)du}{\psi_{n}(i)\log^{\theta(i-1)}(n)e^{-\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}}$
(167) $\displaystyle\hskip
56.9055pt+\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s\right)}{\psi_{n}(i)\log^{\theta(i-1)}(n)e^{-\lambda_{i}^{(n)}\mathfrak{t}^{(n)}_{t-t(i)+\gamma_{n}^{-1}(i)}}}.$
(168)

It allows to write

$\displaystyle\mathbb{P}\left(\sup_{t\in\left[t(i)-\gamma_{n}^{-1}(i),t(i)\right]}\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}{\psi_{n}(i)\log^{\theta(i-1)}(n)}\geq
3\varepsilon\right)$ (169) $\displaystyle\hskip
28.45274pt\leq\mathbb{P}\left(\sup_{t\in\left[t(i)-\gamma_{n}^{-1}(i),t(i)\right]}\Bigg{|}\frac{M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)-M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s\right)}{\psi_{n}(i)\log^{\theta(i-1)}(n)e^{-\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}}\Bigg{|}\geq\varepsilon\right)$
(170) $\displaystyle\hskip
28.45274pt+\mathbb{P}\left(\sup_{t\in\left[t(i)-\gamma_{n}^{-1}(i),t(i)\right]}\frac{\int_{\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s}^{\mathfrak{t}^{(n)}_{t}+s}2\alpha_{i-1}\mu_{i-1}^{(n)}e^{-\lambda_{i}^{(n)}u}Z_{i-1}^{(n)}(u)du}{\psi_{n}(i)\log^{\theta(i-1)}(n)e^{-\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}}\geq\varepsilon\right)$
(171) $\displaystyle\hskip
28.45274pt+\mathbb{P}\left(\sup_{t\in\left[t(i)-\gamma_{n}^{-1}(i),t(i)\right]}\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s\right)}{\psi_{n}(i)\log^{\theta(i-1)}(n)e^{-\lambda_{i}^{(n)}\mathfrak{t}^{(n)}_{t-t(i)+\gamma_{n}^{-1}(i)}}}\geq\varepsilon\right).$
(172)

We have

$\displaystyle\eqref{Eq:
eq3}\leq\mathbb{P}\left(Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s\right)\geq
1\right),$ (173)

because one necessary condition for fulfilling the interesting condition is
that there is at least 1 mutant cell of trait $i$ at time
$\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s$. Then applying Lemma 3.4
gives that (172) converges to $0$.
The convergence to 0 of the term (171) is given by applying the following
Lemma. Notice in particular that $(\varphi_{n}(i),\gamma_{n}(i))$ satisfies
the condition of this lemma.

######  Lemma 3.6.

Let $i\in\mathbb{N}$,
$\gamma_{n}(i)=\frac{\log(n)}{\log(\log(n))\theta(i-1)+\varphi_{n}(i)}$ where
$\varphi_{n}(i)\underset{n\to\infty}{\to}\infty$ such that
$\gamma_{n}(i)\underset{n\to\infty}{\to}\infty$,
$\varphi_{n}(i)\underset{n\to\infty}{\to}\infty$ such that
$\log(n)=o(\psi_{n}(i)\gamma_{n}(i))$, and $s\in\mathbb{R}$. For $i\geq 2$ we
prove that if Proposition 3.1 is true for $i-1$ then

$\displaystyle\mathbb{P}\left(\sup_{t\in[t(i)-\gamma_{n}^{-1}(i),t(i)]}\frac{\int_{\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s}^{\mathfrak{t}^{(n)}_{t}+s}2\alpha_{i-1}\mu_{i-1}^{(n)}e^{-\lambda_{i}^{(n)}u}Z_{i-1}^{(n)}(u)du}{\psi_{n}(i)\log^{\theta(i-1)}(n)e^{-\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}}\geq\varepsilon\right)\underset{n\to\infty}{\longrightarrow}0.$
(174)

For $i=1$ we prove (174) without condition.

###### Proof of Lemma 3.6.

Let

$\displaystyle
a_{t}^{(n)}:=\frac{\int_{\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s}^{\mathfrak{t}^{(n)}_{t}+s}2\alpha_{i-1}\mu_{i-1}^{(n)}e^{-\lambda_{i}^{(n)}u}Z_{i-1}^{(n)}(u)du}{\psi_{n}(i)\log^{\theta(i-1)}(n)e^{-\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}}.$
(175)

Our aim is to prove that for all $\varepsilon>0$

$\displaystyle\mathbb{P}\left(\sup_{t\in\left[t(i)-\gamma_{n}^{-1}(i),t(i)\right]}a_{t}^{(n)}\leq\varepsilon\right)\underset{n\to\infty}{\to}1.$
(176)

(i) Case $i=1$: We have

$\displaystyle a_{t}^{(n)}$
$\displaystyle=\frac{e^{\lambda_{1}^{(n)}\left(\mathfrak{t}_{t}^{(n)}+s\right)}}{\psi_{n}(1)}\int_{\mathfrak{t}^{(n)}_{t(1)-\gamma_{n}^{-1}(1)}+s}^{\mathfrak{t}^{(n)}_{t}+s}2\alpha_{0}\mu_{0}^{(n)}\Bigg{[}W+\left(e^{-\lambda_{0}u}Z_{0}(u)-W\right)$
(177) $\displaystyle\hskip
199.16928pt+\left(e^{-\lambda_{0}^{(n)}u}Z_{0}^{(n)}(u)-e^{-\lambda_{0}u}Z_{0}(u)\right)\Bigg{]}e^{\left(\lambda_{0}^{(n)}-\lambda_{1}^{(n)}\right)u}du.$
(178)

Let us set

$\displaystyle
B_{\widetilde{\varepsilon}}^{(n)}:=\left\\{\sup_{u\in\mathbb{R}^{+}}\Bigg{|}e^{-\lambda_{0}u}Z_{0}(u)-e^{-\lambda_{0}^{(n)}u}Z_{0}^{(n)}(u)\Bigg{|}\leq\widetilde{\varepsilon}\right\\},$
(179) $\displaystyle
C_{x,\widetilde{\varepsilon}}:=\left\\{\sup_{u\in[x,\infty]}|e^{-\lambda_{0}u}Z_{0}(u)-W|\leq\widetilde{\varepsilon}\right\\}.$
(180)

According to Lemma 3.1 and by definition of $W$ (see (70)) we both have that
$\mathbb{P}\left(B_{\widetilde{\varepsilon}}^{(n)}\right)\underset{n\to\infty}{\to}1$
and
$\mathbb{P}\left(C_{\sqrt{\log(n)},\widetilde{\varepsilon}}\right)\underset{n\to\infty}{\to}1$.
Then for $n$ large enough under the event
$B_{\widetilde{\varepsilon}}^{(n)}\cap
C_{\sqrt{\log(n)},\widetilde{\varepsilon}}$ we have

$\displaystyle a_{t}^{(n)}$ $\displaystyle\leq
2\alpha_{0}\left(n^{t(1)}\mu_{0}^{(n)}\right)(W+2\widetilde{\varepsilon})I_{n},$
(181)

where
$I_{n}:=\frac{e^{\lambda_{1}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}}{\psi_{n}(1)n^{t(1)}}\int_{\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(1)}+s}^{\mathfrak{t}^{(n)}_{t}+s}e^{\left(\lambda_{0}-\lambda_{1}^{(n)}\right)u}du.$
In the case $\lambda_{1}<\lambda_{0}$ we have that

$\displaystyle
I_{n}\leq\frac{e^{\lambda_{1}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}}{\psi_{n}(1)n^{t(1)}}\frac{e^{\left(\lambda_{0}-\lambda_{1}^{(n)}\right)\left(\mathfrak{t}^{(n)}_{t}+s\right)}}{\lambda_{0}-\lambda_{1}}=\frac{e^{-\lambda_{0}\mathfrak{t}^{(n)}_{t(1)-t}}e^{\lambda_{0}s}}{\psi_{n}(1)(\lambda_{0}-\lambda_{1})}\leq\frac{e^{\lambda_{0}s}}{\psi_{n}(1)(\lambda_{0}-\lambda_{1})}.$
(182)

In the case $\lambda_{1}=\lambda_{0}$ remembering that
$\lambda_{1}^{(n)}=\lambda_{0}-2\alpha_{1}\mu_{1}^{(n)}$ we obtain

$\displaystyle I_{n}$
$\displaystyle\leq\frac{e^{\lambda_{0}s}e^{-2\alpha_{1}\mu_{1}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}}{\psi_{n}(1)}\frac{e^{2\alpha_{1}\mu_{1}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}-e^{2\alpha_{1}\mu_{1}^{(n)}\left(\mathfrak{t}^{(n)}_{t(1)-\gamma_{n}^{-1}(1)}+s\right)}}{2\alpha_{1}\mu_{1}^{(n)}}$
(183)
$\displaystyle=\frac{e^{\lambda_{0}s}}{\psi_{n}(1)}\frac{1-e^{-2\alpha_{1}\mu_{1}^{(n)}\mathfrak{t}^{(n)}_{t-t(1)+\gamma_{n}^{-1}(1)}}}{2\alpha_{1}\mu_{1}^{(n)}}$
(184)
$\displaystyle\leq\frac{e^{\lambda_{0}s}}{\psi_{n}(1)}\mathfrak{t}^{(n)}_{t-t(1)+\gamma_{n}^{-1}(1)}$
(185)
$\displaystyle\leq\frac{e^{\lambda_{0}s}\log(n)}{\psi_{n}(1)\gamma_{n}(1)\lambda_{0}},$
(186)

where for the second inequality we use the following equation applied with
$a=2\alpha_{1}\mu_{1}^{(n)}>0$ and
$x=\mathfrak{t}^{(n)}_{t-t(1)+\gamma_{n}^{-1}(1)}$

$\displaystyle\forall x\geq 0,\forall a>0,\frac{1-e^{-ax}}{a}\leq x.$ (187)

In any case, since $W$ is a finite random variable (see (71)) we find (176)
and conclude the case $i=1$.
(ii) Case $i\geq 2$: Assume Proposition 3.1 is true for $i-1$. In particular
we have
$\mathbb{P}\left(B_{\widetilde{\varepsilon}}^{(n)}\right)\underset{n\to\infty}{\to}1$
with

$\displaystyle
B_{\widetilde{\varepsilon}}^{(n)}:=\left\\{\sup_{v\in\left[t(i)-\gamma_{n}^{-1}(i),t(i)\right]}\Bigg{|}\frac{Z_{i-1}^{(n)}\left(\mathfrak{t}^{(n)}_{v}+s\right)}{n^{v-t(i-1)}e^{\lambda_{0}s}\log^{\theta(i-1)}(n)}-Ww_{i-1}(v)\Bigg{|}\leq\widetilde{\varepsilon}\right\\}.$
(188)

Using the change of variable $u=\mathfrak{t}_{v}^{(n)}+s$ and that
$t(i-1)=t(i)-\ell(i-1)$, notice that

$\displaystyle
a_{t}^{(n)}=\frac{e^{\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}}{\psi_{n}(i)n^{t(i)}}\int_{t(i)-\gamma_{n}^{-1}(i)}^{t}2\alpha_{i-1}\left(n^{\ell(i-1)}\mu_{i-1}^{(n)}\right)$
(189) $\displaystyle\hskip
113.81102pt\cdot\frac{Z_{i-1}^{(n)}\left(\mathfrak{t}^{(n)}_{v}+s\right)}{n^{v-t(i-1)}e^{\lambda_{0}s}\log^{\theta(i-1)}(n)}e^{\left(\lambda_{0}-\lambda_{i}^{(n)}\right)\left(\mathfrak{t}^{(n)}_{v}+s\right)}\frac{\log(n)}{\lambda_{0}}dv.$
(190)

Using that $w_{i-1}$ is a non decreasing function it comes that under the
event $B_{\widetilde{\varepsilon}}^{(n)}$

$\displaystyle a_{t}^{(n)}$ $\displaystyle\leq
2\alpha_{i-1}\left(n^{\ell(i-1)}\mu_{i-1}^{(n)}\right)\left(Ww_{i-1}(t(i))+\widetilde{\varepsilon}\right)\frac{e^{\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}}{\psi_{n}(i)n^{t(i)}}\int_{\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s}^{\mathfrak{t}^{(n)}_{t}+s}e^{\left(\lambda_{0}-\lambda_{i}^{(n)}\right)u}du.$
(191)

Using similar computations as in (182) and (183), it follows (176). ∎

Now we will prove that (170) converges to 0. We start by introducing two lemma
allowing to control in expectancy both the size of any mutant sub-population
and the quadratic variation associated of the martingale $M^{(n)}_{i}$.
First, a natural upper bound on the mean of the growth of each mutant sub-
population can be easily obtained. This is stated in the next Lemma.

######  Lemma 3.7.

For all $i\in\mathbb{N}_{0}$ it exists $C_{i}>0$ such that for all $u\geq 0$

$\displaystyle\mathbb{E}\left[Z_{i}^{(n)}(u)\right]$ $\displaystyle\leq
C_{i}\mu_{\otimes,i}^{(n)}u^{\theta(i)}e^{\lambda_{0}u},$ (192)

where
$\mu_{\otimes,i}^{(n)}:=\overset{i}{\underset{j=1}{\prod}}\mu_{j-1}^{(n)}$ and
$C_{i}:=\prod_{j=1}^{i}2\alpha_{j-1}\left(\mathbbm{1}_{\\{\lambda_{j}=\lambda_{0}\\}}+\mathbbm{1}_{\\{\lambda_{j}<\lambda_{0}\\}}\frac{1}{\lambda_{0}-\lambda_{j}}\right).$

Notice that there are 3 interesting components. The first one is the
mutational cost to get such mutant cells encoded via the term
$\mu_{\otimes,i}^{(n)}$. Then the second one is given by the contribution over
time of all neutral mutations in the path to the considered mutant population.
And the last one is simply the exponential growth at rate $\lambda_{0}$ given
by the wild-type sub-population.

###### Proof of Lemma 3.7.

First we have that
$\mathbb{E}\left[Z_{0}^{(n)}(u)\right]=e^{\lambda_{0}^{(n)}u}\leq
e^{\lambda_{0}u}$, which is exactly the result for $i=0$.
Then for $i\in\mathbb{N}$ assume that the result is true for $i-1$. Then
taking the expectation of the martingale $M_{i}^{(n)}(u)$ defined in (113) and
using the previous assumption we obtain the following equation

$\displaystyle\mathbb{E}\left[Z_{i}^{(n)}(u)\right]$
$\displaystyle=e^{\lambda_{i}^{(n)}u}\int_{0}^{u}2\alpha_{i-1}\mu_{i-1}^{(n)}e^{-\lambda_{i}^{(n)}s}\mathbb{E}\left[Z_{i-1}^{(n)}(s)\right]ds$
(193) $\displaystyle\leq
e^{\lambda_{i}u}\int_{0}^{u}2\alpha_{i-1}\mu_{i-1}^{(n)}e^{-\lambda_{i}s}\mathbb{E}\left[Z_{i-1}^{(n)}(s)\right]ds$
(194) $\displaystyle\leq
C_{i-1}\mu_{\otimes,i}^{(n)}2\alpha_{i-1}\int_{0}^{u}e^{(\lambda_{0}-\lambda_{i})s}dsu^{\theta(i-1)}e^{\lambda_{i}u}$
(195) $\displaystyle\leq
C_{i-1}\mu_{\otimes,i}^{(n)}2\alpha_{i-1}\left(\mathbbm{1}_{\\{\lambda_{i}=\lambda_{0}\\}}u+\mathbbm{1}_{\\{\lambda_{i}<\lambda_{0}\\}}\frac{1}{\lambda_{0}-\lambda_{i}}e^{\left(\lambda_{0}-\lambda_{i}\right)u}\right)u^{\theta(i-1)}e^{\lambda_{i}u}$
(196)
$\displaystyle=C_{i-1}\mu_{\otimes,i}^{(n)}2\alpha_{i-1}\left(\mathbbm{1}_{\\{\lambda_{i}=\lambda_{0}\\}}+\mathbbm{1}_{\\{\lambda_{i}<\lambda_{0}\\}}\frac{1}{\lambda_{0}-\lambda_{i}}\right)u^{\theta(i)}e^{\lambda_{0}u},$
(197)

which concludes the proof by induction. ∎

Second, using both the expression of the quadratic variation of the martingale
associated to a mutant sub-population given in Equation (114) and the previous
Lemma 3.7, a natural upperbound on its mean is obtained and summed up in the
next Lemma.

######  Lemma 3.8.

Let $0<t_{1}^{(n)}<t_{2}$ and $s\in\mathbb{R}$. It exists $N\in\mathbb{N}$ and
$C>0$ such that for all $n\geq N$ we have

$\displaystyle\mathbb{E}\left[\left\langle
M_{i}^{(n)}\right\rangle_{\mathfrak{t}^{(n)}_{t_{2}}+s}-\left\langle
M_{i}^{(n)}\right\rangle_{\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s}\right]$ (198)
$\displaystyle\leq
C\mu_{\otimes,i}^{(n)}\Bigg{[}\mathbbm{1}_{\\{\lambda_{i}=\lambda_{0}\\}}\frac{e^{-\lambda_{0}s}\left(\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s\right)^{\theta(i)}}{n^{t_{1}^{(n)}}}+\left(\mathfrak{t}^{(n)}_{t_{2}}+s\right)^{\theta(i)}\Bigg{(}\mathbbm{1}_{\\{\lambda_{0}>2\lambda_{i}\\}}e^{\left(\lambda_{0}-2\lambda_{i}\right)\left(\mathfrak{t}^{(n)}_{t_{2}}+s\right)}$
(199) $\displaystyle\hskip
56.9055pt+\mathbbm{1}_{\\{\lambda_{0}=2\lambda_{i}\\}}\left(\mathfrak{t}^{(n)}_{t_{2}}+s\right)+\mathbbm{1}_{\\{\lambda_{i}<\lambda_{0}<2\lambda_{i}\\}}e^{-\left(2\lambda_{i}-\lambda_{0}\right)\left(\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s\right)}\Bigg{)}\Bigg{]}.$
(200)

###### Proof.

In the proof $C$ corresponds to a strictly positive constant that may change
from line to line.
Assume that $\lambda_{i}=\lambda_{0}$. Applying Lemma 3.7, remembering that
$\lambda_{i}^{(n)}=\lambda_{0}-2\alpha_{i}\mu_{i}^{(n)}$, and using that it
exists $N_{1}\in\mathbb{N}$ such that for all $n\geq N_{1}$ we have that
$e^{4\alpha_{i}\mu_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t_{2}}+s\right)}\leq 2$,
we obtain that

$\displaystyle\int_{\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s}^{\mathfrak{t}^{(n)}_{t_{2}}+s}e^{-2\lambda_{i}^{(n)}u}\mathbb{E}\left[Z_{i}^{(n)}(u)\right]du$
$\displaystyle\leq
C\mu_{\otimes,i}^{(n)}\int_{\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s}^{\mathfrak{t}^{(n)}_{t_{2}}+s}u^{\theta(i)}e^{-\lambda_{0}u}du.$
(201)

Using an integration by parts we obtain that

$\displaystyle\int_{\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s}^{\mathfrak{t}^{(n)}_{t_{2}}+s}u^{\theta(i)}e^{-\lambda_{0}u}du\leq\frac{1}{\lambda(0)}\left(\mathfrak{t}^{(n)}_{t^{(n)}_{1}}+s\right)^{\theta(i)}e^{-\lambda_{0}\left(\mathfrak{t}^{(n)}_{t^{(n)}_{1}}+s\right)}+\frac{1}{\lambda_{0}}\int_{\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s}^{\mathfrak{t}^{(n)}_{t_{2}}+s}u^{\theta(i)-1}e^{-\lambda_{0}u}du.$
(202)

Then using $\theta(i)$ integration by parts, it exists $N_{2}\in\mathbb{N}$
such that for $n\geq N_{2}$ we have

$\displaystyle\int_{\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s}^{\mathfrak{t}^{(n)}_{t_{2}}+s}u^{\theta(i)}e^{-\lambda_{0}u}du\leq
C\frac{e^{-\lambda_{0}s}}{n^{t_{1}^{(n)}}}\left(\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s\right)^{\theta(i)}.$
(203)

It follows that for $n\geq\max(N_{1},N_{2})$

$\displaystyle\int_{\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s}^{\mathfrak{t}^{(n)}_{t_{2}}+s}e^{-2\lambda_{i}^{(n)}u}\mathbb{E}\left[Z_{i}^{(n)}(u)\right]du$
$\displaystyle\leq
C\frac{e^{-\lambda_{0}s}}{n^{t_{1}^{(n)}}}\mu_{\otimes,i}^{(n)}\left(\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s\right)^{\theta(i)}.$
(204)

Because the vertex $i$ is assumed to be neutral it follows that
$\theta(i-1)=\theta(i)-1$. Using similar computation as the latter one, it
exists $N_{3}\in\mathbb{N}$ such that for $n\geq N_{3}$ we have

$\displaystyle\int_{\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s}^{\mathfrak{t}^{(n)}_{t_{2}}+s}\mu_{i-1}^{(n)}e^{-2\lambda_{i}^{(n)}u}\mathbb{E}\left[Z_{i-1}^{(n)}(u)\right]du\leq
C\frac{e^{-\lambda_{0}s}}{n^{t_{1}^{(n)}}}\mu_{\otimes,i}^{(n)}\left(\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s\right)^{\theta(i)-1}.$
(205)

It follows that for all $n\geq\max(N_{1},N_{2},N_{3})$ we have

$\displaystyle\mathbb{E}\left[\left\langle
M_{i}^{(n)}\right\rangle_{\mathfrak{t}^{(n)}_{t_{2}}+s}-\left\langle
M_{i}^{(n)}\right\rangle_{\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s}\right]$
$\displaystyle\leq
C\frac{e^{-\lambda_{0}s}}{n^{t_{1}^{(n)}}}\mu_{\otimes,i}^{(n)}\left(\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s\right)^{\theta(i)}.$
(206)

We now deal with the case $\lambda_{i}<\lambda_{0}$ by applying the same
strategy. We obtain

$\displaystyle\int_{\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s}^{\mathfrak{t}^{(n)}_{t_{2}}+s}e^{-2\lambda_{i}^{(n)}u}\mathbb{E}\left[Z_{i}^{(n)}(u)\right]du$
(207) $\displaystyle\leq
C\mu_{\otimes,i}^{(n)}\int_{\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s}^{\mathfrak{t}^{(n)}_{t_{2}}+s}u^{\theta(i)}e^{\left(\lambda_{0}-2\lambda_{i}\right)u}du$
(208) $\displaystyle\leq
C\mu_{\otimes,i}^{(n)}\left(\mathfrak{t}^{(n)}_{t_{2}}+s\right)^{\theta(i)}\Bigg{[}\mathbbm{1}_{\\{\lambda_{0}>2\lambda_{i}\\}}e^{\left(\lambda_{0}-2\lambda_{i}\right)\left(\mathfrak{t}^{(n)}_{t_{2}}+s\right)}$
(209) $\displaystyle\hskip
113.81102pt+\mathbbm{1}_{\\{\lambda_{0}=2\lambda_{i}\\}}t^{(n)}\left(\mathfrak{t}^{(n)}_{t_{2}}+s\right)+\mathbbm{1}_{\\{\lambda_{i}<\lambda_{0}<2\lambda_{i}\\}}e^{-\left(2\lambda_{i}-\lambda_{0}\right)\left(\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s\right)}\Bigg{]}.$
(210)

Then remembering that $\theta(i-1)=\theta(i)$, we get that

$\displaystyle\int_{\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s}^{\mathfrak{t}^{(n)}_{t_{2}}+s}\mu_{i-1}^{(n)}e^{-2\lambda_{i}^{(n)}u}\mathbb{E}\left[Z_{i-1}^{(n)}(u)\right]du$
(211) $\displaystyle\leq
C\mu_{\otimes,i}^{(n)}\left(\mathfrak{t}^{(n)}_{t_{2}}+s\right)^{\theta(i)}\Bigg{[}\mathbbm{1}_{\\{\lambda_{0}>2\lambda_{i}\\}}e^{\left(\lambda_{0}-2\lambda_{i}\right)\left(\mathfrak{t}^{(n)}_{t_{2}}+s\right)}$
(212) $\displaystyle\hskip
113.81102pt+\mathbbm{1}_{\\{\lambda_{0}=2\lambda_{i}\\}}t^{(n)}\left(\mathfrak{t}^{(n)}_{t_{2}}+s\right)+\mathbbm{1}_{\\{\lambda_{i}<\lambda_{0}<2\lambda_{i}\\}}e^{-\left(2\lambda_{i}-\lambda_{0}\right)\left(\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s\right)}\Bigg{]}.$
(213)

At the end we have

$\displaystyle\mathbb{E}\left[\left\langle
M_{i}^{(n)}\right\rangle_{\mathfrak{t}^{(n)}_{t_{2}}+s}-\left\langle
M_{i}^{(n)}\right\rangle_{\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s}\right]$
$\displaystyle\leq
C\mu_{\otimes,i}^{(n)}\left(\mathfrak{t}^{(n)}_{t_{2}}+s\right)^{\theta(i)}\cdot\Bigg{[}\mathbbm{1}_{\\{\lambda_{0}>2\lambda_{i}\\}}e^{\left(\lambda_{0}-2\lambda_{i}\right)\left(\mathfrak{t}^{(n)}_{t_{2}}+s\right)}$
(214)
$\displaystyle+\mathbbm{1}_{\\{\lambda_{0}=2\lambda_{i}\\}}\left(\mathfrak{t}^{(n)}_{t_{2}}+s\right)+\mathbbm{1}_{\\{\lambda_{i}<\lambda_{0}<2\lambda_{i}\\}}e^{-\left(2\lambda_{i}-\lambda_{0}\right)\left(\mathfrak{t}^{(n)}_{t_{1}^{(n)}}+s\right)}\Bigg{]}.$
(215)

∎

Now we can prove that (170) converges to 0. Using the Maximal inequality, see
[10] Chapter VI page 72, applied to the supermartingale
$\left[\frac{M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)-M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s\right)}{\psi_{n}(i)\log^{\theta(i-1)}(n)e^{-\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}}\right]_{t\geq
t(i)-\gamma_{n}^{-1}(i)}$ it follows

$\displaystyle\eqref{Eq:
eq1}\leq\frac{3}{\varepsilon\psi_{n}(i)\log^{\theta(i-1)}(n)}\sup_{t\in\left[t(i)-\gamma_{n}^{-1}(i),t(i)\right]}e^{\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}\mathbb{E}\left[\left\langle
M_{i}^{(n)}\right\rangle_{\mathfrak{t}^{(n)}_{t}+s}-\left\langle
M_{i}^{(n)}\right\rangle_{\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s}\right]^{\frac{1}{2}}.$
(216)

First notice that the function

$t\mapsto
f^{(n)}(t):=e^{\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}\mathbb{E}\left[\left\langle
M_{i}^{(n)}\right\rangle_{\mathfrak{t}^{(n)}_{t}+s}-\left\langle
M_{i}^{(n)}\right\rangle_{\mathfrak{t}^{(n)}_{t(i)-\gamma_{n}^{-1}(i)}+s}\right]^{\frac{1}{2}}$

is a non-decreasing function implying that

$\displaystyle\sup_{t\in\left[t(i)-\gamma_{n}^{-1}(i),t(i)\right]}f^{(n)}(t)=f^{(n)}(t(i)).$
(217)

In the case $\lambda_{i}=\lambda_{0}$, using that it exists a constant $C$
such that $e^{\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)}\leq
Cn^{t(i)}e^{\lambda_{0}s}$ and according to Lemma 3.8 applied with
$t_{1}^{(n)}=t(i)-\gamma_{n}^{-1}(i)$ and $t_{2}=t(i)$ we have that

$\displaystyle f^{(n)}(t(i))\leq
C\left(e^{\lambda_{0}s}\left(n^{t(i)}\mu_{\otimes,i}^{(n)}\right)\right)^{\frac{1}{2}}n^{\frac{\gamma_{n}^{-1}(i)}{2}}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)^{\frac{\theta(i)}{2}}.$
(218)

Notice that

$\displaystyle
n^{t(i)}\mu_{\otimes,i}^{(n)}=\prod_{j=1}^{i}n^{\ell(j-1)}\mu_{j-1}^{(n)}\underset{n\to\infty}{\longrightarrow}\prod_{j=1}^{i}\mu_{j-1}<\infty.$
(219)

Then for $n$ large enough, and remembering that $\theta(i)=\theta(i-1)+1$ we
have

$\displaystyle\eqref{Eq: eq1}\leq
C\left(\frac{e^{\frac{\log(n)}{\gamma_{n}(i)}}\log(n)}{\psi_{n}^{2}(i)\log^{\theta(i-1)}(n)}\right)^{\frac{1}{2}}.$
(220)

And we have

$\displaystyle\frac{e^{\frac{\log(n)}{\gamma_{n}(i)}}\log(n)}{\psi_{n}^{2}(i)\log^{\theta(i-1)}(n)}=\frac{e^{\varphi_{n}(i)}\log(n)}{\psi_{n}^{2}(i)}.$
(221)

We obtain that (170) converges to 0 by hypothesis on $\psi_{n}(i)$.
In the case $\lambda_{i}<\lambda_{0}$, using that it exists a constant $C>0$
such that $e^{\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)}\leq
Ce^{\lambda_{i}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)}$ and according to
Lemma 3.8 applied with $t_{1}^{(n)}=t(i)-\gamma_{n}^{-1}(i)$ and $t_{2}=t(i)$
we have that

$\displaystyle f^{(n)}(t(i))$ $\displaystyle\leq
Ce^{\lambda_{i}s}\left(n^{t(i)}\mu_{\otimes,i}^{(n)}\right)^{\frac{1}{2}}\Bigg{(}\mathbbm{1}_{\\{\lambda_{0}>2\lambda_{i}\\}}n^{\frac{\lambda_{0}-2\lambda_{i}}{\lambda_{0}}t(i)}e^{\left(\lambda_{0}-2\lambda_{i}\right)s}$
(222)
$\displaystyle+\mathbbm{1}_{\\{\lambda_{0}=2\lambda_{i}\\}}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)+\mathbbm{1}_{\\{\lambda_{i}<\lambda_{0}<2\lambda_{i}\\}}n^{-\frac{2\lambda_{i}-\lambda_{0}}{\lambda_{0}}\left(t(i)-\gamma_{n}^{-1}(i)\right)}e^{-(2\lambda_{i}-\lambda_{0})s}\Bigg{)}^{\frac{1}{2}}$
(223) $\displaystyle\cdot
n^{-\frac{\lambda_{0}-2\lambda_{i}}{2\lambda_{0}}t(i)}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)^{\frac{\theta(i)}{2}}.$
(224)

Then for $n$ large enough, and remembering that $\theta(i)=\theta(i-1)$ we
have

$\displaystyle\eqref{Eq: eq1}\leq
C\left(\mathbbm{1}_{\\{\lambda_{0}>2\lambda_{i}\\}}+\mathbbm{1}_{\\{\lambda_{0}=2\lambda_{i}\\}}\log(n)+\mathbbm{1}_{\\{\lambda_{i}<\lambda_{0}<2\lambda_{i}\\}}e^{\frac{2\lambda_{i}-\lambda_{0}}{\lambda_{0}}\frac{\log(n)}{\gamma_{n}(i)}}\right)^{\frac{1}{2}}\left(\frac{1}{\psi_{n}^{2}(i)\log^{\theta(i-1)}(n)}\right)^{\frac{1}{2}}.$
(225)

Then by hypothesis on $(\psi_{n}(i),\gamma_{n}(i))$ it follows that (170)
converges to 0. ∎

Step 3: Convergence to 0 of (127):
Notice that for all $t\geq t(i)$

$\displaystyle\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}{n^{t-t(i)}\log^{\theta(i)}(n)e^{\lambda_{0}s}}=\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)e^{-\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}-Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)e^{-\lambda_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)}}{n^{-t(i)}\log^{\theta(i)}(n)e^{\left(\lambda_{0}-\lambda_{i}^{(n)}\right)\left(\mathfrak{t}^{(n)}_{t}+s\right)}}$
(226) $\displaystyle\hskip
85.35826pt+\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)}{\log^{\theta(i)}(n)e^{\left(\lambda_{0}-\lambda_{i}^{(n)}\right)\mathfrak{t}^{(n)}_{t-t(i)}}e^{\lambda_{0}s}}$
(227)
$\displaystyle=\frac{M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)-M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)+\int_{\mathfrak{t}^{(n)}_{t(i)}+s}^{\mathfrak{t}^{(n)}_{t}+s}2\alpha_{i-1}\mu_{i-1}^{(n)}Z_{i-1}^{(n)}(u)e^{-\lambda_{i}^{(n)}u}du}{n^{-t(i)}\log^{\theta(i)}(n)e^{\left(\lambda_{0}-\lambda_{i}^{(n)}\right)\left(\mathfrak{t}^{(n)}_{t}+s\right)}}$
(228) $\displaystyle\hskip
85.35826pt+\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)}{\log^{\theta(i)}(n)e^{\left(\lambda_{0}-\lambda_{i}^{(n)}\right)\mathfrak{t}^{(n)}_{t-t(i)}}e^{\lambda_{0}s}}.$
(229)

Then it allows to write

$\displaystyle\mathbb{P}\left(\sup_{t\in\left[t(i),T\right]}\Bigg{|}\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)}{n^{t-t(i)}\log^{\theta(i)}(n)e^{\lambda_{0}s}}-Ww_{i}(t)\Bigg{|}\geq
3\varepsilon\right)$ (230) $\displaystyle\hskip
85.35826pt\leq\mathbb{P}\left(\sup_{t\in\left[t(i),T\right]}\Bigg{|}\frac{M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)-M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)}{n^{-t(i)}\log^{\theta(i)}(n)e^{\left(\lambda_{0}-\lambda_{i}^{(n)}\right)\left(\mathfrak{t}^{(n)}_{t}+s\right)}}\Bigg{|}\geq\varepsilon\right)$
(231) $\displaystyle\hskip
85.35826pt+\mathbb{P}\left(\sup_{t\in\left[t(i),T\right]}\Bigg{|}\frac{\int_{\mathfrak{t}^{(n)}_{t(i)}+s}^{\mathfrak{t}^{(n)}_{t}+s}2\alpha_{i-1}\mu_{i-1}^{(n)}Z_{i-1}^{(n)}(u)e^{-\lambda_{i}^{(n)}u}du}{n^{-t(i)}\log^{\theta(i)}(n)e^{\left(\lambda_{0}-\lambda_{i}^{(n)}\right)\left(\mathfrak{t}^{(n)}_{t}+s\right)}}-Ww_{i}(t)\Bigg{|}\geq\varepsilon\right)$
(232) $\displaystyle\hskip
85.35826pt+\mathbb{P}\left(\sup_{t\in\left[t(i),T\right]}\frac{Z_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)}{\log^{\theta(i)}(n)e^{\left(\lambda_{0}-\lambda_{i}^{(n)}\right)\mathfrak{t}^{(n)}_{t-t(i)}}e^{\lambda_{0}s}}\geq\varepsilon\right).$
(233)

We will show that every terms (231), (232) and (233) converge to $0$ when $n$
goes to infinity.
Concerning the term (231), we use first that
$\lambda_{0}\geq\lambda_{i}^{(n)}$ to simplify the denominator. Then we apply
Doob’s inequality to the martingale
$\left(M_{i}^{(n)}\left(\mathfrak{t}_{t}^{(n)}+s\right)-M_{i}^{(n)}\left(\mathfrak{t}_{t(i)}^{(n)}+s)\right)\right)_{t\geq
t(i)}$, and we use that if $M$ is a square integrable martingale then
$\mathbb{E}\left[(M(t)-M(s))^{2}\right]=\mathbb{E}[M^{2}(t)-M^{2}(s)]=\mathbb{E}[\langle
M\rangle_{t}-\langle M\rangle_{s}]$. Then using Equation (114), it follows

$\displaystyle\mathbb{P}\left(\sup_{t\in\left[t(i),T\right]}\Bigg{|}\frac{M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)-M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)}{n^{-t(i)}\log^{\theta(i)}(n)e^{\left(\lambda_{0}-\lambda_{i}^{(n)}\right)\left(\mathfrak{t}^{(n)}_{t}+s\right)}}\Bigg{|}\geq\varepsilon\right)$
(234)
$\displaystyle\leq\mathbb{P}\left(\sup_{t\in\left[t(i),T\right]}\Bigg{|}\frac{M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)-M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)}{n^{-t(i)}\log^{\theta(i)}(n)}\Bigg{|}\geq\varepsilon\right)$
(235)
$\displaystyle\leq\frac{4n^{2t(i)}}{\varepsilon^{2}\log^{2\theta(i)}(n)}\mathbb{E}\left[\left\\{M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{T}+s\right)-M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)\right\\}^{2}\right]$
(236)
$\displaystyle=\frac{4n^{2t(i)}}{\varepsilon^{2}\log^{2\theta(i)}(n)}\mathbb{E}\left[\left\langle
M_{i}^{(n)}\right\rangle_{\mathfrak{t}^{(n)}_{T}+s}-\left\langle
M_{i}^{(n)}\right\rangle_{\mathfrak{t}^{(n)}_{t(i)}+s}\right].$ (237)

Applying Lemma 3.8 at times $t_{1}^{(n)}=t(i)$ and $t_{2}=T$ we obtain that

$\displaystyle\mathbb{E}\left[\left\langle
M_{i}^{(n)}\right\rangle_{\mathfrak{t}^{(n)}_{T}+s}-\left\langle
M_{i}^{(n)}\right\rangle_{\mathfrak{t}^{(n)}_{t(i)}+s}\right]\leq
C\frac{e^{-\lambda_{0}s}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)^{\theta(i)}\mu_{\otimes,i}^{(n)}}{n^{t(i)}}.$
(238)

Then combining (234) and (238) we get

$\displaystyle\mathbb{P}\left(\sup_{t\in\left[t(i),T\right]}\Bigg{|}\frac{M_{(i}^{(n)}\left(\mathfrak{t}^{(n)}_{t}+s\right)-M_{i}^{(n)}\left(\mathfrak{t}^{(n)}_{t(i)}+s\right)}{n^{-t(i)}\log^{\theta(i)}(n)e^{\left(\lambda_{0}-\lambda_{i}^{(n)}\right)\left(\mathfrak{t}^{(n)}_{t}+s\right)}}\Bigg{|}\geq\varepsilon\right)\leq\frac{4Ce^{-\lambda_{0}s}}{\varepsilon^{2}\log^{\theta(i)}(n)}\left(\frac{\mathfrak{t}^{(n)}_{t(i)}+s}{\log(n)}\right)^{\theta(i)}n^{t(i)}\mu_{\otimes,i}^{(n)}$
(239) $\displaystyle\underset{n\to\infty}{\longrightarrow}0,$ (240)