# The Symmetric Coalescent and Wright-Fisher models with bottlenecks

Adrián González Casanova, Verónica Miró Pina, Arno Siri-Jégousse
_Corresponding author:_<EMAIL_ADDRESS>
###### Abstract

We define a new class of $\Xi$-coalescents characterized by a possibly
infinite measure over the non negative integers. We call them symmetric
coalescents since they are the unique family of exchangeable coalescents
satisfying a symmetry property on their coagulation rates: they are invariant
under any transformation that consists of moving one element from one block to
another without changing the total number of blocks. We illustrate the
diversity of behaviors of this family of processes by introducing and studying
a one parameter subclass, the $(\beta,S)$-coalescents. We also embed this
family in a larger class of $\Xi$-coalescents arising as the limit genealogies
of Wright-Fisher models with bottlenecks. Some convergence results rely on a
new Skorokhod type metric, that induces the Meyer-Zheng topology, which allows
us to study the scaling limit of non-Markovian processes using standard
techniques.

##  1 Introduction

###  1.1 Wright-Fisher models with demographic bottlenecks

Since it was proposed in 1982, the Kingman coalescent [18] has become a key
tool in population genetics. It can describe the limit genealogy of classical
models such as the Wright-Fisher and the Moran model. It has proven to be
robust to modifications of these models’ assumptions (such as constant
population size or random mating) and thus arises as the genealogy of a broad
class of population models. However, it does not model well genealogies from
certain populations, e.g. with a skewed offspring distribution, which are
captured by coalescents with (simultaneous) multiple collisions [25].

Modeling populations with varying population size has been of great interest,
for example to infer the human population history [19, 34]. Several variations
of the Wright-Fisher model with fluctuating population size have been studied,
in which the population size changes but remains of order $N$. It has been
shown that in many cases, the genealogy converges to a continuous time-
rescaling of the Kingman coalescent (see for example [14, 16, 15]). More
recently, Freund [11] studied the case of Cannings models with highly variant
offspring number (whose genealogy is usually described by a
$\Lambda$-coalescent) in which the population size fluctuates (but remains of
the order of $N$) and has shown that the genealogy converges to a time-
rescaled $\Lambda$-coalescent.

In Section 6.1 of [5], Birkner et al. consider a population undergoing
recurrent demographic bottlenecks. We call a bottleneck an event that reduces
substantially the population size and that may last for one or several
generations. They suggest that the genealogy is described by a discontinuous
time-rescaling of the Kingman coalescent, more precisely a Kingman coalescent
where time is rescaled by a subordinator, and which is in fact a coalescent
with simultaneous multiple collisions. But they only consider the case where
the population size during the bottleneck is small compared to $N$ but still
tends to infinity as $N\to\infty$.

To our knowledge, the case of drastic fluctuations, in which the population
size during the bottleneck does not tend to infinity as $N\to\infty$, has not
been studied yet. In this article, we are going to study different types of
bottlenecks, with different scalings for the population sizes inside and
outside the bottleneck, and different lengths. We will establish a
classification of the limiting genealogies obtained in the different settings
and give some intuitions on how to relate the different processes. To do so,
we define a class of models that can be called Wright-Fisher models with
demographic bottlenecks.

######  Definition 1.1 (The Wright-Fisher model with bottlenecks).

The Wright-Fisher model with bottlenecks (parametrized by $N\in\mathbb{N}$)
has varying population size, which is given by a sequence of random variables
$\\{R^{N}_{g}\\}_{g\in\mathbb{Z}_{+}}$ taking values in $[N]=\\{1,\dots,N\\}$.
It is the random graph $(V,E)$ where
$V=\\{(i,g):i\in[R_{g}^{N}],g\in\mathbb{Z}_{+}\\}$, each individual $(i,g)\in
V$ chooses her parent uniformly amongst the $R_{g-1}^{N}$ individuals of
generation $g-1$, and the set of edges is $E:=\\{((j,g-1)(i,g)):(j,g-1)$ is
the parent of $(i,g),g\in\mathbb{Z}_{+}\\}$.

The case $\mathbb{P}(R^{N}_{g}=N)=1,\forall g\in\mathbb{Z}_{+}$, is the
classical Wright-Fisher model and it is well known that, when the time is
rescaled by $N$ and $N\to\infty$, the genealogy of a sample of $n$ individuals
is described by the Kingman coalescent. We are led to ask ourselves under
which conditions on $\\{R^{N}_{g}\\}_{g\in\mathbb{Z}_{+}}$ does the genealogy
still converge to a Kingman coalescent, and if it does not, what type of
coalescents describe the genealogy of a population that has undergone
bottlenecks.

We are going to study different types of Wright-Fisher models with
bottlenecks, with different types of laws for the sequence
$\\{R^{N}_{g}\\}_{g\in\mathbb{Z}_{+}}$. Inspired by [5] (Section 6, p. 57), we
are going to describe the demographic history of the population by three
random sequences of i.i.d. positive real numbers: $\\{s_{i,N}\\}_{i\in N}$,
$\\{l_{i,N}\\}_{i\in N}$ and $\\{b_{i,N}\\}_{i\in N}$. The sequence of
population sizes $\\{R^{N}_{g}\\}_{g\in\mathbb{Z}_{+}}$ is given by

$R^{N}_{g}\ =\ \left\\{\begin{array}[]{ll}b_{m,N}N\textrm{ if
}\sum_{i=1}^{m-1}(s_{i,N}+l_{i,N})+s_{m,N}<g\leq\sum_{i=1}^{m}(s_{i,N}+l_{i,N})\\\
N\textrm{ otherwise.}\end{array}\right.$

This means that the population size stays at $N$ for $s_{i,N}$ generations and
then it is reduced to $b_{i,N}N$ for $l_{i,N}$ generations. At the end of the
bottleneck, the population reaches $N$ again and it stays until the next
event. Note that we have assumed that the decline and the re-growth of the
population size are instantaneous.

We call $b_{i,N}$ the intensity of the $i$-th bottleneck and we distinguish
between:

  * •

Soft bottlenecks, where $b_{i,N}\to 0$ in distribution but $Nb_{i,N}\to\infty$
in distribution i.e. the population size during the bottleneck is small
compared to $N$, but still large in absolute numbers.

  * •

Drastic bottlenecks, where $b_{i,N}\to 0$ in distribution and
$Nb_{i,N}<\infty$ in distribution (as $N\to\infty$) i.e. the population size
during the bottleneck is very small compared to $N$, and remains finite in the
limiting scenario, when the population size outside the bottlenecks is
infinite.

We call $l_{i,N}$ the duration of the $i$-th bottleneck. We will distinguish
between short bottlenecks, that last for only one generation and long
bottlenecks that last for several generations. In both cases, we will assume
that there exists $\alpha\in(0,1]$ such that $l_{i,N}N^{\alpha}\to 0$ as
$N\to\infty$, in distribution, i.e. when time is re-scaled by $N^{\alpha}$ the
duration of the bottleneck is negligible. Finally, we call $s_{i,N}$ the
periodicity of the bottlenecks and again, we distinguish between frequent
bottlenecks when, $s_{i,N}/N\to 0$ in distribution as $N\to\infty$ and rare
bottlenecks otherwise.

As we shall see, coalescents with simultaneous multiple collisions arise as
the limiting genealogies for the Wright-Fisher model with bottlenecks. In the
case of short drastic bottlenecks, the genealogies are described by a new
family of coalescents that we define and study.

###  1.2 A new family of $\Xi$-coalescents

Coalescents with simultaneous multiple collisions ($\Xi$-coalescents, [30, 25,
2, 3]) form the widest class of exchangeable coagulating Markov chains with
values in the set of partitions of $\mathbb{N}$. Mathematically, they give a
nice connection with de Finetti’s representation of exchangeable partitions
and they exhibit a rich variety of behaviors. Biologically, they describe the
genealogy of a large class of population models and their study provides some
statistical tools for inference. Schweinsberg [30] showed that any
exchangeable coalescent is characterized by a finite measure $\Xi$ on the
ranked infinite simplex

$\Delta=\\{\mathbf{\zeta}=(\zeta_{1},\zeta_{2}.\dots),\
\zeta_{1}\geq\zeta_{2}\geq\dots\geq 0,\ \sum_{i=1}^{\infty}\zeta_{i}\leq
1\\}.$

Its dynamics are described as follows. We decompose $\Xi$ into a ‘Kingman
part’ and a ‘simultaneous multiple collisions’ part, i.e.
$\Xi=a\delta_{(0,0,\dots)}+\Xi^{0}$ with $a\in[0,\infty)$ and
$\Xi^{0}(\\{(0,0,\dots)\\})=0$. A $[b,(k_{1},\dots,k_{r}),s]$-collision is a
merger of $b$ blocks into $r$ new blocks and $s$ unchanged blocks. Each new
block contains $k_{1},\dots,k_{r}\geq 2$ original blocks, so that
$\sum_{i=1}^{r}k_{i}=b-s$. Note that the order of the $k_{1},\dots,k_{r}$ does
not matter. Each $[b,(k_{1},\dots,k_{r}),s]$-collision happens at some fixed
rate

$\lambda_{b,(k_{1},\dots,k_{r}),s}=\ a\
\mathds{1}_{\\{r=1,s=b-1\\}}+\int_{\Delta}\sum_{l=0}^{s}\binom{s}{l}(1-\sum_{i\geq
1}\zeta_{i})^{s-l}\sum_{i_{1}\neq\dots\neq
i_{r+l}}\zeta_{i_{i}}^{k_{1}}\dots\zeta_{i_{r}}^{k_{r}}\zeta_{i_{r+1}}\dots\zeta_{i_{r+l}}\frac{\Xi^{0}(d\zeta)}{(\zeta,\zeta)}$
(1.1)

where $(\zeta,\zeta):=\sum_{i\geq 1}\zeta_{i}^{2}$. This complicated formula
is resulting from colliding original blocks according to a Kingman’s paintbox
associated with a partition $\zeta$ drawn from the $\sigma$-finite measure
$\Xi^{0}(d\zeta)/(\zeta,\zeta)$. This complexity justifies the necessity to
consider subclasses of exchangeable coalescents that are easier to study.

When $\Xi$ only puts weight on the subset of mass-partitions having only one
positive element, formula (1.1) reduces considerably as there is now no
possibility to obtain simultaneous collisions. The resulting subclass of
exchangeable coalescents is that of coalescents with multiple collisions
($\Lambda$-coalescents, [27, 28]). Their elegant theory, built around their
one to one correspondence with finite measures in $[0,1]$, turned this family
into the most studied class of coalescent processes for twenty years now. In
particular, $Beta$-coalescents [31] provide a one-parameter family of
$\Lambda$-coalescents, more convenient to study and better calibrated for
statistical applications in population genetics. This model is now validated
by the biological community [10, 33, 26].

However, we find fewer results about $\Xi$-coalescents in the literature and
their applications in biology are rarer. We can yet cite the Poisson-Dirichlet
coalescent [29, 24] and the Beta-Xi family [4, 6] that provide promising
models. The reason for this is probably the difficulty arising from the
complex formulation of (1.1). A first step to simplify it is to consider a
measure $\Xi$ on

$\Delta^{*}=\\{\mathbf{\zeta}=(\zeta_{1},\zeta_{2}.\dots),\
\zeta_{1}\geq\zeta_{2}\geq\dots\geq 0,\ \sum_{i=1}^{\infty}\zeta_{i}=1\\}.$

In this case the transition rates simplify. We can now consider
$[b,(k_{1},\dots,k_{r})]$-collisions where $b$ blocks merge into $r$ blocks,
each one containing $k_{1},\dots,k_{r}\geq 1$ original blocks. Each
$[b,(k_{1},\dots,k_{r})]$-collision happens at some fixed rate

$\lambda_{b,(k_{1},\dots,k_{r})}=\ a\
\mathds{1}_{\\{r=b-1,k_{1}=2\\}}+\int_{\Delta^{*}}\sum_{i_{1}\neq\dots\neq
i_{r}}\zeta_{i_{i}}^{k_{1}}\dots\zeta_{i_{r}}^{k_{r}}\frac{\Xi^{0}(d\zeta)}{(\zeta,\zeta)}.$
(1.2)

In this paper we describe a simple family of $\Xi$-coalescents: the symmetric
coalescent, which arises as the limiting genealogy for Wright-Fisher models
with short drastic bottlenecks. The reason for its name is that the
distribution of the tree obtained from a symmetric coalescent is invariant
under the transformation that involves cutting one branch from one node and
pasting it somewhere else in the tree, at the same height (see Figure 1 for an
illustration). In other words, as a partition-valued process, the symmetric
coalescent is invariant under the transformation that consists of displacing
one element from one block to another (without changing the number of non-
empty blocks).

Figure 1: In the symmetric coalescent these three labelled trees have the same
probability. The second one is obtained from the first one by cutting and
pasting the purple branch to a different node. The third one can be obtained
from the second one by displacing the blue branch from one position to
another. In the $S$-coalescent,
$\lambda_{6,(4,1,1)}=\lambda_{6,(2,3,1)}=\lambda_{6,(2,2,2)}$.

######  Definition 1.2.

The symmetric coalescents are the exchangeable coalescents whose transition
rates satisfy the following symmetry property: for every $b>1$, $2\leq r<b$
and for every $k_{1},\ldots,k_{r}$ and $k^{\prime}_{1},\ldots,k^{\prime}_{r}$
such that $\sum_{i=1}^{r}k_{i}=\sum_{i=1}^{r}k^{\prime}_{i}=b$,

$\lambda_{b,(k_{1},\dots,k_{r})}=\lambda_{b,(k^{\prime}_{1},\dots,k^{\prime}_{r})}.$

In the sequel we will consider the symmetric elements of $\Delta^{*}$. Let
$\xi^{0}:=(0,0,\dots)$ and, for $k\in\mathbb{N}$,

$\xi^{k}:=(\frac{1}{k},\dots,\frac{1}{k},0,\dots)$

and we denote the set of the symmetric elements of $\Delta^{*}$ by
$\Delta^{sym}:=\\{\xi^{k},k\in\mathbb{N}_{0}\\}$. Our first result establishes
a correspondence between symmetric coalescents and measures on a simple set,
being here $\mathbb{Z}_{+}:=\mathbb{N}\cup\\{0\\}.$

######  Theorem 1.3.

A coalescent is symmetric if and only if there exists a measure $F$ on
$\mathbb{Z}_{+}$ such that

$F(0)<\infty\ \textrm{ and }\ \sum_{k\geq 1}\frac{F(k)}{k}<\infty$ (1.3)

and such that its characterizing measure $S$ on $\Delta$ only puts weight on
$\Delta^{sym}$ and

$S(\xi^{k})\ =\ \left\\{\begin{array}[]{ll}\frac{F(k)}{k}&\textrm{ if
}k\in\mathbb{N}\\\ F(0)&\textrm{ if }k=0\end{array}\right..$ (1.4)

Condition (1.3) ensures that the measure $S$ is finite which is a necessary
and sufficient condition for a $\Xi$-coalescent to be well defined. Observe
that for $k\in\mathbb{N}$, $(\xi^{k},\xi^{k})=\frac{1}{k}$, so the rate of a
$k$ merger is $F(k)$. Mimicking the common notations we will speak about
$S$-coalescents.

Before going into further detail, we start by recalling a useful tool, which
is Kingman’s paintbox construction of $\Xi$-coalescents ([18]). We will only
discuss the case when $\Xi$ is concentrated on $\Delta^{*}$. Each element in
$\Delta^{*}$ can be seen as a tiling of (0,1), where the sizes of the
subintervals are $\zeta_{1},\zeta_{2},\dots$. The $\Xi$-coalescent can be
constructed as follows: when there are $b$ blocks, for every
$\zeta\in\Delta^{*}$, at rate $\Xi(d\zeta)/(\zeta,\zeta)$, we choose the
tiling associated with $\zeta$, then we throw $b$ uniform random variables in
$(0,1)$, each one associated with one block, and all blocks within one
subinterval merge. In the case of the symmetric coalescent, the paintbox
construction can be reformulated as follows: when there are $b$ blocks, at
rate $F(k)$, we distribute $b$ balls into $k$ boxes and blocks corresponding
to balls that are in the same box merge. For more details we refer the reader
to the first chapter of [1]. This construction allows us to obtain a nice
explicit formula for the transition rates.

######  Proposition 1.4.

For each $b\geq 2$ and $k_{1},\ldots,k_{r}$ such that $\sum_{i=1}^{r}k_{i}=b$,
we have

$\lambda_{b,(k_{1},\dots,k_{r})}\ =\ a\ \mathds{1}_{\\{r=b-1,k_{1}=2\\}}\ +\
\sum_{k\geq r}F(k)\frac{k!}{(k-r)!}\frac{1}{k^{b}},$

where $a=F(0)$.

This result is obtained from (1.1) as follows. The term $F(k)$ comes from
$\Xi^{0}(d\zeta)/(\zeta,\zeta)$, while $k!/(k-r)!$ is the number of choices of
$i_{1},\dots,i_{r}$ and in this case
$\zeta_{i_{i}}^{k_{1}}\dots\zeta_{i_{r}}^{k_{r}}$ equals $1/k^{b}$. In other
words, at each jump time, if the number of boxes is $k$ (chosen with respect
to $F$), we choose $r$ ordered boxes and we allocate the $b$ balls to these
$r$ boxes ($k_{1}$ balls to the first box, $k_{2}$ to the second one, etc…).

Finally let us consider $\\{N_{t}\\}_{t\geq 0}$, the block-counting process of
the symmetric coalescent and, for $i>j$, let us denote by $q_{ij}$ its
transition rate from $i$ to $j$. Our next result is the symmetric coalescent
version of Proposition 2.1 in [12]. Let $W^{k,b}$ be the random variable
corresponding to the number of non-empty boxes when allocating randomly $b$
balls into $k$ boxes, whose distribution can be found in [8], proof of Theorem
3.6.10, page 172.

######  Proposition 1.5.

We have

$q_{ij}\ =\ a\binom{i}{2}\mathds{1}_{\\{j=i-1\\}}\ +\ \sum_{k\geq
1}F(k)\mathbb{P}(W^{k,i}=j)$

with

$\mathbb{P}(W^{k,i}=j)\ =\
\binom{k}{j}\left(\frac{j}{k}\right)^{i}\sum_{r=0}^{j}(-1)^{r}\binom{j}{r}\left(1-\frac{r}{j}\right)^{i}.$

The fact that the characterizing measure $S$ (or $F$) only puts weight on
elements of $\Delta^{sym}$ simplifies a lot the global picture of the
coalescence tree, even when starting from an infinite population. In
particular, an $S$-coalescent is almost surely finite after the first
coalescence (that is not a ‘Kingman type’ coalescence). However, the symmetric
coalescent can come down from infinity. Denoting by $\\{N_{t}\\}_{t\geq 0}$
the block-counting process of the symmetric coalescent and assuming that
$N_{0}=\infty$, recall that a coalescent comes down from infinity if for every
$t>0$, $N_{t}<\infty$ almost surely. Observing that the time of the first (non
‘Kingman type’) coalescence event is exponentially distributed, with parameter
$\sum_{k=1}^{\infty}\frac{S(\xi^{k})}{(\xi^{k},\xi^{k})}=\sum_{k=1}^{\infty}kS(\xi^{k})$,
it is straightforward to get the next result.

######  Proposition 1.6.

An $S$-coalescent comes down from infinity if and only if

$S(\xi^{0})>0\ \textrm{ or }\ \sum_{k\geq 1}kS(\xi^{k})=\sum_{k\geq
1}F(k)=\infty.$

An interesting family of symmetric coalescents, that we will call
$(\beta,S)$-coalescents, contains those characterized by $F(k)=k^{-\beta}$,
for $\beta>0$ (so that condition (1.3) is satisfied). By Proposition 1.6, a
$(\beta,S)$-coalescent comes down from infinity if and only if $0<\beta\leq
1$. We now focus on the total coalescence rate when there are $n$ lineages,
which is given by

$\lambda_{n}=\sum_{r=1}^{n-1}\ \sum_{k_{1},\dots,k_{r},\sum
k_{i}=n}\mathcal{N}(n,(k_{1},\dots,k_{r}))\lambda_{n,(k_{1},\dots,k_{r})},$
(1.5)

where $\mathcal{N}(n,(k_{1},\dots,k_{r}))$ is the number of different
simultaneous choices of a $k_{1}$-tuple, a $k_{2}$-tuple,… and a $k_{r}$-tuple
from a set of $b$ elements. An explicit formula for this number can be found
in [30], display $(3)$.

######  Proposition 1.7.

For the $(\beta,S)$-coalescent, with $\beta\in\ (0,1)$, we have

$\lim_{n\to\infty}n^{2(\beta-1)}\lambda_{n}=\frac{2^{\beta-1}\Gamma(\beta)}{1-\beta}.$

For the $(1,S)$-coalescent, we have

$\lim_{n\to\infty}\frac{\lambda_{n}}{\log n}=2.$

It is interesting to compare these asymptotics with other classical
coalescents. For example, when $\beta\to 0$, the total coalescence rate
becomes very close to the total coalescence rate of the Kingman coalescent,
which is of order $n^{2}$. When $\beta\in(0,1/2]$ the total coalescence rate
is very close to that of a $Beta(2-2\beta,2\beta)$-coalescent, see Lemma 2.2
in [7]. In particular, the rates of the $(1/2,S)$-coalescent have the same
order than those of the Bolthausen-Sznitman coalescent.

###  1.3 Genealogies of Wright-Fisher models with bottlenecks

In this paper, we study four types of Wright-Fisher models with bottlenecks
and their genealogies. We establish the relations between forwards and
backwards models via moment duality results. In particular, when the
bottlenecks are short, drastic and rare and $b_{i,N}N$ is distributed as
$F^{0}$, a measure on $\mathbb{N}$, we prove (see Theorem 5.4) that the
scaling limit, in the sense of weak convergence in the Skorokhod topology, of
a subpopulation frequency is given by a Wright-Fisher diffusion with jumps

$dX_{t}\ =\ \sqrt{X_{t}(1-X_{t})}dB_{t}\ +\
\int_{\mathbb{N}}\int_{[0,1]^{\mathbb{N}}}\frac{1}{k}\sum_{i=1}^{k}\left(\mathds{1}_{\\{u_{i}\leq
X_{t^{-}}\\}}-X_{t^{-}}\right)\hat{N}(dt,dk,du),$

where $\\{B_{t}\\}_{t\geq 0}$ is a standard Brownian motion and $\hat{N}$ is a
compensated Poisson measure on
$(0,\infty)\times\mathbb{N}\times[0,1]^{\mathbb{N}}$ with intensity $ds\otimes
F^{0}(k)\otimes du$, where $du$ is the Lebesgue measure on
$[0,1]^{\mathbb{N}}$. The jump term can be interpreted as follows. At rate
$F^{0}(k)$ there is a bottleneck in which only $k$ individuals survive. The
term ‘$\mathds{1}_{\\{u_{i}\leq X_{t^{-}}\\}}$’ is the probability that an
individual chooses a type 1 parent (when choosing her parent uniformly from
the generation before the bottleneck), and therefore
$\frac{1}{k}\sum_{i=1}^{k}\mathds{1}_{\\{u_{i}\leq X_{t^{-}}\\}}$ is the
frequency of type 1 individuals after the bottleneck. As we shall see in
Section 2.3, this equation has a unique strong solution that is moment dual to
the block-counting process of the symmetric coalescent characterized by
$F=\delta_{0}+F^{0}$. Duality relations between $\Xi$-coalescents and Wright-
Fisher diffusions with jumps were established in [5]. Moment duality implies
that the process counting the number of ancestors to a sample of individuals
in the Wright-Fisher model with short drastic bottlenecks converges to the
block-counting process of the symmetric coalescent, in the sense of finite
dimensional distributions. In Section 3.1 of [13] the authors show that
convergence in the $J_{1}$ Skorokhod topology of the forward frequency process
to the solution of the above SDE implies convergence in $J_{1}$ of the process
counting the number of ancestors to a sample.

A similar strategy is used in the case of Wright-Fisher models with long
bottlenecks. However, the situation is different since the frequency process
is not Markovian anymore, as the transition rates depend on whether the
population is undergoing a bottleneck or not. Nevertheless, we still obtain a
scaling limit, but in the sense of convergence in measure (as defined in
Section 3) over the Skorokhod space, to a diffusion with jumps. Intuitively,
only a measure zero set of points prevents $J_{1}$ convergence. These are
exactly the accumulation points of the times at which a bottleneck occurred in
the discrete model. The method for proving this convergence relies on a new
Skorokhod type metric that allows us to prove convergence in measure using
standard arguments. We prove that the diffusion with jumps is moment dual to
the block-counting process of a $\Xi$-coalescent. In the case of long drastic
bottlenecks, it is the drastic bottleneck coalescent (see Definition 4.1) and
in the case of long soft bottlenecks it is the subordinated Kingman coalescent
introduced in [5] and studied in Section 5.2. Again, moment duality implies
convergence, in the sense of finite dimensional distributions, of the process
counting the number of ancestors of a sample.

In the case of Wright-Fisher models with soft drastic bottlenecks, a different
strategy is used. Using Möhle’s theorem [25], we obtain that, under an
appropriate time re-scaling, the (partition-valued) ancestral process
converges to a time-changed Kingman coalescent. Again, moment duality implies
the convergence in the sense of finite dimensional distributions of the
frequency process to a Wright-Fisher diffusion. Table 1 summarizes these
results.

| Drastic | Soft
---|---|---
Short |  $S$-coalescent | Continuous time-rescaling of the Kingman coalescent
Long | Drastic bottleneck coalescent | Subordinated Kingman coalescent
Table 1: Limiting genealogies for the different types of Wright-Fisher models
with bottlenecks.

###  1.4 Outline

We start the core of this article by a complete study of the symmetric
coalescent. More precisely, in Section 2.1, we prove Theorem 1.4. In Section
2.2, we study asymptotics of the total coalescent rates (Proposition 1.7) and
the tree length in the special case of $(\beta,S)$-coalescents. In Section
2.3, we establish a first duality result between the $S$-coalescent and the
Wright-Fisher diffusion with short drastic bottlenecks. In Section 3 we
introduce the new Skorokhod type metric, that will be used in the last two
sections, which are devoted to the study of other models with bottlenecks and
their genealogies: Section 4 for long drastic bottlenecks and Section 5 for
soft bottlenecks, where time-changed Kingman coalescents appear as limiting
genealogies.

##  2 The symmetric coalescent

We will start by considering bottlenecks that are drastic and short i.e.
bottlenecks that only last for one generation and in which the population size
during the bottleneck does not tend to infinity as $N\to\infty$. More
precisely we consider the following model (that is a special case of
Definition 1.1).

######  Definition 2.1 (Wright-Fisher model with short drastic bottlenecks).

Fix $\alpha\in(0,1]$, $N\in\mathbb{N}$, $k^{(N)}\in(0,N^{\alpha})$ and $F^{0}$
a probability measure on $\mathbb{N}$. Let $\\{F_{g}\\}_{g\in\mathbb{Z}_{+}}$
be a sequence of i.i.d. random variables of law $F^{0}$. Also, let
$\\{B^{N}_{g}\\}_{g\in\mathbb{Z}_{+}}$ be a sequence of i.i.d. Bernoulli
random variables of parameter $k^{(N)}/N^{\alpha}$. The Wright-Fisher model
with short drastic bottlenecks is such that the sequence of population sizes
$\\{R^{N}_{g}\\}_{g\in\mathbb{Z}_{+}}$ is given by

$\ R^{N}_{g}=N(1-B_{g}^{N})+\min(N,F_{g})B_{g}^{N}.$

######  Remark 2.2.

In this case, the bottlenecks are short and if the $i$-th bottleneck takes
place during generation $g$, $b_{i,N}N=\min(N,F_{g})$, which does not tend to
infinity when $N$ goes to infinity, so the bottlenecks are drastic. In
addition, $s_{i,N}$, the time between two bottlenecks follows a geometric
distribution of parameter $k^{(N)}/N^{\alpha}$, so if $k^{(N)}=O(1)$, the
expectation of $s_{i,N}/N$ is of order $N^{\alpha}/N$. Thus, when $\alpha<1$,
the bottlenecks are frequent and when $\alpha=1$ the bottlenecks are rare; the
main consequence of this is that in the case of rare bottlenecks there is a
Kingman/Wright-Fisher component, while in the case of frequent bottlenecks
there is not enough time for the Kingman part to be a part of the scaling
limit.

As we will prove, when $N\to\infty$ and time is rescaled by $N^{\alpha}$, the
genealogy of this model is described by the symmetric coalescent. It is now
time to study this process.

###  2.1 Characterization

Let us start with the proof of Theorem 1.4.

###### Proof of Theorem 1.4.

From (1.4), we decompose $F$ (resp. $S$) into a ‘Kingman part’ and a
‘simultaneous multiple collisions’ part, i.e. $F=a\delta_{0}+F^{0}$ where
$a:=F(0)\geq 0$ and $F^{0}(0)=0$ (resp. $S=a\delta_{(0,0,\dots)}+S^{0}$).

We start by proving that any $\Xi$-coalescent that is characterized by a
measure $S$ on $\Delta^{sym}$ as above is symmetric. We fix $b\geq 2$ and
$k_{1},\ldots,k_{r}$ and $k^{\prime}_{1},\ldots,k^{\prime}_{r}$ such that
$\sum_{i=1}^{r}k_{i}=\sum_{i=1}^{r}k^{\prime}_{i}=b$. From Theorem 2 in [30],
the transition rates can be written as follows:

$\displaystyle\lambda_{b,(k_{1},\dots,k_{r})}$
$\displaystyle=a\mathds{1}_{\\{r=b-1\\}}+\int_{\Delta}\sum_{i_{1}\neq\dots\neq
i_{r}}\zeta^{k_{1}}_{i_{1}}\dots\zeta^{k_{r}}_{i_{r}}\
\frac{S^{0}(d\zeta)}{(\zeta,\zeta)}$
$\displaystyle=a\mathds{1}_{\\{r=b-1\\}}+\sum_{j=r}^{\infty}\frac{j!}{(j-r)!}\left(\frac{1}{j}\right)^{b}F^{0}(j)$
$\displaystyle=\lambda_{b,(k^{\prime}_{1},\dots,k^{\prime}_{r})}.$

Conversely, suppose that a $\Xi$-coalescent satisfies the symmetric condition
on its transition rates. We write $\Xi=a\delta_{(0,0,\dots)}+\Xi^{0}$. For any
$\zeta\in\Delta$, we set $\zeta_{0}=1-\sum_{i=1}^{\infty}\zeta_{i}$. We define

$Z=\\{\zeta\in\Delta,\ \exists j,i,\ \zeta_{i}>\zeta_{j}>0\\}$

and we assume that $\Xi^{0}(Z)>0$. Using Theorem 2 in [30], we have

$\displaystyle\lambda_{4,(2,2)}\ $ $\displaystyle=\
\int_{\Delta}\sum_{i_{1}\neq
i_{2}}\zeta_{i_{1}}^{2}\zeta_{i_{2}}^{2}\frac{\Xi^{0}(d\zeta)}{(\zeta,\zeta)}\
=\ 2\int_{\Delta}\sum_{i_{1}<i_{2}}\
\zeta_{i_{1}}^{2}\zeta_{i_{2}}^{2}\frac{\Xi^{0}(d\zeta)}{(\zeta,\zeta)}$

and

$\displaystyle\lambda_{4,(3,1)}\ $ $\displaystyle=\
\int_{\Delta}\left(\sum_{i_{1}\neq
i_{2}}\zeta_{i_{1}}^{3}\zeta_{i_{2}}+\zeta_{0}\sum_{j}\zeta_{j}^{3}\right)\frac{\Xi^{0}(d\zeta)}{(\zeta,\zeta)}$
$\displaystyle\geq\int_{\Delta}\left(\sum_{i_{1}<i_{2}}\zeta_{i_{1}}^{3}\zeta_{i_{2}}+\sum_{i_{1}<i_{2}}\zeta_{i_{1}}\zeta_{i_{2}}^{3}\right)\frac{\Xi^{0}(d\zeta)}{(\zeta,\zeta)}.$

So,

$\displaystyle\lambda_{4,(3,1)}-\lambda_{4,(2,2)}\ $
$\displaystyle\geq\int_{\Delta}\left(\sum_{i_{1}<i_{2}}\zeta_{i_{1}}^{3}\zeta_{i_{2}}+\sum_{i_{1}<i_{2}}\zeta_{i_{1}}\zeta_{i_{2}}^{3}-2\sum_{i_{1}<i_{2}}\zeta_{i_{1}}^{2}\zeta_{i_{2}}^{2}\right)\frac{\Xi^{0}(d\zeta)}{(\zeta,\zeta)}$
$\displaystyle=\int_{\Delta}\sum_{i_{1}<i_{2}}\zeta_{i_{1}}\zeta_{i_{2}}(\zeta_{i_{1}}-\zeta_{i_{2}})^{2}\frac{\Xi^{0}(d\zeta)}{(\zeta,\zeta)}$
$\displaystyle=\int_{Z}\sum_{i_{1}<i_{2}}\zeta_{i_{1}}\zeta_{i_{2}}(\zeta_{i_{1}}-\zeta_{i_{2}})^{2}\frac{\Xi^{0}(d\zeta)}{(\zeta,\zeta)}>0,$

as the integrand is equal to zero on $\Delta\setminus Z$ and strictly positive
on $Z$. This cannot be true (as the coalescent is symmetric). So we need
$\Xi^{0}(Z)=0$, i.e. $\Xi^{0}$ can only take positive values on
$\Delta\setminus Z$, i.e. elements of $\Delta$ such that there exists $0<u\leq
1$ with $\zeta=(u,u,\dots,u,0,0,\dots)$.

Now, we consider the set

$Z_{0}=\\{\zeta\in\Delta\setminus Z,\ \zeta_{0}>0\\}$

and we assume that $\Xi^{0}(Z)=0$ and $\Xi^{0}({Z_{0}})>0$. We have

$\lambda_{4,(2,2)}=\int_{\Delta\setminus Z}\sum_{i_{1}\neq
i_{2}}\zeta_{i_{1}}^{2}\zeta_{i_{2}}^{2}\frac{\Xi^{0}(d\zeta)}{(\zeta,\zeta)}$

and

$\lambda_{4,(3,1)}=\int_{\Delta\setminus Z}\left(\sum_{i_{1}\neq
i_{2}}\zeta_{i_{1}}^{3}\zeta_{i_{2}}+\zeta_{0}\sum_{j}\zeta_{j}^{3}\right)\frac{\Xi^{0}(d\zeta)}{(\zeta,\zeta)}.$

Recall that, if $\zeta\in\Delta\setminus Z$, then $\forall i\geq
1,\zeta_{i}=\zeta_{1}$ or $\zeta_{i}=0$, so $\sum_{i_{1}\neq
i_{2}}\zeta_{i_{1}}^{2}\zeta_{i_{2}}^{2}=\sum_{i_{1}\neq
i_{2}}\zeta_{i_{1}}^{3}\zeta_{i_{2}}$. So
$\lambda_{4,(2,2)}=\lambda_{4,(3,1)}$ if and only if $\Xi^{0}(Z_{0})=0$, which
means that $\Xi^{0}$ can only put weight on elements of $(\Delta\setminus
Z)\setminus Z_{0}$, i.e. elements of $\Delta^{sym}$. This completes the proof.
∎

As we shall see in Section 2.3, the symmetric coalescent characterized by a
measure $F=a\delta_{0}+F^{0}$, where $F^{0}$ is a probability measure,
describes the genealogy of a Wright-Fisher model with short drastic
bottlenecks parametrized by $\alpha\in(0,1]$, $N$, $k^{(N)}=1$ and $F^{0}$, in
the limit when $N\to\infty$. The case $a=0$ corresponds to frequent
bottlenecks ($\alpha<1$) and the case $a=1$ corresponds to rare bottlenecks
($\alpha=1$). In fact, when time is rescaled by $N^{\alpha}$, if the
bottlenecks are frequent, in the limiting genealogy we only see coalescent
events taking place during the bottlenecks, whereas if the bottlenecks are
rare, there is a ‘Kingman part’ in the limiting genealogy, corresponding to
coalescence events taking place outside the bottlenecks. We will also discuss
in this section a model where the limit genealogical process is a symmetric
coalescent characterized by a measure $F$ that is not finite. Indeed, we show
that, when $N\to\infty$, the Wright-Fisher model with short drastic
bottlenecks converges to a diffusion with jumps that is moment dual to the
block-counting process of the symmetric coalescent.

###  2.2 Tree length and total coalescence rate of $(\beta,S)$-coalescents

We now focus on the family of $(\beta,S)$-coalescents. In this case, the total
coalescence rate (1.5) is

$\displaystyle\lambda_{n}\ =\
\sum_{k=1}^{\infty}k^{-\beta}\mathbb{P}(\mathcal{C}^{k}_{n}),$ (2.6)

where $\mathcal{C}^{k}_{n}$ is the event that, in the paintbox construction
with $k$ boxes and $n$ balls, there are at least two balls that are allocated
to the same box. For $n>k$, $\mathbb{P}(\mathcal{C}^{k}_{n})=1$ and for $n\leq
k$,

$\mathbb{P}(\mathcal{C}^{k}_{n})=1-\prod_{i=2}^{n}\frac{k+1-i}{k}.$

In fact, the probability of $\mathcal{C}^{k}_{n}$ is 1 minus the probability
that $n$ successive balls are allocated to distinct boxes, which can be
computed in the following way: the first ball is allocated to any box and
then, for $2\leq i\leq n$, the $i$-th ball is allocated to one of the $k+1-i$
empty boxes. We are now ready to prove Proposition 1.7.

###### Proof of Proposition 1.7.

We first treat the case $\beta\in(0,1)$. Fix $0<\epsilon<1$. We divide (2.6)
into two parts:

$\lambda_{n}=\sum_{k=1}^{\lfloor
n^{1+\epsilon}\rfloor-1}k^{-\beta}\mathbb{P}(\mathcal{C}^{k}_{n})+\sum_{k=\lfloor
n^{1+\epsilon}\rfloor}^{\infty}k^{-\beta}\mathbb{P}(\mathcal{C}^{k}_{n}).$

For the second term, we have

$\displaystyle\sum_{k=\lfloor
n^{1+\epsilon}\rfloor}^{\infty}k^{-\beta}\left(1-\prod_{i=1}^{n-1}\left(1-\frac{i}{k}\right)\right)$
$\displaystyle\sim\sum_{k=\lfloor
n^{1+\epsilon}\rfloor}^{\infty}k^{-\beta}\left(1-\exp\left(-\sum_{i=1}^{n-1}\frac{i}{k}\right)\right)$
$\displaystyle\sim\sum_{k=\lfloor
n^{1+\epsilon}\rfloor}^{\infty}k^{-\beta}\left(1-\exp\left(-\frac{n^{2}}{2k}\right)\right)$
$\displaystyle\sim
n^{2-2\beta}\int_{0}^{\infty}x^{-\beta}\left(1-\exp\left(-\frac{1}{2x}\right)\right)dx$
$\displaystyle=\frac{2^{\beta-1}\Gamma(\beta)}{1-\beta}n^{2(1-\beta)},$

where the last equality is obtained by integrating by parts and using the
inverse-gamma distribution.

For the first term, observe that

$\sum_{k=1}^{\lfloor
n^{1+\epsilon}\rfloor}k^{-\beta}\mathbb{P}(\mathcal{C}^{k}_{n})\leq\sum_{k=1}^{\lfloor
n^{1+\epsilon}\rfloor}k^{-\beta}\sim n^{(1+\epsilon)(1-\beta)},$

which is negligible compared to the second term.

Let us now suppose that $\beta=1$. We divide (2.6) into three parts (recall
that $\mathbb{P}(\mathcal{C}^{k}_{n})=1$ when $k\leq n$).

$\lambda_{n}=\sum_{k=1}^{n-1}k^{-1}+\sum_{k=n}^{\lfloor
n^{1+\epsilon}\rfloor-1}k^{-1}\mathbb{P}(\mathcal{C}^{k}_{n})+\sum_{k=\lfloor
n^{1+\epsilon}\rfloor}^{\infty}k^{-1}\mathbb{P}(\mathcal{C}^{k}_{n}).$

The first term is obviously equivalent to $\log n$. The second term is clearly
smaller than $\epsilon\log n$. Let us now find a lower bound

$\displaystyle\sum_{k=n}^{\lfloor
n^{1+\epsilon}\rfloor-1}k^{-1}\left(1-\prod_{i=1}^{n-1}\left(1-\frac{i}{k}\right)\right)$
$\displaystyle\geq\sum_{k=n}^{\lfloor
n^{1+\epsilon}\rfloor-1}k^{-1}\left(1-\exp\left(\sum_{i=1}^{n-1}\left(-\frac{i}{k}\right)\right)\right)$
$\displaystyle\sim\int_{n^{-1}}^{n^{\epsilon-1}}x^{-1}\left(1-\exp\left(-\frac{1}{2x}\right)\right)dx$
$\displaystyle=\int_{n^{1-\epsilon}}^{n}y^{-1}\left(1-\exp\left(-\frac{y}{2}\right)\right)dy$
$\displaystyle=\log
n\left(1-\exp\left(-\frac{n}{2}\right)\right)-(1-\epsilon)\log
n\left(1-\exp\left(-\frac{n^{1-\epsilon}}{2}\right)\right)$
$\displaystyle+\frac{1}{2}\int_{n^{1-\epsilon}}^{n}\log
y\exp\left(-\frac{y}{2}\right)dy$ $\displaystyle\sim\epsilon\log n.$

For the third term, we use similar computations as in the case $\beta<1$,

$\displaystyle\sum_{k=\lfloor
n^{1+\epsilon}\rfloor}^{\infty}k^{-1}\left(1-\prod_{i=1}^{n-1}\left(1-\frac{i}{k}\right)\right)$
$\displaystyle\sim\int_{n^{\epsilon-1}}^{\infty}x^{-1}\left(1-\exp\left(-\frac{1}{2x}\right)\right)dx$
$\displaystyle=\int_{0}^{n^{1-\epsilon}}y^{-1}\left(1-\exp\left(-\frac{y}{2}\right)\right)dy$
$\displaystyle=(1-\epsilon)\log
n\left(1-\exp\left(-\frac{n^{1-\epsilon}}{2}\right)\right)+\frac{1}{2}\int_{0}^{n^{1-\epsilon}}\log
y\exp\left(-\frac{y}{2}\right)dy$ $\displaystyle\sim(1-\epsilon)\log n.$

This ends the proof. ∎

This result on the total coalescence rate allows us to give a first estimate
of the tree length of the $(\beta,S)$-coalescent. Let $L_{n}$ be the sum of
the lengths of all the branches of the tree obtained from a
$(\beta,S)$-coalescent started with $n$ lineages and stopped at the first time
when there is only one lineage.

######  Corollary 1.

For $\beta\in(0,1)$, there exist two positive constants $C^{\prime}_{\beta}$
and $c^{\prime}_{\beta}$, that only depend on $\beta$ (and not on $n$), such
that for $n$ large enough,

$c^{\prime}_{\beta}n^{(2\beta-1)\vee 0}\ \leq\ \mathbb{E}(L_{n})\ \leq\
C^{\prime}_{\beta}n^{(2\beta)\wedge 1}.$

For $\beta=1$, there exists a constant $C^{\prime}_{1}$ such that

$\frac{n}{2\log n}(1+o(1))\ \leq\ \mathbb{E}(L_{n})\ \leq\ C^{\prime}_{1}n.$

As we can see in Figure 2, this corollary provides better estimates of the
tree length when $\beta$ is close to 1 or close to 0.

Figure 2: Illustration of Corollary 1. The orange lines are $y=2x-1$ and
$y=2x$, and the green line is $y=1$. The blue area is the region where
$\log(\mathbb{E}(L_{n}))$ is located.

###### Proof.

We start by proving that, for any $\beta\in[0,1]$, the expected tree length is
at most of order $n$. We consider the $S^{1}$-coalescent, characterized by
$S^{1}(\xi^{k})=\mathds{1}_{\\{k=1\\}}$. First, one can easily show that the
tree length of the $S^{1}$-coalescent is of order $n$ (it is a star-shaped
coalescent). Second, the rate of events of size 1 in the $S^{1}$-coalescent
and in the $(\beta,S)$-coalescent is the same and, for $k>1$,
$S^{1}(\xi^{k})=0$ while, in the $(\beta,S)$-coalescent,
$S(\xi^{k})=k^{\beta}>0$. It is not hard to construct a coupling between the
two processes in such a way that the length of the $S^{1}$ coalescent is
always larger than the length of the $(\beta,S)$-coalescent.

The expectation of the time to the first coalescence when there are $k$
lineages is $1/\lambda_{k}$, so ${k}/{\lambda_{k}}$ is the expected length of
a tree started with $k$ lineages and stopped at the first coalescence event.
So, for $\beta\in(0,1)$, we have

$\mathbb{E}(L_{n})\ \leq\ \sum_{k=2}^{n}\frac{k}{\lambda_{k}}.$

Recall that, in a coalescent where only two blocks can coalesce at a time (for
example the Kingman coalescent), the sum on the right hand side would be the
expected length, but in a coalescent with simultaneous multiple collisions we
do not observe all the states $\\{2,\dots,n\\}$ for the block-counting process
so it is only an upper bound. Using Proposition 1.7 for $\beta<1$, there
exists a constant $c$ such that

$\displaystyle\mathbb{E}(L_{n})\ $
$\displaystyle\leq\frac{2^{\beta-1}\Gamma(\beta)}{1-\beta}\
\sum_{k=2}^{n}k^{2\beta-1}(1+o(1))$ $\displaystyle\leq c\
\int_{1}^{n}t^{2\beta-1}dt=\frac{c}{2\beta}(n^{2\beta}-1),$

which completes the proof of this first step.

For $\beta\in(0,1]$, for the lower bound, we have

$\mathbb{E}(L_{n})\ \geq\ \frac{n}{\lambda_{n}},$

which is the length of the tree stopped at the first coalescence event. When
$0<\beta<1/2$ this lower bound is not interesting, as it is of order
$n^{2\beta-1}$ and it decreases with $n$. But $\mathbb{E}(L_{n})$ can always
be bounded from below by a positive constant, which completes the proof. ∎

###  2.3 Duality with the Wright-Fisher model with short drastic bottlenecks

We consider the Wright-Fisher model with short drastic bottlenecks from
Definition 2.1. Imagine that there are two types of individuals, $0$ and $1$,
and each individual inherits the type of her parent. We denote by
$\\{X^{N}_{g}\\}_{g\in\mathbb{Z}}$ the process corresponding to the frequency
of type $1$ individuals in the population i.e., for any $g\in\mathbb{Z}_{+}$,

$X^{N}_{g}=\frac{\sum_{i=1}^{R_{g}^{N}}\mathds{1}_{\\{(i,g)\textrm{ is of type
1}\\}}}{R_{g}^{N}}.$

As in the classical Wright-Fisher model, given $R^{N}_{g+1}$ and $X^{N}_{g}$,
$R_{g+1}^{N}X^{N}_{g+1}$ follows a binomial distribution of parameters
$R_{g+1}^{N}$ and $X^{N}_{g}$. In the following, ‘$\Longrightarrow$’ denotes
weak convergence in the Skorokhod on topology $D([0,1],\mathbb{R}_{+})$.

######  Theorem 2.3.

Let $F^{0}$ be a measure on $\mathbb{N}$ that fulfills condition (1.3). Fix
$\alpha\in(0,1]$ and $\gamma\in(0,\alpha/2)$. We consider the probability
measure $F_{\gamma}^{N}$ defined by

$F_{\gamma}^{N}:=\frac{\sum_{k=1}^{\lfloor
N^{\gamma}\rfloor}F^{0}(k)\delta_{k}}{\sum_{k=1}^{\lfloor
N^{\gamma}\rfloor}F^{0}(k)}.$

Consider the sequence of processes $\\{X^{N}\\}_{N\in\mathbb{N}}$, such that
$X^{N}=\\{X^{N}_{g}\\}_{g\in\mathbb{Z}_{+}}$ is the frequency process
associated with the Wright-Fisher model with short drastic bottlenecks
parametrized by $\alpha$, $N$, $k^{(N)}=\sum_{k=1}^{\lfloor
N^{\gamma}\rfloor}F^{0}(k)$ and $F_{\gamma}^{N}$ (from Definition 2.1). Then,

$\\{X^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{t\geq
0}\underset{N\to\infty}{\Longrightarrow}\\{X_{t}\\}_{t\geq 0},$

where $\\{X_{t}\\}_{t\geq 0}$ is the unique strong solution of the SDE

$dX_{t}\ =\ \mathds{1}_{\\{\alpha=1\\}}\sqrt{X_{t}(1-X_{t})}dB_{t}\ +\
\int_{\mathbb{N}}\int_{[0,1]^{\mathbb{N}}}\frac{1}{k}\sum_{i=1}^{k}\left(\mathds{1}_{\\{u_{i}\leq
X_{t^{-}}\\}}-X_{t^{-}}\right)\hat{N}(dt,dk,du),$ (2.7)

where $\\{B_{t}\\}_{t\geq 0}$ is a standard Brownian motion and $\hat{N}$ is a
compensated Poisson measure on
$(0,\infty)\times\mathbb{N}\times[0,1]^{\mathbb{N}}$ with intensity $ds\otimes
F^{0}(k)\otimes du$, where $du$ is the Lebesgue measure on
$[0,1]^{\mathbb{N}}$.

The same result holds if we consider $F^{0}$, a probability measure on
$\mathbb{N}$ and the Wright-Fisher model with short drastic bottlenecks
parametrized by $\alpha$, $N$, $k^{(N)}=1$ and $F^{0}$.

######  Remark 2.4.

The definition of $F_{\gamma}^{N}$ ensures that $F_{\gamma}^{N}$ is a
probability measure on $\mathbb{N}$ and $k^{(N)}/N^{\alpha}\in[0,1]$, so the
Wright-Fisher model with short drastic bottlenecks is well-defined, at least
for $N$ large enough. In fact, if $F^{0}$ satisfies condition (1.3), for $k$
large enough $F(k)<k$, so $k^{(N)}\leq N^{2\gamma}+C$, where $C$ is a
constant.

In words, the frequency process associated with the Wright-Fisher model with
short drastic bottlenecks converges to a diffusion with jumps that is similar
to the frequency process associated with $\Xi$-Fleming-Viot process (where the
characterising measure $\Xi$ is the measure $S$ on $\Delta^{sym}$ that can be
obtained from $F^{0}$ as in Theorem 1.4). When the bottlenecks are frequent
($\alpha<1$), the limiting process is a pure jump process whereas, when the
bottlenecks are rare ($\alpha=1$), we also have a diffusion term, which is a
Wright-Fisher diffusion and corresponds to the evolution of the population
outside the bottlenecks. Before proving Theorem 2.3 we shall make sure that a
solution to Equation (2.7) exists.

######  Lemma 2.5.

For any measure $F^{0}$ in $\mathbb{N}$ that satisfies condition (1.3) and any
$\alpha\in(0,1]$, there exists a unique strong solution to the SDE (2.7).

###### Proof.

This result is a direct consequence of Lemma 3.6 in [13] (which is itself a
consequence of Theorem 5.1 in [20]), applied to the measure $S$ on
$\Delta^{sym}$ obtained from $F^{0}$ as in Proposition 1.4 and a drift
coefficient equal to 0. ∎

We are now ready to prove Theorem 2.3.

###### Proof of Theorem 2.3.

The proof follows closely the proof of Proposition 3.4 in [13]. The idea is to
prove the convergence of the generator of $\\{X^{N}_{\lfloor
N^{\alpha}t\rfloor}\\}_{t\geq 0}$ to the generator of $\\{X_{t}\\}_{t\geq 0}$.
Provided this claim is true, we can use Theorem 19.25 and 19.28 of [17] to
prove the weak convergence in the Skorokhod topology.

From Lemma 2.5, $\\{X_{t}\\}_{t\geq 0}$ exists and has generator
$\mathcal{A}$. Its domain contains twice differentiable functions and for a
function $f\in C^{2}[0,1]$ and $x\in[0,1]$, we have

$\displaystyle\mathcal{A}f(x)\ =$ $\displaystyle\
\mathds{1}_{\\{\alpha=1\\}}\frac{1}{2}x(1-x)f^{\prime\prime}(x)\ +\
\sum_{k\geq
1}F^{0}(k)\mathbb{E}\left(f\left(\frac{\sum_{i=1}^{k}B_{i}^{x}}{k}\right)-f(x)\right),$
(2.8)

where the $B_{i}^{x}$’s are independent Bernoulli random variables of
parameter $x$ and the second term is the generator of a $\Xi$-Fleming-Viot
process, see for example formula (5.6) in [5] (applied to the measure $S$
associated with $F^{0}$).

For every $N\in\mathbb{N}$, let $\mathcal{U}^{N}$ be the transition operator
associated with $X^{N}$ and define the operator

$\mathcal{A}^{N}:=N^{\alpha}(\mathcal{U}^{N}-I),$ (2.9)

where $I$ is the identity operator (see Theorem 19.28 in [17]).
$\mathcal{A}^{N}$ is referred to as the discrete generator of
$\\{X^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{t\geq 0}$. For any function $f\in
C^{2}[0,1]$ in $x\in[0,1]$ we have

$\displaystyle\mathcal{A}^{N}f(x)\ =$ $\displaystyle\
(1-\frac{\sum_{k=1}^{\lfloor
N^{\gamma}\rfloor}F^{0}(k)}{N^{\alpha}})N^{\alpha}\mathbb{E}\left(f\left(\frac{\sum_{i=1}^{N}B_{i}^{x}}{N}\right)-f(x)\right)$
(2.10) $\displaystyle+$ $\displaystyle\ N^{\alpha}\frac{\sum_{k=1}^{\lfloor
N^{\gamma}\rfloor}F^{0}(k)}{N^{\alpha}}\sum_{k=1}^{\lfloor
N^{\gamma}\rfloor}\frac{F^{0}(k)}{\sum_{k=1}^{\lfloor
N^{\gamma}\rfloor}F^{0}(k)}\mathbb{E}\left(f\left(\frac{\sum_{i=1}^{\min(N,k)}B_{i}^{x}}{\min(N,k)}\right)-f(x)\right).$
(2.11)

First, we study part (2.10). Following Remark 2.4, the prefactor converges to
1. When $\alpha=1$, it is well known that (2.10) converges uniformly as
$N\to\infty$ to $\frac{1}{2}x(1-x)f^{\prime\prime}(x)$, which is the generator
of the Wright-Fisher diffusion (see for example Chapter 2 and Theorem 3.6 in
[9]). When $\alpha<1$ this term becomes of order $N^{\alpha-1}$ and therefore
converges to $0$. Second, it is easy to see that part (2.11) converges when
$N\to\infty$ to the second term of $\mathcal{A}$ in (2.8). Combining these two
results, we have $\mathcal{A}^{N}f\to\mathcal{A}f$ uniformly. ∎

Let us consider $\\{N_{t}\\}_{t\geq 0}$, the block-counting process of the
symmetric coalescent characterized by
$F=\mathds{1}_{\\{\alpha=1\\}}\delta_{0}+F^{0}$ (or by the associated measure
$S$ on $\Delta^{sym}$, as defined in Proposition 1.4). We have the following
duality relation between the block-counting process of the symmetric
coalescent and $\\{X_{t}\\}_{t\geq 0}$, the unique strong solution of (2.7).

######  Theorem 2.6.

For every $x\in[0,1],\ n\in\mathbb{N}$, we have

$\mathbb{E}(X_{t}^{n}|X_{0}=x)\ =\ \mathbb{E}(x^{N_{t}}|N_{0}=n).$

This is a special case of Proposition 3.8 in [13], but we find the proof of
this particular case instructive and for the sake of completeness we include
it here.

###### Proof.

Recall that, from Proposition 1.5, for any function
$h:\mathbb{N}\to\mathbb{R}$, $\mathcal{G}$ the infinitesimal generator of the
block-counting process $\\{N_{t}\\}_{t\geq 0}$ is given by

$\mathcal{G}h(n):=a\binom{n}{2}\left(h(n-1)-h(n)\right)\ +\ \sum_{k\geq
1}F^{0}(k)\sum_{j=1}^{n-1}\mathbb{P}(W^{k,n}=j)\left(h(j)-h(n)\right).$ (2.12)

We first consider the case $\alpha<1$. We use Lemma 4.1 in [13], which states
that generator $\mathcal{A}$ of $\\{X_{t}\\}_{t\geq 0}$, applied to a function
$f\in C^{2}[0,1]$ admits the following representation

$\displaystyle\mathcal{A}f(x)\ $ $\displaystyle=\
\frac{S(\Delta_{S})}{2}\mathbb{E}\left[\frac{(-x+\sum_{i=1}^{\infty}Z_{i}B_{i}^{x})(\sum_{i=1}^{\infty}Z_{i}B_{i}^{x})}{(\sum_{i=1}^{\infty}Z_{i}^{2})}f^{\prime\prime}(x(1-V)+UV\sum_{i=1}^{\infty}Z_{i}B_{i}^{x})\right]$

where $Z=(Z_{1},Z_{2},\dots)$ is $S$-distributed,
$\\{B_{i}^{x}\\}_{i\in\mathbb{N}}$ is a sequence of i.i.d. Bernoulli random
variables with parameter $x$, $U$ is uniform in $[0,1]$, $V$ is $Beta(2,1)$ in
$[0,1]$ and $Z,\\{B_{i}^{x}\\},U$ and $V$ are independent. Using the
definition of $S$, this can be rewritten as

$\displaystyle\mathcal{A}f(x)\ $ $\displaystyle=\ \frac{\sum_{k\geq
1}F^{0}(k)/k}{2}\
\mathbb{E}\left[\frac{(-x+\frac{1}{K}\sum_{i=1}^{K}B_{i}^{x})(\frac{1}{K}\sum_{i=1}^{K}B_{i}^{x})}{(1/K)}f^{\prime\prime}(x(1-V)+UV\frac{1}{K}\sum_{i=1}^{K}B_{i}^{x})\right]$

where the expectation is taken with respect to $K$, a random variable such
that for $k\geq 1$,

$\ \mathbb{P}(K=k)=\frac{F^{0}(k)/k}{\sum_{k\geq 1}F^{0}(k)/k}.$

Integrating by parts,

$\displaystyle\mathcal{A}f(x)\ $ $\displaystyle=\ \frac{\sum_{k\geq
1}F^{0}(k)/k}{2}\
\mathbb{E}\left[\frac{(-x+\frac{1}{K}\sum_{i=1}^{K}B_{i}^{x})}{V(1/K)}\left(f^{\prime}(x(1-V)+V\frac{1}{K}\sum_{i=1}^{K}B_{i}^{x})-f^{\prime}(x(1-V))\right)\right].$

Conditioning on the value of $V$, we have

$\displaystyle\mathbb{E}\left[\frac{(-x+\frac{1}{K}\sum_{i=1}^{K}B_{i}^{x})}{V(1/K)}f^{\prime}(x(1-V))\
\right]=0.$

Again, following closely the proof of Lemma 4.1 in [13], we calculate the
expectation with respect to $V$,

$\displaystyle\mathcal{A}f(x)\ $ $\displaystyle=\ \frac{\sum_{k\geq
1}F^{0}(k)/k}{2}\
\mathbb{E}\left[\frac{(-x+\frac{1}{K}\sum_{i=1}^{K}B_{i}^{x})}{V(1/K)}f^{\prime}(x(1-V)+V\frac{1}{K}\sum_{i=1}^{K}B_{i}^{x})\right]$
$\displaystyle=\ \frac{\sum_{k\geq 1}F^{0}(k)/k}{2}\
\mathbb{E}\left[\int_{0}^{1}K\frac{(-x+\frac{1}{K}\sum_{i=1}^{K}B_{i}^{x})}{s}f^{\prime}(x(1-s)+\frac{s}{K}\sum_{i=1}^{K}B_{i}^{x})2sds\right]$
$\displaystyle=\ \left({\sum_{k\geq 1}F^{0}(k)/k}\right)\
\mathbb{E}\left[K\int_{0}^{1}(-x+\frac{1}{K}\sum_{i=1}^{K}B_{i}^{x})f^{\prime}(x(1-s)+\frac{s}{K}\sum_{i=1}^{K}B_{i}^{x})ds\right]$
$\displaystyle=\ \left({\sum_{k\geq 1}F^{0}(k)/k}\right)\
\mathbb{E}\left[K\left(f(\frac{1}{K}\sum_{i=1}^{K}B_{i}^{x})-f(x)\right)\right],$

where the last equality comes by integration by parts. Now, we consider the
function $h$ on $[0,1]\times\mathbb{N}$ such that $h(x,n)=x^{n}$. We fix
$n\in\mathbb{N}$ and we apply the generator $\mathcal{A}$ to $h$, seen as a
function of $x$,

$\displaystyle\mathcal{A}h(x)\ $ $\displaystyle=\ \left({\sum_{k\geq
1}F^{0}(k)/k}\right)\
\mathbb{E}\left[K\left((\frac{1}{K}\sum_{i=1}^{K}B_{i}^{x})^{n}-x^{n}\right)\right]$
$\displaystyle=\ \left({\sum_{k\geq 1}F^{0}(k)/k}\right)\
\mathbb{E}\left[K\left(\frac{1}{K^{n}}\sum_{k_{1}+\dots+k_{K}=n}\binom{n}{k_{1},\dots,k_{K}}\prod_{i=1}^{K}(B_{i}^{x})^{k_{i}}-x^{n}\right)\right]$
$\displaystyle=\ \left({\sum_{k\geq 1}F^{0}(k)/k}\right)\
\mathbb{E}\left[K\left(\sum_{k_{1}+\dots+k_{K}=n}\binom{n}{k_{1},\dots,k_{K}}\frac{1}{K^{n}}\left(x^{\sum_{i=1}^{K}\mathds{1}_{\\{k_{i}>0\\}}}-x^{n}\right)\right)\right],$
(2.13)

where in the last line we use the fact that for
$k_{1},\dots,k_{K}\in\mathbb{N}$,

$\mathbb{E}\left(\prod_{i=1}^{K}(B_{i}^{x})^{k_{i}}\right)=x^{\sum_{i=1}^{K}\mathds{1}_{\\{k_{i}>0}\\}}.$

Consider the random variable $W^{k,n}$ defined in Proposition 1.5. For any
$\kappa\in\mathbb{N}$, we have

$\displaystyle\sum_{k_{1}+\dots+k_{\kappa}=n}\binom{n}{k_{1},\dots,k_{\kappa}}\frac{1}{\kappa^{n}}\left(x^{\sum_{i=1}^{\kappa}\mathds{1}_{\\{k_{i}>0}\\}}-x^{n}\right)$
$\displaystyle=\mathbb{E}\left[x^{W^{\kappa,n}}-x^{n}\right]\ =:\ E_{\kappa}.$
(2.14)

So we have,

$\displaystyle\mathcal{A}h(x)\ $ $\displaystyle=\ \left({\sum_{k\geq
1}F^{0}(k)/k}\right)\ \mathbb{E}\left[KE_{K}\right]$ $\displaystyle=\
\left({\sum_{k\geq 1}F^{0}(k)/k}\right)\ \sum_{k\geq
1}\frac{F^{0}(k)/k}{\sum_{k\geq 1}F^{0}(k)/k}kE_{k}$
$\displaystyle=\sum_{k\geq 1}F^{0}(k)E_{k}$ $\displaystyle=\sum_{k\geq
1}F^{0}(k)\sum_{j=1}^{n}\mathbb{P}(W^{k,n}=j)\left(x^{j}-x^{n}\right)=\mathcal{G}h(n),$

where in the last line $h$ is seen as a function of $n$ and $\mathcal{G}$ is
the generator of the block-counting process of the symmetric coalescent
defined in (2.12), in the case $a=0$. This completes the proof for the case
$\alpha<1$.

If $\alpha=1$ and $f$ is defined as previously, we have

$\displaystyle\mathcal{A}f(x)\ =\ \mathcal{A}_{1}f(x)+\mathcal{A}_{2}f(x)$

where $\mathcal{A}_{2}$ is the infinitesimal generator of $\\{X_{t}\\}_{t\geq
0}$ for the case $\alpha<1$ and $\mathcal{A}_{1}$ is the generator of the
Wright-Fisher diffusion. Let $\\{K_{t}\\}_{t\geq 0}$ be the block-counting
process of the Kingman coalescent and $\\{Y_{t}\\}_{t\geq 0}$ be the classical
Wright-Fisher diffusion. We recall the well known moment duality between these
two processes,

$\mathbb{E}(Y_{t}^{n}|Y_{0}=x)\ =\ \mathbb{E}(x^{K_{t}}|K_{0}=n).$ (2.15)

This implies that

$\displaystyle\mathcal{A}_{1}h(x,n)\ $ $\displaystyle=\
\frac{1}{2}x(1-x)\frac{\partial^{2}h(x,n)}{\partial x}$ $\displaystyle=\
\binom{n}{2}\left(h(x,n-1)-h(x,n)\right),$

where the last line corresponds to the generator of $\\{K_{t}\\}_{t\geq 0}$.
Combining this with the result for the case $\alpha<1$, we have

$\displaystyle\mathcal{A}h(x,n)\ $ $\displaystyle=\
\binom{n}{2}\left(x^{n-1}-x^{n}\right)+\sum_{k\geq 1}F^{0}(k)\sum_{j\leq
k}\mathbb{P}(W^{k,i}=j)\left(x^{j}-x^{n}\right)$
$\displaystyle=\mathcal{G}h(x,n),$

where $\mathcal{G}$ is the infinitesimal generator of the block-counting
process of the symmetric coalescent for the case $a=1$. This completes the
proof. ∎

##  3 A topological interlude

As explained in the introduction, in the case of the Wright-Fisher model with
long drastic bottlenecks and long soft bottlenecks, we also want to prove the
convergence of the frequency process to a diffusion with jumps, which is
moment dual to the block-counting process of a bottleneck coalescent. In these
cases, the frequency process is not Markovian (the transition rates depend on
whether the population is undergoing a bottleneck or not) and can have
important fluctuations during the bottlenecks (see Figure 4 for an
illustration), that prevent the convergence in the Skorokhod $J_{1}$ (and
$M_{1}$) topology. However, the points that prevent this convergence are
exactly the accumulation points of the times at which a bottleneck occurred in
the discrete models, that have Lebesgue measure 0 when time is rescaled by
$N^{\alpha}$ and $N$ tends to infinity.

For $T>0$, we denote by $D[0,T]$ the space of real-valued càdlàg functions
defined on $[0,T]$. We will introduce a new metric for convergence in measure
on $D[0,T]$ (see Meyer and Zheng [21]). A sequence of càdlàg functions
converges in measure to another càdlàg function if for any $\epsilon>0$, the
Lebesgue measure of the set of points for which the distance to the limit is
higher than $\epsilon$ converges to 0. We say that a sequence of stochastic
processes converges in measure to another process on $D[0,T]$ if it converges
weakly, as probability measures on $D[0,T]$, in the topology induced by
convergence in measure. Lemma 1 in [21] states that convergence is measure is
equivalent to convergence in the pseudopath space i.e. that $x_{n}\to x$ in
measure on $D[0,T]$ if and only if for any continuous bounded function
$g:[0,T]\times\mathbb{R}\mapsto\mathbb{R}$,

$\displaystyle\lim_{n\rightarrow\infty}\int_{0}^{T}g(s,x_{n}(s))ds$
$\displaystyle=$ $\displaystyle\int_{0}^{T}g(s,x(s))ds,$

and $x_{n}(T)\to x(T)$. This is also known as convergence in the Meyer-Zheng
topology.

Our metric has the advantage of being a slight modification of the Skorokhod
$J_{1}$ distance [32] and thus some techniques to manipulate it are well
known. Our method can be generalized to the study of other processes that can
have strong fluctuations (sparks) only in a set of timepoints that has
Lebesgue measure 0 in the limit. Let us denote by $||\cdot||$ the uniform norm
on $D[0,T]$.

######  Definition 3.1.

Fix $T>0$. Let $x_{1},x_{2}\in D[0,T]$ and $\mathcal{I}$ the set of finite
unions of càdlàg intervals on $[0,T]$ i.e.

$\mathcal{I}=\\{I=\cup_{i=1}^{n}[a_{i},b_{i}):n\in\mathbb{N},0\leq
a_{i}<b_{i}\leq T\\}.$

We say that a function is a Skorokhod reparameterisation of time (SRT) if it
is strictly increasing, continuous with a continuous inverse and we define

$\mathcal{F}=\\{f:[0,T]\mapsto[0,T]:f\text{ is a SRT}\\}.$

Let $\lambda$ be the Lebesgue measure on $[0,T]$ and $Id$ the identity map in
$[0,T]$. For any $x_{1},x_{2}\in D[0,T]$, define

$d_{\lambda}(x_{1},x_{2})=\inf_{A\in\mathcal{I},f\in\mathcal{F}}\\{||\mathds{1}_{A}(x_{1}-x_{2}\circ
f)||\vee||Id-f||\vee\lambda([0,T]/A)\vee|x_{1}(T)-x_{2}(T)|\\}.$

The topology induced by $d_{\lambda}$ is separable. Indeed, the set of
$\mathbb{Q}$-valued staircase functions with discontinuities in $\mathbb{Q}$
is a countable and dense set. However the resulting metric space is not
complete. As a counterexample, one can think of the sequence
$f_{n}=\sum_{i=1}^{n}(-1)^{i}\mathds{1}_{[1/(i+1),1/i)}$ and its behavior at
0.

######  Proposition 3.2.

The mapping $d_{\lambda}$ is a metric in the Skorokhod space $D[0,T]$ for
every $T>0$ and if $\lim_{n\rightarrow 0}d_{\lambda}(x_{n},x)=0$, then
$x_{n}\rightarrow x$ in measure on the Skorokhod space.

###### Proof.

It is clear that, if $x_{1}=x_{2}$, then $d_{\lambda}(x_{1},x_{2})=0$. Now let
us prove the converse. Assume that $x_{1}\neq x_{2}$. If
$|x_{1}(T)-x_{2}(T)|>0$ then $d_{\lambda}(x_{1},x_{2})>0$. Now, if
$|x_{1}(T)-x_{2}(T)|=0$, then there exists a point $\tau\in[0,T)$ such that
$|x_{1}(\tau)-x_{2}(\tau)|>0$. Let us call $y=|x_{1}-x_{2}|$ and observe that
$y\in D[0,T]$. Using the right continuity of $y$ at the point $\tau$, we know
that there exist $\epsilon,\delta>0$ such that $\forall
t\in[\tau,\tau+\delta),\ y(t)>\epsilon$. Let $A\in\mathcal{I}$. If
$[\tau,\tau+\delta)\subset A^{c}$ then $\lambda(A^{c})>\delta$. Otherwise,
$||\mathds{1}_{A}(x_{1}-x_{2})||>\epsilon$. Now, take $f\in\mathcal{F}$ such
that $||Id-f||<\delta/2$, then there exists $\bar{\tau}\in[\tau,\tau+\delta)$
such that for all $t\in[\bar{\tau},\bar{\tau}+\delta/2)$,
$\bar{y}(t)=|x_{1}(t)-x_{2}(f(t))|>\epsilon.$ Repeating the argument, we
conclude that $d_{\lambda}(x_{1},x_{2})>\min\\{\delta/2,\epsilon\\}$, and thus
$x_{1}=x_{2}$ if and only if $d_{\lambda}(x_{1},x_{2})=0$.

Symmetry follows by the same arguments used by Skorokhod in the case of
$J_{1}$ [32]. In fact $f\in\mathcal{F}$ implies that $f(0)=0$ and $f(T)=T$ and
that $f^{-1}\in\mathcal{F}$.Using this observation, it is easy to see that
$d_{\lambda}(x_{1},x_{2})=d_{\lambda}(x_{2},x_{1})$.

To show that the triangle inequality holds, let $x_{1},x_{2},x_{3}\in D[0,T]$
and observe that for any $A,B\in\mathcal{I}$ and $f_{2},f_{3}\in\mathcal{F}$
it holds that $A\cup B\in\mathcal{I}$ and $f_{1}:=f_{3}\circ
f_{2}\in\mathcal{F}.$ The triangle inequality follows from four observations
(most of them due to Skorokhod [32]).

First,

$\displaystyle||\mathds{1}_{A\cup B}(x_{1}-x_{2}\circ f_{1})||$
$\displaystyle\leq$ $\displaystyle||\mathds{1}_{A\cup B}(x_{1}-x_{3}\circ
f_{2})||+||\mathds{1}_{A\cup B}(x_{3}\circ f_{2}-x_{2}\circ f_{3}\circ
f_{2})||$ $\displaystyle\leq$ $\displaystyle||\mathds{1}_{A}(x_{1}-x_{3}\circ
f_{2})||+||\mathds{1}_{B}(x_{3}-x_{2}\circ f_{3})||.$

Second,

$\displaystyle||Id-f_{1}||$ $\displaystyle\leq$ $\displaystyle||Id-
f_{2}||+||Id-f_{3}||.$

Third,

$\displaystyle\lambda((A\cup B)^{c})$ $\displaystyle\leq$
$\displaystyle\lambda(A^{c})+\lambda(B^{c}).$

And finally,

$\displaystyle|x_{1}(T)-x_{2}(T)|\leq|x_{1}(T)-x_{3}(T)|+|x_{3}(T)-x_{2}(T)|.$

Putting all these observations together, we conclude that the triangle
inequality holds, and thus $d_{\lambda}$ is a metric.

Finally, we need to prove that convergence according to $d_{\lambda}$ implies
convergence in the Meyer-Zheng topology. Let $x,x_{1},x_{2},...\in D[0,T]$ and
let $g:[0,T]\times\mathbb{R}\mapsto\mathbb{R}$ be a continuous bounded
function. Assume that $\lim_{n\rightarrow\infty}d_{\lambda}(x_{n},x)=0.$ For
any $n\in\mathbb{N}$, there exists $\epsilon_{n}\to 0$, $A_{n}\in\mathcal{I}$
and $f_{n}\in\mathcal{F}$ such that

$||\mathds{1}_{A_{n}}(x_{n}-x\circ f_{n})||\vee||Id-
f_{n}||\vee\lambda(A_{n}^{c})\vee|x(T)-x_{n}(T)|<d_{\lambda}(x_{n},x)+\epsilon_{n},$

which implies that

$\lim_{n\to\infty}||\mathds{1}_{A_{n}}(x_{n}-x\circ f_{n})||\vee||Id-
f_{n}||\vee\lambda(A_{n}^{c})\vee|x(T)-x_{n}(T)|=0.$

Since $g$ is continuous, bounded and $||Id-f_{n}||\to 0$, we have

$\displaystyle\lim_{n\to\infty}\int_{0}^{T}g(s,x_{n}(s))ds$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\int_{A_{n}}g(s,x_{n}(s))ds+\lim_{n\to\infty}\int_{A_{n}^{c}}g(s,x_{n}(s))ds.$

First, as $d_{\lambda}(x_{n},x)\to 0$ and $\epsilon_{n}\to 0$ , we have
$\lambda(A_{n}^{c})\to 0$ and $g$ is bounded, so

$\displaystyle\lim_{n\to\infty}\int_{A_{n}^{c}}g(s,x_{n}(s))ds=0.$

Second,

$\displaystyle\int_{A_{n}}g(s,x_{n}(s))ds=\int_{A_{n}}g(f_{n}(s),x(f_{n}(s)))ds+\int_{A_{n}}\left(g(s,x_{n}(s))-g(f_{n}(s),x(f_{n}(s)))\right)ds,$

where

$\displaystyle|\int_{A_{n}}\left(g(s,x_{n}(s))-g(f_{n}(s),x(f_{n}(s)))\right)ds|\leq\sup_{s\in
A_{n}}\\{|g(s,x_{n}(s))-g(f_{n}(s),x(f_{n}(s))|\\}T,$

and as $g$ is bounded, continuous, $A_{n}$ is relatively compact and $||Id-
f_{n}||\to 0$,

$\displaystyle\lim_{n\to\infty}\sup_{s\in
A_{n}}\\{|g(s,x_{n}(s))-g(f_{n}(s),x(f_{n}(s))|\\}=0.$

Finally, using the fact that $f_{n}^{-1}\in\mathcal{F}$, and
$\lim_{n\to\infty}||Id-f_{n}||=0$, which implies that
$\lim_{n\to\infty}||Id-f^{-1}_{n}||=0$,

$\displaystyle\lim_{n\to\infty}\int_{A_{n}}g(f_{n}(s),x(f_{n}(s)))ds$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\int_{f_{n}^{-1}(A_{n})}g(t,x(t))(f_{n}^{-1})^{\prime}(t)dt$
$\displaystyle=$ $\displaystyle\int_{0}^{T}g(t,x(t))dt.$

Combining these facts, we conclude that $x_{n}\rightarrow x$ in the pseudopath
space, which completes the proof. ∎

######  Remark 3.3.

Convergence in $J_{1}$ implies convergence according to $d_{\lambda}$. To see
this, take $A=[0,T]$.

Convergence in the sense of $d_{\lambda}$ is close to convergence in measure.
However, it is easy to see that it is not equivalent. The reason is that the
difference of the functions in the point $\\{T\\}$ is crucial in order to have
convergence according to $d_{\lambda}$ and $\\{T\\}$ is clearly a set of
Lebesgue measure zero. It seems feasible to modify $d_{\lambda}$ in order to
have the equivalence, but we decide to stay with $d_{\lambda}$ as it is
because it is a minimalistic modification of the Skorokhod $J_{1}$ distance.
Simply removing the term $|x_{1}(T)-x_{2}(T)|$ would cause that the paths that
differ only on the last point would be at distance zero and then $d_{\lambda}$
would only be a pseudometric.

##  4 Coalescents with drastic bottlenecks

###  4.1 The drastic bottleneck coalescent

Now we consider bottlenecks that are drastic but can last for several
generations. As we shall see, when the bottlenecks last for more than one
generation the genealogy is not described by a symmetric coalescent anymore.

######  Definition 4.1 (Wright-Fisher model with long drastic bottlenecks).

Fix $\alpha\in(0,1]$, $\eta>0$, $N\in\mathbb{N}$ and $F^{0}$ and $\mathrm{L}$
two probability measures in $\mathbb{N}$. Let $\\{F_{i}\\}_{i\in\mathbb{N}}$
be a sequence of i.i.d. random variables of law $F^{0}$. Let
$\\{l_{i,N}\\}_{i\in\mathbb{N}}$ and $\\{s_{i,N}\\}_{i\in\mathbb{N}}$ be two
sequences of independent positive random variables such that for all $i\geq
1,\ l_{i,N}$ converges in distribution to $\mathrm{L}$ and $s_{i,N}$ follows a
geometric distribution of parameter $\eta/N^{\alpha}$. In the Wright-Fisher
model with long drastic bottlenecks the sequence of population sizes
$\\{R^{N}_{g}\\}_{g\in\mathbb{Z}_{+}}$ is given by

$R^{N}_{g}\ =\ \left\\{\begin{array}[]{ll}\min(F_{m},N)\textrm{ if
}\sum_{i=1}^{m-1}(s_{i,N}+l_{i,N})+s_{m,N}<g\leq\sum_{i=1}^{m}(s_{i,N}+l_{i,N})\\\
N\textrm{ otherwise}\end{array}\right.$

######  Remark 4.2.

As $\mathrm{L}$ does not depend on $N$, $\mathrm{L}/N^{\alpha}\to 0$ in
distribution, which ensures that even if the bottlenecks last for several
generations, their duration is negligible when time is rescaled by
$N^{\alpha}$. Also, in that time scale, when $N\to\infty$, the distribution of
the time between two bottlenecks converges to an exponential distribution of
parameter $\eta$.

In the limit when $N\to\infty$, the genealogy of this model can be described
by the drastic bottleneck coalescent that we now define. As for the symmetric
coalescent, the idea is that, in the genealogy there is a ‘Kingman part’
corresponding to what happens outside the bottlenecks (where the population
size goes to infinity) and a ‘simultaneous multiple collisions’ part
corresponding to what happens during the bottlenecks.

To define this type of event, we start by fixing $k,g\in\mathbb{N}$ and
considering the (partition-valued) ancestral process of a classical Wright-
Fisher model with constant population size $k$, running for $g$ generations.
The blocks obtained are given labels in $[k]$. The block labelled $i$ contains
all the descendants of individual $(i,0)$. We then define the following random
variables:

  * •

$K^{k,g,b}$ is the number of ancestors of a sample of size $b\leq k$.

  * •

$(A^{k,g}_{1},\dots,A^{k,g}_{k})$ are the family sizes: $A^{k,g}_{i}$ is the
size of the block labelled $i$ and $A^{k,g}_{i}=0$ if there is no block
labelled $i$. In other words, $A^{k,g}_{i}$ is the number of descendants of
individual $i$ after $g$ generations. We denote by $\mathrm{A}^{k,g}$ the
distribution, in $E^{k}=\\{(i_{1},\dots.i_{k}),\sum i_{j}=k\\}$, of
$(A^{k,g}_{1},\dots,A^{k,g}_{k})$.

  * •

Let $V^{k,n}_{i}$ denote the number of balls allocated to box $i$ in the
paintbox construction of the symmetric coalescent. We define a biased version
of $\mathrm{A}^{k,g}$ as follows: For $n\in\mathbb{N}\cup\infty$ and
$i\in\\{1,\dots,k\\}$,

$\bar{A}^{k,g,n}_{i}\ =\ \sum_{j=1}^{k}V^{k,n}_{j}\mathds{1}_{\\{j\textrm{
belongs to the block labelled }i\\}}.$

######  Definition 4.3 (The drastic bottleneck coalescent).

Fix $F^{0}$ and $\mathrm{L}$ two probability measures in $\mathbb{N}$ and
$\eta>0$. The drastic bottleneck coalescent is defined by the following
transition rates. For each $b\geq 2$ and $k_{1},\ldots,k_{r}$ such that
$\sum_{i=1}^{r}k_{i}=b$, each $[b,(k_{1},\dots,k_{r})]$-collision happens at
rate

$\displaystyle\lambda_{b,(k_{1},\dots,k_{r})}\ =$ $\displaystyle\ a\
\mathds{1}_{\\{r=b-1,k_{1}=2\\}}$ $\displaystyle+\ $
$\displaystyle\mathcal{N}(n,(k_{1},\dots,k_{r}))^{-1}\eta\sum\limits_{k\geq
r}F^{0}(k)\sum\limits_{g\geq 1}\mathrm{L}(g)\sum\limits_{i_{1}\neq\dots\neq
i_{r}}\mathbb{P}(\bar{A}^{k,g-1,b}_{i_{1}}=k_{1},\dots,\bar{A}^{k,g-1,b}_{i_{r}}=k_{r}).$

In words, there are $b$ lineages and a bottleneck of size $k$ and duration $g$
occurs. First each one of the $b$ lineages chooses a parent amongst the $k$
individuals of the last generation of the bottleneck. There remain $W^{k,b}$
lineages (each one containing $V^{k,b}_{1},\dots,V^{k,b}_{k}$ original
lineages). Then the bottleneck still lasts for $g-1$ more generations, in
which the lineages merge as in a Wright-Fisher model with population size $k$
(see Figure 3 for an illustration).

Figure 3: A $[6,(4,2)]$-collision ilustrated. Before the bottleneck, only
‘Kingman type’ mergers occur. Each of the 6 remaining lineages is allocated to
one of the 5 individuals of the last generation of the bottleneck, and we have
$V^{5,6}=(3,0,1,0,2)$. Then, the system evolves for 3 more generations as a
Wright-Fisher model. The family sizes are $A^{5,4}_{1}=3,\ A^{5,4}_{2}=0,\
A^{5,4}_{3}=1,\ A^{5,4}_{4}=1$ and $A^{5,4}_{5}=0$. The biased family sizes
are $\bar{A}^{5,4,6}_{1}=4,\ \bar{A}^{5,4,6}_{4}=2$ and
$\bar{A}^{5,4,6}_{i}=0$ otherwise.

As a consequence, we have the following result.

######  Proposition 4.4.

The block-counting process of the drastic bottleneck coalescent has the
following transition rates

$q_{ij}\ =\ a\binom{i}{2}\mathds{1}_{\\{j=i-1\\}}+\eta\sum_{k\geq
r}F^{0}(k)\sum_{g\geq 1}\mathrm{L}(g)\mathbb{P}(K^{k,g-1,W^{k,i}}=j).$

As in the previous section, we can define $\mathcal{\bar{G}}$, the
infinitesimal generator of the block-counting process
$\\{\bar{N}_{t}\\}_{t\geq 0}$. For any function $h:\mathbb{N}\to\mathbb{R}$,
we have

$\mathcal{\bar{G}}h(n):=a\binom{n}{2}\left(h(n-1)-h(n)\right)\ +\
\eta\sum_{k\geq 1}F^{0}(k)\sum_{g\geq
1}\mathrm{L}(g)\sum_{j=1}^{n-1}\mathbb{P}(K^{k,g-1,W^{k,n}}=j)\left(h(j)-h(n)\right).$
(4.16)

######  Remark 4.5.

As in the previous section, we can describe the drastic bottleneck coalescent
using a paintbox construction. When the bottleneck lasts for $g$ generations,
it would correspond to iterating $g$ times the paintbox construction of the
symmetric coalescent.

###  4.2 Duality with the Wright-Fisher model with long drastic bottlenecks

Now, we consider the Wright-Fisher model with long drastic bottlenecks from
Definition 4.1, with two types of individuals. We denote by
$\\{\bar{X}^{N}_{g}\\}_{g\in\mathbb{N}}$ the frequency process associated with
that model. In the following, ‘$\overset{d_{\lambda}}{\Longrightarrow}$’
denotes weak convergence in the topology induced by $d_{\lambda}$.

######  Theorem 4.6.

Fix $\alpha\in(0,1]$ and $\eta>0$. Let $F^{0}$ and $\mathrm{L}$ be two
probability measures in $\mathbb{N}$. Consider the sequence of processes
$\\{\bar{X}^{N}\\}_{N\in\mathbb{N}}$, such that
$\bar{X}^{N}:=\\{\bar{X}^{N}_{g}\\}_{g\in\mathbb{Z}_{+}}$ is the frequency
process associated with the Wright-Fisher model with long drastic bottlenecks
parametrized by $\alpha,\ \eta,\ N$, $F^{0}$ and $\mathrm{L}$ (see Definition
4.1). Then, for any $T>0$, in $D[0,T]$,

$\\{\bar{X}^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq T}\
\overset{d_{\lambda}}{\underset{N\to\infty}{\Longrightarrow}}\
\\{\bar{X}_{t}\\}_{0\leq t\leq T},$

where $\\{\bar{X}_{t}\\}_{0\leq t\leq T}$ is the unique strong solution of the
SDE

$d\bar{X}_{t}\ =\
\mathds{1}_{\\{\alpha=1\\}}\sqrt{\bar{X}_{t}(1-\bar{X}_{t})}dB_{t}\ +\
\int_{U_{0}}\sum_{i=1}^{k}\frac{a_{i}}{k}\left(\mathds{1}_{\\{u_{i}\leq\bar{X}_{t^{-}}\\}}-\bar{X}_{t^{-}}\right)\bar{N}(dt,dk,dg,da,du),$
(4.17)

where $\\{B_{t}\\}_{t\geq 0}$ is a standard Brownian motion and $\bar{N}$ is a
compensated Poisson measure on $(0,\infty)\times U_{0}$ with
$U_{0}=\mathbb{N}\times\mathbb{N}\times E^{k}\times[0,1]^{\mathbb{N}}$.
$\bar{N}$ has intensity $\eta ds\otimes
F^{0}(dk)\otimes\mathrm{L}(dg)\otimes\mathrm{A}^{k,g-1}(da)\otimes du$, where
$du$ is the Lebesgue measure on $[0,1]^{\mathbb{N}}$ and $\mathrm{A}^{k,g-1}$
is the distribution in $E^{k}$ of the family sizes in a classical Wright-
Fisher model with population size $k$, after $g-1$ generations (as defined in
Section 4.1), i.e. $\mathrm{A}^{k,g-1}$ is the distribution of a vector whose
$i$-th coordinate corresponds to the number of descendants of individual $i$
after $g-1$ generations in a classical Wright-Fisher model with population
size $k$.

######  Remark 4.7.

This result implies that $\\{\bar{X}^{N}_{\lfloor
N^{\alpha}t\rfloor}\\}_{0\leq t\leq T}$ converges to $\\{\bar{X}_{t}\\}_{0\leq
t\leq T}$ in measure on the Skorokhod space. As already mentioned in the
introduction (and illustrated in Figure 4), it is impossible to have
convergence in the $J_{1}$ or $M_{1}$ Skorokhod topology.

Figure 4:  (a) represents a realization of the Wright-Fisher model with long
drastic bottlenecks in which the usual population size is $N=8$. We observe
that $F_{1}=2$, $s_{1,8}=2$ and $l_{1,8}=2$, meaning that in generation $2$ a
bottleneck that reduces the population to $2$ individuals, starts and last for
$2$ generations. Similarly, $F_{2}=4$, $s_{2,8}=3$, $l_{2,8}=5$ and
$s_{3,8}\geq 4$. In (c), the frequency process (associated with (a)) is
colored according to the population size: red outside the bottlenecks and blue
during the bottlenecks. In (b) we observe the result of collapsing the
bottlenecks in the Wright-Fisher model with long drastic bottlenecks
represented in (a). We observe that a lot of the fluctuations of the frequency
process are lost. In (d), the red line is the frequency process and the blue
excursions, that were present in (c), are lost due to the collapsing of the
bottlenecks. These blue excursions (sparks) make it impossible to have
convergence in $J_{1}$ or $M_{1}$ to the diffusion with jumps that is the
solution of (4.17).

To understand why the jump term of (4.17) takes this form, we need to think
about what happens during a bottleneck of size $k$ and length $g$. When the
bottleneck begins, only $k$ individuals of the infinite population survive.
The term ‘$\mathds{1}_{\\{u_{i}\leq\bar{X}_{t^{-}}\\}}$’ comes from the fact
that individual $i$ is of type 1 with probability $\bar{X}_{t^{-}}$ and of
type $0$ with probability $1-\bar{X}_{t^{-}}$. This generation corresponds to
time $0$ for the bottleneck. Then the bottleneck lasts for another $g-1$
generations and individual $i$ has $a_{i}$ descendants which are all of the
same type as her. See Figure 5 for an illustration of this jump term.

Figure 5: The jump part of (4.17) illustrated. Type 1 individuals are
represented in pink and type 0 individuals in green.

Again, before proving Theorem 4.6 we shall make sure that a solution to
Equation (4.17) exists.

######  Lemma 4.8.

For any probability measures $F^{0}$ and $\mathrm{L}$ in $\mathbb{N}$ and any
$\alpha\in(0,1],\ \eta>0$, there exists a unique strong solution to the SDE
(4.17).

######  Remark 4.9.

To prove this result, one could rewrite the stochastic differential equation
(4.17) in terms of a measure $\Xi$ on $\Delta$ that would depend on $F^{0}$,
$\mathrm{L}$ and $\mathrm{A}^{k,g-1}$ (in fact for $k\in\mathbb{N}$ and
$(a_{1},\dots,a_{k})$ drawn from the distribution $\mathrm{A}^{k,g-1}$, the
re-ordering of $(a_{1}/k,\dots,a_{k}/k)$ is an element of $\Delta^{*}$). Then
Lemma 4.8 would follow from Lemma 3.6 in [13] (and Theorem 4.11 would follow
form Proposition 3.8 in the same reference). However, we have decided not to
modify (4.17) and we give a proof that is more instructive, as it is connected
to the parameters of the Wright-Fisher model with long drastic bottlenecks and
sheds light on the connection between this Wright-Fisher model and the drastic
bottleneck coalescent.

###### Proof.

We use Theorem 5.1 in [20]. In particular, we need to verify conditions (3.a),
(3.b) and (5.a) of that paper. Condition (3.a) is trivial, as in our case the
drift coefficient is equal to 0. To prove condition (3.b), we have to prove
that there exists a constant $K$ such that for every $x,y\in[0,1]$,

$\displaystyle\int_{\mathbb{N}}\int_{\mathbb{N}}F^{0}(k)\mathrm{L}(g)\mathbb{E}\left[\left(\sum_{i=1}^{k}\frac{a_{i}}{k}\left(\mathds{1}_{\\{u_{i}\leq
x\\}}-x-\mathds{1}_{\\{u_{i}\leq y\\}}+y\right)\right)^{2}\right]$
$\displaystyle+\mid\sqrt{x(1-x)}-\sqrt{y(1-y)}\mid\ \leq\ K\mid x-y\mid.$
(4.18)

First, we use the fact that

$\displaystyle\mid\sqrt{x(1-x)}-\sqrt{y(1-y)}\mid\leq 4\mid x-y\mid,$

see for example claim (26) in [13]. Second, without lost of generality we
assume that $x>y$ and we have

$\displaystyle\int_{\mathbb{N}}\int_{\mathbb{N}}F^{0}(k)\mathrm{L}(g)\mathbb{E}\left[\left(\sum_{i=1}^{k}\frac{a_{i}}{k}\left(\mathds{1}_{\\{u_{i}\leq
x\\}}-x-\mathds{1}_{\\{u_{i}\leq y\\}}+y\right)\right)^{2}\right]$
$\displaystyle=\int_{\mathbb{N}}\int_{\mathbb{N}}F^{0}(k)\mathrm{L}(g)\mathbb{E}\left[\sum_{i=1}^{k}\frac{a_{i}^{2}}{k^{2}}\left((B_{i}^{x}-B_{i}^{y})-(x-y)\right)^{2}\right],$

where the $B_{i}^{x}$’s and the $B_{i}^{y}$’s are (dependent) Bernoulli random
variables of parameter $x$ and $y$ respectively. Using the fact that
$(B_{i}^{x}-B_{i}^{y})$ is Bernoulli of parameter $x-y$ we have:

$\displaystyle\int_{\mathbb{N}}\int_{\mathbb{N}}F^{0}(k)\mathrm{L}(g)\mathbb{E}\left[\sum_{i=1}^{k}\frac{a_{i}^{2}}{k^{2}}\left((B_{i}^{x}-B_{i}^{y})-(x-y)\right)^{2}\right]$
$\displaystyle\leq((x-y)(1-(x-y))\
\int_{\mathbb{N}}\int_{\mathbb{N}}F^{0}(k)\mathrm{L}(g)\mathbb{E}\left(\sum_{i=1}^{k}\frac{a_{i}^{2}}{k^{2}}\right)$
$\displaystyle\leq(x-y),$

where we used the facts that $\frac{a_{i}}{k}\leq 1$,
$\sum_{i=1}^{k}{a_{i}}=k$ and $F^{0}$ and $\mathrm{L}$ are probability
measures. This proves claim (4.18). Finally, condition (5.a) in [20] is
verified because, using similar arguments as before,

$\displaystyle\int_{\mathbb{N}}\int_{\mathbb{N}}F^{0}(k)\mathrm{L}(g)\mathbb{E}\left[\left(\sum_{i=1}^{k}\frac{a_{i}}{k}\left(\mathds{1}_{\\{u_{i}\leq
x\\}}-x\right)\right)^{2}\right]+x(1-x)\ \leq\ 2.$ (4.19)

This implies that we can apply Theorem 5.1 in [20] and conclude the existence
and uniqueness of strong solution of (4.17). ∎

######  Remark 4.10.

The fact that $F^{0}$ and $\mathrm{L}$ are probability measures guarantees the
existence of a unique strong solution to (4.17), but a sufficient condition is
that

$\int_{\mathbb{N}}\int_{\mathbb{N}}F^{0}(k)\mathrm{L}(g)\mathbb{E}\left(\sum_{i=1}^{k}\frac{a_{i}^{2}}{k^{2}}\right)\
<\ \infty,$

where $(a_{1},\dots,a_{k})$ is distributed as $\mathrm{A}^{k,g-1}$.

We are now ready to prove Theorem 4.6. The strategy of the proof is the
following. We want to prove the convergence of the sequence of processes
$\\{\bar{X}^{N}\\}_{N\in\mathbb{N}}$, which are not Markovian, to a diffusion
with jumps. In Step 1 we will construct an auxiliary process $V^{N}$ that
corresponds to collapsing each bottleneck into one single time step into
$\bar{X}^{N}$ and that is Markovian. We will prove that
$d_{\lambda}(\\{\bar{X}^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq
T},\\{V^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq T})\rightarrow 0$ in
probability. In Step 2, we prove, using standard techniques for Markov
processes, that $\\{V^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq T}$
converges weakly to $\\{\bar{X}_{t}\\}_{0\leq t\leq T}$, as processes in
$D[0,T]$ endowed with Skorokhod’s $J_{1}$ topology. Combining the two steps,
the conclusion is straightforward.

This method can be generalized to any case in which convergence in the
Skorokhod topology is prevented by strong fluctuations that happen in a set of
times that has Lebesgue measure 0 in the limit.

###### Proof of Theorem 4.6.

Step 1. We say that a generation is in a bottleneck and write $g\in B^{N}$ if
for some $m$,
$\underline{t}_{m}:=\sum_{i=1}^{m-1}(s_{i,N}+l_{i,N})+s_{m,N}<g\leq\bar{t}_{m}:=\sum_{i=1}^{m}(s_{i,N}+l_{i,N})$.
Let $g_{i}$ be the $i$-th generation that is not in a bottleneck. Define

$V^{N}_{i}=\bar{X}^{N}_{g_{i}}.$

Note that while $\bar{X}^{N}$ is not a Markov process (as one needs to know if
there is a bottleneck or not to calculate the transition probabilities),
$V^{N}$ is a Markov process, and it is constructed from $\bar{X}^{N}$ simply
by collapsing the entire bottleneck into one step. Now, consider the random
$R^{N}_{g}$-measurable projection $\pi^{N}:\mathbb{R}_{+}\mapsto\mathbb{N}$
defined as $\pi^{N}(t)=\max\\{g\in\mathbb{N}:g<t,g\in(B^{N})^{c}\\}$ and we
define the process $\\{Z^{N}_{t}\\}_{t\geq 0}$ by

$Z^{N}_{t}=\bar{X}^{N}_{\pi^{N}(t)}.$

The difference between $V^{N}_{\lfloor t\rfloor}$ and $Z^{N}_{t}$ is that
$V^{N}_{\lfloor t\rfloor}$ can move every unit of time, while $Z^{N}_{t}$
stays still during the duration of a bottleneck (see Figure 6 for an
illustration).

Taking $f\in\mathcal{F}$ to be the linear function having slope 1 whenever
$\lfloor t\rfloor$ is not in a bottleneck and having slope
$({\bar{t}_{m}-\underline{t}_{m}})$ whenever $\lfloor t\rfloor$ is inside the
$m$-th bottleneck, we conclude that $||V^{N}_{\lfloor
N^{\alpha}t\rfloor}-Z^{N}_{f(\lfloor N^{\alpha}t\rfloor)}||=0$ (see Figure 6)
and thus

$d_{\lambda}(\\{V^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq
T},\\{Z^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq
T})\leq||Id-f||\leq\frac{\sum_{m=1}^{\infty}(\bar{t}_{m}-\underline{t}_{m})\mathds{1}_{\\{\bar{t}_{m}\leq
TN^{\alpha}\\}}}{N^{\alpha}}.$

Observing that the latter converges to 0 in probability when
$N\rightarrow\infty$, we conclude that

$d_{\lambda}(\\{V^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq
T},\\{Z^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq T})\rightarrow
0\textrm{ in probability}.$

Now, observe that taking
$A^{N}=\cup_{m=1}^{i_{T}^{N}}[\underline{t}_{m},\bar{t}_{m})$ where
$i_{T}^{N}=\sup\\{m:\underline{t}_{m}<TN^{\alpha}\\}$ we have

$||\mathds{1}_{(A^{N})^{c}}(Z^{N}_{\lfloor
N^{\alpha}t\rfloor}-\bar{X}^{N}_{\lfloor N^{\alpha}t\rfloor})||=0\textrm{ in
probability}.$

and thus

$d_{\lambda}(\\{\bar{X}^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq
T},\\{Z^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq T})\rightarrow
0\textrm{ in probability},$

which completes this step.

Figure 6: The curves represent realizations of $X^{N}$, $V^{N}$, $Z^{N}$ and
$Z^{N}\circ{f}$ respectively. The first curve is colored according to the
population size: red outside the bottlenecks and blue during the bottlenecks.

Step 2. As in the proof of Theorem 2.3, the idea is to prove the convergence
of the generator of $\\{V^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{t\geq 0}$, to
the generator of $\\{\bar{X}_{t}\\}_{t\geq 0}$. Provided this claim is true,
we can use Theorem 19.25 and 19.28 of [17] to prove the weak convergence of
$\\{V^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq T}$ towards
$\\{\bar{X}_{t}\\}_{0\leq t\leq T}$.

From Lemma 4.8, $\\{\bar{X}_{t}\\}_{t\geq 0}$ exists and has generator
$\mathcal{\bar{A}}$. Its domain contains twice differentiable functions and
for a function $f\in C^{2}[0,1]$ and $x\in[0,1]$, we have

$\mathcal{\bar{A}}f(x)\ =\
\mathds{1}_{\\{\alpha=1\\}}\frac{1}{2}x(1-x)f^{\prime\prime}(x)\ +\
\eta\sum_{k\geq 1}F^{0}(k)\sum_{g\geq
1}\mathrm{L}(g)\mathcal{\bar{A}}^{k,g}f(x),$ (4.20)

with

$\mathcal{\bar{A}}^{k,g}f(x)=\sum_{i=0}^{k}\mathbb{P}(\bar{Y}^{k,g,x}=i/k)\left(f(i/k)-f(x)\right),$

where $\bar{Y}^{k,g,x}$ is a random variable such that
$\bar{Y}^{k,g,x}=\sum_{i=1}^{k}\frac{a_{i}}{k}\mathds{1}_{\\{u_{i}\leq x\\}}$,
where $(a_{1},\dots,a_{k})$ is distributed as $\mathrm{A}^{k,g-1}$ and
$(u_{1},\ldots,u_{k})$ is uniformly distributed in $[0,1]^{k}$. This generator
can be interpreted in the same way as the jump part of (4.17), see Figure 5.

The discrete generator $\mathcal{\bar{A}}^{N}$ of $\\{V^{N}_{\lfloor
N^{\alpha}t\rfloor}\\}_{t\geq 0}$ (defined as in (2.9)), applied to a function
$f\in C^{2}[0,1]$ in $x\in[0,1]$ can be written as

$\displaystyle\mathcal{\bar{A}}^{N}f(x)=\
N^{\alpha}(1-\frac{\eta}{N^{\alpha}})\mathbb{E}\left(f\left(\frac{\sum_{i=1}^{N}B^{x}_{i}}{N}\right)-f(x)\right)$
$\displaystyle+\eta\sum_{k\geq 1}F^{0}(k)\sum_{g\geq
1}\mathbb{P}(l_{1,n}=g)\sum_{i=0}^{k}\mathbb{P}\left(\bar{Y}^{\min(N,k),g,x}=\frac{i}{\min(N,k)}\right)\left(f(\frac{i}{\min(N,k)})-f(x)\right).$

The first term corresponds to the generator of a classical Wright-Fisher model
(outside the bottlenecks). As already mentioned, it is well-known that when
$\alpha=1$, this term converges when $N\to\infty$ to
$\frac{1}{2}x(1-x)f^{\prime\prime}(x)$, which is the generator of the Wright-
Fisher diffusion. When $\alpha<1$ this term becomes of order $N^{\alpha-1}$
and therefore converges to $0$. The second term corresponds to what happens
during a bottleneck (and again, it can be interpreted using Figure 5). Recall
that $\mathrm{L}/N^{\alpha}\to 0$ in distribution, i.e. in the new time scale
the bottlenecks are instantaneous. As $l_{1,N}$ converges in distribution to
$\mathrm{L}$, the second term in the generator converges when $N\to\infty$ to
the second term of $\mathcal{\bar{A}}$ (see (4.20)). Combining these two
results, we have $\mathcal{\bar{A}}^{N}f\to\mathcal{\bar{A}}f$ uniformly. This
implies that $\\{V^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq T}$
converges weakly in the Skorokhod $J_{1}$ topology to
$\\{\bar{X}_{t}\\}_{0\leq t\leq T}$. Since convergence in $J_{1}$ implies
convergenec in $d_{\lambda}$ we have the desired result. ∎

We fix $\alpha\in(0,1],\ \eta>0$ and $F^{0}$ and $\mathrm{L}$ two probability
measures in $\mathbb{N}$. Let us consider $\\{\bar{N}_{t}\\}_{t\geq 0}$, the
block-counting process of the drastic bottleneck coalescent characterized by
$\alpha$, $\eta$, $F^{0}$ and $\mathrm{L}$. As in the previous section, we are
going to prove a duality relation between this block-counting process and
$\\{\bar{X}_{t}\\}_{t\geq 0}$, the unique strong solution of (4.17) (with the
same parameters).

######  Theorem 4.11.

For every $x\in[0,1],\ n\in\mathbb{N}$, we have

$\mathbb{E}(\bar{X}_{t}^{n}|\bar{X}_{0}=x)\ =\
\mathbb{E}(x^{\bar{N}_{t}}|\bar{N}_{0}=n).$

###### Proof.

We start by recalling a moment duality between the frequency process of a
classical Wright-Fisher model with population size $k$, started at $x$,
denoted by $\\{Y^{k,g,x}\\}_{g\in\mathbb{N}}$, and the number of blocks in the
associated ancestry process of a sample of size $n$,
$\\{K^{k,g,n}\\}_{g\in\mathbb{N}}$, which was established by Möhle
(Proposition 3.5 in [23]). We consider the function $h$ on
$[0,1]\times\mathbb{N}$ such that $h(x,n)=x^{n}$. We have

$\mathbb{E}(h(Y^{k,g,x},n))\ =\ \mathbb{E}(h(x,K^{k,g,n}))$

i.e.,

$\sum_{i=0}^{k}\mathbb{P}(Y^{k,g,x}=i/k)h(i/k,n)\ =\
\sum_{b=1}^{n}\mathbb{P}(K^{k,g,n}=b)h(x,b).$ (4.21)

We start by considering the case $\alpha<1$. Using the Markov property of the
Wright-Fisher model with bottlenecks (i.e. the fact that the $u_{i}$’s are
independent from the $a_{i}$’s) we can rewrite $\bar{Y}^{k,g,x}$ so that for
every $n\in\mathbb{N}$, the generator $\mathcal{\bar{A}}$ applied to $h$ (seen
as a function of $x$) is

$\displaystyle\mathcal{\bar{A}}h(x,n)$ $\displaystyle=\eta\sum_{k\geq
1}F^{0}(k)\sum_{g\geq
1}\mathrm{L}(g)\sum_{j=0}^{k}\sum_{y=0}^{k}\mathbb{P}(\sum_{i=1}^{k}\mathds{1}_{\\{u_{i}\leq
x\\}}=y)\mathbb{P}(Y^{k,g-1,y/k}=\frac{j}{k})(h(\frac{j}{k},n)-h(x,n)).$

Using the Markov property of $\\{Y^{k,g,y/k}\\}_{g\in\mathbb{N}}$ we have that

$\sum_{j=0}^{k}\mathbb{P}(Y^{k,g-1,y/k}=\frac{j}{k})=\sum_{j=0}^{k}\sum_{p=0}^{k}\mathbb{P}(Y^{k,g-2,y/k}=\frac{p}{k})\mathbb{P}(Y^{k,1,p/k}=\frac{j}{k}).$

Then, by definition of the classical Wright-Fisher model, we have that
$Y^{k,1,p/k}$ has the same distribution as
$\frac{1}{k}\sum_{i=1}^{k}B_{i}^{p/k}$, where the $B_{i}^{p/k}$’s are
Bernoulli variables of parameter $p/k$, so using exactly the same computations
as in (2.13) and (2.14), we have that

$\displaystyle\mathbb{E}\left((Y^{k,1,p/k})^{n}\right)\ =\
\mathbb{E}\left((p/k)^{W^{k,n}}\right),$

so,

$\displaystyle\sum_{j=0}^{k}\mathbb{P}(Y^{k,g-1,y/k}=\frac{j}{k})h(\frac{j}{k},n)$
$\displaystyle=\sum_{p=0}^{k}\mathbb{P}(Y^{k,g-2,y/k}=\frac{p}{k})\sum_{j=0}^{k}\mathbb{P}(Y^{k,1,p/k}=\frac{j}{k})h(\frac{j}{k},n)$
$\displaystyle=\sum_{p=0}^{k}\mathbb{P}(Y^{k,g-2,y/k}=\frac{p}{k})\sum_{m=1}^{n}\mathbb{P}(W^{k,n}=m)h(\frac{p}{k},m)$
$\displaystyle=\sum_{m=1}^{n}\mathbb{P}(W^{k,n}=m)\sum_{b=1}^{k}\mathbb{P}(K^{k,g-2,m}=b)h(\frac{y}{k},b)$
$\displaystyle=\sum_{b=1}^{k}\mathbb{P}(K^{k,g-2,W^{k,n}}=b)h(\frac{y}{k},b),$

where, from the second to third line we used the duality relation (4.21).
Replacing into the expression of the generator $\mathcal{\bar{A}}$, we have
that

$\displaystyle\mathcal{\bar{A}}h(x,n)$ $\displaystyle=\eta\sum_{k\geq
1}F^{0}(k)\sum_{g\geq
1}\mathrm{L}(g)\sum_{y=0}^{k}\mathbb{P}(\sum_{i=1}^{k}\mathds{1}_{\\{u_{i}\leq
x\\}}=y)\sum_{b=1}^{k}\mathbb{P}(K^{k,g-2,W^{k,n}}=b)(h(\frac{y}{k},b)-h(x,n)).$

Recall that, if $x\in\\{0,1/k,\dots,1\\}$, the distribution of
$\frac{1}{k}\sum_{i=1}^{k}\mathds{1}_{\\{u_{i}\leq x\\}}$ is exactly the
distribution of $Y^{k,1,x}$, so we can apply (4.21) to $Y^{k,1,x}$ and
$K^{k,1,b}$. However, we need to prove that a similar relation exists when
$x\in[0,1]\setminus\\{0,1/k,\dots,1\\}$. In fact, using again the same
computations as in (2.13) and (2.14), we have that

$\displaystyle\mathbb{E}\left((\frac{1}{k}\sum_{i=1}^{k}\mathds{1}_{\\{u_{i}\leq
x\\}})^{b}\right)\ =\ \mathbb{E}\left(x^{W^{k,b}}\right),$

and we let the reader convince herself that $W^{k,b}$ has the same
distribution as $K^{k,1,b}$. Finally,

$\displaystyle\mathcal{\bar{A}}h(x,n)$ $\displaystyle=\eta\sum_{k\geq
1}F^{0}(k)\sum_{g\geq
1}\mathrm{L}(g)\sum_{b=1}^{k}\mathbb{P}(K^{k,g-2,W^{k,n}}=b)\sum_{a=1}^{k}\mathbb{P}(K^{k,1,b}=a)(h(x,a)-h(x,n))$
$\displaystyle=\eta\sum_{k\geq 1}F^{0}(k)\sum_{g\geq
1}\sum_{a=1}^{k}\mathbb{P}(K^{k,g-1,W^{k,n}}=a)\left(h(x,a)-h(x,n)\right)\ =\
\mathcal{\bar{G}}h(x,n),$

where in the last line we used the Markov property of
$\\{K^{k,g,n}\\}_{g\in\mathbb{N}}$ and $\mathcal{\bar{G}}$ is the generator of
$\\{\bar{N}_{t}\\}_{t\geq 0}$ (defined in 4.16) applied to $h$, seen as a
function of $n$.

If $\alpha=1$ we have

$\displaystyle\mathcal{\bar{A}}h(x,n)\ =\
\mathcal{A}_{1}h(x,n)+\mathcal{\bar{A}}_{2}h(x,n)$

where $\mathcal{\bar{A}}_{2}$ is the infinitesimal generator of
$\\{\bar{X}_{t}\\}_{t\geq 0}$ for the case $\alpha<1$ and $\mathcal{A}_{1}$ is
the generator of the Wright-Fisher diffusion. The proof follows by using the
moment duality between the Wright-Fisher diffusion and the Kingman coalescent,
i.e.

$\ \mathcal{A}_{1}h(x,n)=\ \binom{n}{2}\left(h(x,n-1)-h(x,n)\right).$

This, combined with the case $\alpha<1$ completes the proof. ∎

##  5 Coalescents with soft bottlenecks

###  5.1 Kingman coalescents with continuous time rescaling

In this section we consider bottlenecks that are soft and short. More
precisely, we consider a Wright-Fisher model with bottlenecks where the
sequence of population sizes $\\{R^{N}_{g}\\}_{g\in\mathbb{Z}_{+}}$ is such
that

$R^{N}_{g}=\lfloor Nr_{g}\rfloor+1$

where $\\{r_{g}\\}_{g\in\mathbb{Z}_{+}}$ is a sequence of i.i.d. random
variables with a certain law on $[0,1)$ that we denote by $R$. We review here
some examples, in which the genealogy is a time rescaled Kingman coalescent.
The proofs are based on Möhle’s theorem [25] that can be rewritten as follows.
Let us denote by $\\{\Pi^{N}_{t}\\}_{t\geq 0}$ the ancestral process
describing the genealogy of a Wright-Fisher model with bottlenecks
parametrized by $N$ and $\\{R^{N}_{g}\\}_{g\in\mathbb{N}}$ and let
$\\{\Pi_{t}\\}_{t\geq 0}$ be the standard Kingman coalescent.

######  Proposition 5.1 (Möhle’s theorem for the Wright-Fisher model with
bottlenecks).

Consider

$C_{N}=\sum_{i=1}^{N}\frac{1}{i}\mathbb{P}(R^{N}_{1}=i)\textrm{ and
}D_{N}=\sum_{i=1}^{N}\frac{1}{i^{2}}\mathbb{P}(R^{N}_{1}=i).$

If

$C_{N}\underset{N\to\infty}{\longrightarrow}0\textrm{ and }\
D_{N}/C_{N}\underset{N\to\infty}{\longrightarrow}0,$

then $\\{\Pi^{N}_{\lfloor t/C_{N}\rfloor}\\}_{t\geq 0}$ converges to
$\\{\Pi_{t}\\}_{t\geq 0}$ in the sense of finite dimensional distributions.

Example 1. We start by considering the case where there exists $\epsilon>0$
such that $\mathbb{P}(R>\epsilon)=1$, then the genealogy of the model
converges to a constant time rescaling of the Kingman coalescent. To prove it
we use Möhle’s theorem. In fact,

$C_{N}=\sum_{i=1}^{N}\frac{1}{i}\mathbb{P}(R^{N}_{1}=i)=\sum_{i=\lfloor\epsilon
N\rfloor}^{N}\frac{1}{i}\mathbb{P}(R\in[\frac{i-1}{N},\frac{i}{N}))\in[\frac{1}{N},\frac{1}{\epsilon
N}]$

and

$D_{N}=\sum_{i=1}^{N}\frac{1}{i^{2}}\mathbb{P}(R^{N}_{1}=i)=\sum_{i=\lfloor\epsilon
N\rfloor}^{N}\frac{1}{i^{2}}\mathbb{P}(R\in[\frac{i-1}{N},\frac{i}{N}))\in[\frac{1}{N^{2}},\frac{1}{(\epsilon
N)^{2}}].$

So $C_{N}\rightarrow 0$, $D_{N}/C_{N}\rightarrow 0$ and we can apply
Proposition 5.1.

Example 2. We assume that $R$ is a uniform random variable in $[0,1]$, i.e.
for $g\in\mathbb{Z}_{+}$ $R_{g}^{N}$ is uniformly chosen in $\\{1,\dots,N\\}$.
Informally, this means that going backward in time, one has to wait, on
average, for $N$ generations until the population size is reduced to only 1
individual. Any ancestral lineages present at that time need to coalesce into
one. However, the limiting genealogy is still a Kingman coalescent. In fact,

$C_{N}\ =\ \frac{1}{N}\sum_{i=1}^{N}\frac{1}{i}\sim\frac{\log N}{N}$

and

$D_{N}\ =\ \frac{1}{N}\sum_{i=1}^{N}\frac{1}{i^{2}}=O(1/N).$

So $C_{N}\rightarrow 0$, $D_{N}/C_{N}\rightarrow 0$ and Proposition 5.1
implies that $\\{\Pi^{N}_{\lfloor tN/\log N\rfloor}\\}_{t\geq
0}\rightarrow\\{\Pi_{t}\\}_{t\geq 0}$ in the sense of finite dimensional
distributions.

###  5.2 Subordinated Kingman coalescents

In this section we consider bottlenecks that are soft but that last for
several generations. As in Section 4.1, we are going to assume that, in the
limiting model, the times between two bottlenecks are exponentially
distributed. But this time we are going to assume that the bottlenecks are
soft i.e. $b_{i,N}\to 0$ but $Nb_{i,N}\to\infty$. We present an example
inspired by Birkner et al. [5], in which the limiting genealogy is a
subordinated Kingman coalescent.

######  Definition 5.2 (Wright-Fisher model with long soft bottlenecks).

Fix $\alpha\in(0,1]$, $\eta>0$, $N\in\mathbb{N}$ and $\mathrm{L}_{\gamma}$ a
probability measure on $\mathbb{R}_{+}$. Let $\\{b_{i,N}\\}_{i\in\mathbb{N}}$,
$\\{l_{i,N}\\}_{i\in\mathbb{N}}$ and $\\{s_{i,N}\\}_{i\in\mathbb{N}}$ be three
sequences of independent positive random variables. For any $i\in\mathbb{N}$,
assume that $b_{i,N}\to 0$ in distribution, that
$l_{i,N}/(Nb_{i,N})\to\gamma_{i}$ in distribution, where $\gamma_{i}$ is a
random variable with law $\mathrm{L}_{\gamma}$, and that $s_{i,N}$ follows a
geometric distribution of parameter $\eta/N^{\alpha}$. In the Wright-Fisher
model with long drastic bottlenecks, the sequence of population sizes
$\\{R^{N}_{g}\\}_{g\in\mathbb{Z}_{+}}$ is given by

$R^{N}_{g}\ =\ \left\\{\begin{array}[]{ll}b_{m,N}N\textrm{ if
}\sum_{i=1}^{m-1}(s_{i,N}+l_{i,N})+s_{m,N}<g\leq\sum_{i=1}^{m}(s_{i,N}+l_{i,N})\\\
N\textrm{ otherwise}\end{array}\right..$

As suggested by Birkner et al. [5], we show in Section 5.3 that when
$N\to\infty$ and time is rescaled by $N^{\alpha}$, the genealogy is described
by $\\{\Pi_{S_{t}}\\}_{t\geq 0}$ where $\\{S_{t}\\}_{t\geq 0}$ is a
subordinator (a compound Poisson process with Lévy measure $\eta{L}_{\gamma}$
and drift 1). Proposition 6.3 in [5] states that this subordinated Kingman
coalescent is in fact a $\Xi$-coalescent with characterizing measure

$\Xi=a\delta_{(0,0,\dots)}+\Xi_{KS}\\\ $

where

$\Xi_{KS}(d\zeta)=(\zeta,\zeta)\int_{(0,\infty)}\sum_{j=1}^{\infty}\mathbb{P}(K_{\sigma}=j)\eta\mathrm{L}_{\gamma}(d\sigma)D_{j}(d\zeta),$
(5.22)

and $K_{t}$ is the number of lineages at time $t>0$ in the standard Kingman
coalescent starting with $K_{0}=\infty$ and $D_{j}$ is the law of the re-
ordering of a ($j$-dimensional) Dirichlet $(1,\dots,1)$ random vector
according to decreasing size. This result can be interpreted as follows: the
simultaneous multiple collisions part in the measure $\Xi$ corresponds to the
way the lineages coalesce during the bottlenecks. As the population size
during the bottleneck still goes to infinity, its evolution is still given by
a Kingman coalescent and it lasts for a time distributed according to
$\mathrm{L}_{\gamma}$. The frequencies of the remaining blocks have a
Dirichlet distribution (see for example Corollary 2.1 in [1]).

######  Remark 5.3.

In an informal sense, this model can be seen as a limiting scenario for the
model presented in Section 4.1 when the population size during the bottleneck
goes to infinity. In fact, in the model from Definition 5.2 we have

$\lambda_{b,(k_{1},\dots,k_{r})}=\mathds{1}_{\\{r=b-1,k_{1}=2\\}}+\mathcal{N}(n,(k_{1},\dots,k_{r}))^{-1}\eta\int_{(0,\infty)}{L}_{\gamma}(d\sigma)\mathbb{P}(X^{\sigma,b}=(k_{1},\dots,k_{r}))$

where $X^{\sigma,b}$ is the vector of the sizes of the blocks of a Kingman
coalescent at time $\sigma$, starting with $b$ blocks (without taking into
account the ordering). The latter can be understood as a $k=\infty$ version of
Proposition 4.4.

###  5.3 Duality between the Wright-Fisher model with long soft bottlenecks
and the subordinated Kingman coalescent

Finally, we consider the Wright-Fisher model with long soft bottlenecks from
Definition 5.2, with two types of individuals. We denote by
$\\{\hat{X}^{N}_{g}\\}_{g\in\mathbb{N}}$ the frequency process associated with
that model.

######  Theorem 5.4.

Let $\mathrm{L}_{\gamma}$ be a probability measure on $\mathbb{R}_{+}$. Fix
$\alpha\in(0,1]$ and $\eta>0$. Consider the sequence of processes
$\\{\hat{X}^{N}\\}_{N\in\mathbb{N}}$, such that
$\hat{X}^{N}=\\{\hat{X}^{N}_{g}\\}_{g\in\mathbb{Z}_{+}}$ is the frequency
process associated with the Wright-Fisher model with long soft bottlenecks
parametrized by $\alpha$, $\eta$, $N$ and $\mathrm{L}_{\gamma}$ (see
Definition 5.2). Then, for all $T>0$, in $D[0,T]$,

$\\{\hat{X}^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq
T}\overset{d_{\lambda}}{\underset{N\to\infty}{\Longrightarrow}}\\{\hat{X}_{t}\\}_{0\leq
t\leq T},$

where $\\{\hat{X}_{t}\\}_{t\geq 0}$ is the strong solution of the SDE

$d\hat{X}_{t}\ =\
\mathds{1}_{\\{\alpha=1\\}}\sqrt{\hat{X}_{t}(1-\hat{X}_{t})}dB_{t}\ +\
\int_{[0,1]}\int_{[0,1]}\sum_{i\geq
1}{\zeta_{i}}\left(\mathds{1}_{\\{u_{i}\leq\hat{X}_{t^{-}}\\}}-\hat{X}_{t^{-}}\right)\tilde{N}(dt,d\zeta,du),$
(5.23)

where $\\{B_{t}\\}_{t\geq 0}$ is a standard Brownian motion and $\hat{N}$ is a
compensated Poisson measure on
$(0,\infty)\times\Delta\times[0,1]^{\mathbb{N}}$. $\hat{N}$ has intensity
$\eta ds\otimes\frac{\Xi_{KS}(d\zeta)}{(\zeta,\zeta)}\otimes du$, where $du$
is the Lebesgue measure on $[0,1]^{\mathbb{N}}$ and $\Xi_{KS}$ is the
characterizing measure of the subordinated Kingman coalescent, defined in
(5.22).

Again, before proving Theorem 5.4 we shall make sure that a solution to
Equation (5.23) exists.

######  Lemma 5.5.

For any probability measure $\mathrm{L}_{\gamma}$ in $\mathbb{R}_{+}$ and any
$\alpha\in(0,1],\ \eta>0$, there exists a unique strong solution to the SDE
(5.23).

###### Proof.

See Proposition 3.4 in [13]. ∎

We are now ready to prove Theorem 5.4.

###### Proof of Theorem 5.4.

We use the same strategy as in the proof of Theorem 4.6. Again, if $g_{i}$ is
the $i$-th generation that is not in a bottleneck, we define

$\hat{V}^{N}_{i}=\hat{X}^{N}_{g_{i}}.$

Following Step 1 in the proof of Theorem 4.6 we have

$d_{\lambda}(\\{\hat{X}^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq
T},\\{\hat{V}^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{0\leq t\leq T})\rightarrow
0\textrm{ in probability}.$

Again, we need to prove the convergence of the generator of
$\\{\hat{V}^{N}_{\lfloor N^{\alpha}t\rfloor}\\}_{t\geq 0}$, to the generator
of $\\{\hat{X}_{t}\\}_{t\geq 0}$.

From Lemma 5.5, $\\{\hat{X}_{t}\\}_{t\geq 0}$ exists and has generator
$\mathcal{\hat{A}}$. Its domain contains twice differentiable functions and
for a function $f\in C^{2}[0,1]$ and $x\in[0,1]$, we have

$\displaystyle\mathcal{\hat{A}}f(x)\ =$ $\displaystyle\
\mathds{1}_{\\{\alpha=1\\}}\frac{1}{2}x(1-x)f^{\prime\prime}(x)\ +\
\int_{\Delta}\frac{\Xi_{KS}(d\zeta)}{(\zeta,\zeta)}\mathbb{E}\left(f\left(\sum_{i\geq
1}\zeta_{i}B_{i}^{x}\right)-f(x)\right),$ (5.24)

where the $B_{i}^{x}$’s are Bernoulli random variables of parameter $x$ and
the second term is the generator of the frequency process associated with a
$\Xi_{KS}$-Fleming-Viot process, see for example formula (5.6) in [5].

The discrete generator $\mathcal{\hat{A}}^{N}$ of $\\{\hat{V}^{N}_{\lfloor
N^{\alpha}t\rfloor}\\}_{t\geq 0}$ (defined as in (2.9)), applied to a function
$f\in C^{2}[0,1]$ in $x\in[0,1]$ can be written as

$\displaystyle\mathcal{\hat{A}}^{N}f(x)\ =$ $\displaystyle\
N^{\alpha}(1-\frac{\eta}{N^{\alpha}})\mathbb{E}\left(f\left(\frac{\sum_{i=1}^{N}B_{i}^{x}}{N}\right)-f(x)\right)$
(5.25) $\displaystyle+$ $\displaystyle\
N^{\alpha}\frac{\eta}{N^{\alpha}}\sum_{k\geq
1}\mathbb{P}(b_{i,N}N=k)\sum_{g\geq
1}\mathbb{P}(l_{1,N}=g)\sum_{i=0}^{k}\mathbb{P}(\bar{Y}^{k,g,x}=i/k)\left(f(i/k)-f(x)\right),$
(5.26)

which can be interpreted in the same way as the generator
$\mathcal{\bar{A}}^{N}$. Again, when $\alpha=1$, part (5.25), corresponds to
the generator of a classical Wright-Fisher model and converges when
$N\to\infty$ to $\frac{1}{2}x(1-x)f^{\prime\prime}(x)$, which is the generator
of the Wright-Fisher diffusion. When $\alpha<1$ this term becomes of order
$N^{\alpha-1}$ and therefore converges to $0$.

Part (5.26) corresponds to what happens during a bottleneck (we recall that
$\mathrm{L}_{\gamma}/N^{\alpha}\to 0$ in distribution i.e. in the new time
scale the bottlenecks are instantaneous). It is well-known that
$\\{Y^{k,\lfloor kt\rfloor,x}\\}_{t\geq 0}$ converges in distribution, in the
Skorokhod topology to the Wright-Fisher diffusion $\\{Y_{t}\\}_{t\geq 0}$ with
$Y_{0}=x$ (see for example Chapter 2 in [9]). In a similar way we can prove
that $\\{\bar{Y}^{k,\lfloor kt\rfloor,x}\\}_{t\geq 0}$ converges in
distribution, in the Skorokhod topology to the same process. In fact,
$\\{\bar{Y}^{k,\lfloor kt\rfloor,x}\\}_{t\geq 0}$ has the distribution of the
frequency process of a Wright-Fisher model, with a random initial condition.
This, combined with the assumptions that $b_{1,N}N\to\infty$ and
$l_{1,N}/(Nb_{1,N})\to\mathrm{L}_{\gamma}$ in distribution, implies that

$\displaystyle\mathcal{\hat{A}}^{N}f(x)\ {\longrightarrow}$ $\displaystyle\
\mathds{1}_{\\{\alpha=1\\}}\frac{1}{2}x(1-x)f^{\prime\prime}(x)+\eta\int_{\mathbb{R}_{+}}\mathrm{L}_{\gamma}(d\sigma)\int_{[0,1]}\mathbb{P}(Y_{\sigma}\in
dy|Y_{0}=x)\left(f(y)-f(x)\right).$ (5.27)

Finally, to compute $\mathbb{P}(Y_{\sigma}\in dy|Y_{0}=x)$, we use the duality
relation between the Wright-Fisher diffusion and the Kingman coalescent
(2.15). More precisely, to compute the probability that the proportion of type
$1$ individuals is $y$ at time $\sigma$, we can follow backwards in time the
ancestry of the whole population. The number of ancestors is given by a
Kingman coalescent started at $K_{0}=\infty$. If $K_{\sigma}=j$, each one of
the $j$ ancestors is of type $1$ with probability $x$ and the fraction of the
population (at time $\sigma$) that is issued from each one of the $j$
ancestors is given by a Dirichlet distribution $D_{j}$. This means that

$\displaystyle\eta\int_{\mathbb{R}_{+}}\mathrm{L}_{\gamma}(d\sigma)\mathbb{P}(Y_{\sigma}\in
dy|Y_{0}=x)$
$\displaystyle=\eta\int_{\mathbb{R}_{+}}\mathrm{L}_{\gamma}(d\sigma)\sum_{j\geq
1}\mathbb{P}(K_{\sigma}=j)\int_{\Delta}D_{j}(\zeta)\mathbb{P}(\sum_{i\geq
1}\zeta_{i}B_{i}^{x}\in dy)$ (5.28)
$\displaystyle=\int_{\Delta}\frac{\Xi_{KS}}{(\zeta,\zeta)}(d\zeta)\mathbb{P}(\sum_{i\geq
1}\zeta_{i}B_{i}^{x}\in dy),$

and replacing into (5.27), we have that $\mathcal{\hat{A}}^{N}$ converges to
$\mathcal{\hat{A}}$ uniformly. This implies that $\\{V^{N}_{\lfloor
N^{\alpha}t\rfloor}\\}_{0\leq t\leq T}$ converges weakly in the Skorokhod
$J_{1}$ topology to $\\{\bar{X}_{t}\\}_{0\leq t\leq T}$. Since convergence in
$J_{1}$ implies convergenec in $d_{\lambda}$ we have the desired result. ∎

We fix $\alpha\in(0,1],\ \eta>0$ and $\mathrm{L}_{\gamma}$ a probability
measure on $\mathbb{R}_{+}$. Let us consider $\\{\hat{N}_{t}\\}_{t\geq 0}$,
the block-counting process of the subordinated Kingman coalescent
characterized by $\alpha$, $\eta$ and $\mathrm{L}_{\gamma}$. As in the
previous sections, we are going to prove a moment duality property between the
block-counting process and the diffusion with jumps $\\{\hat{X}_{t}\\}_{t\geq
0}$ defined above (with the same parameters).

######  Theorem 5.6.

For every $x\in[0,1],\ n\in\mathbb{N}$, we have

$\mathbb{E}(\hat{X}_{t}^{n}|\hat{X}_{0}=x)\ =\
\mathbb{E}(x^{\hat{N}_{t}}|\hat{N}_{0}=n).$

###### Proof.

We only consider the case $\alpha<1$, (as the extension to the case $\alpha=1$
can be done exactly as in Section 2.3). Let $h(x,n)=x^{n}$, seen as a function
of $x$. Using (5.28), for every $n\in\mathbb{N}$, the generator
$\mathcal{\hat{A}}$ applied to $h$ (seen as a function of $x$) can be
rewritten as

$\displaystyle\mathcal{\hat{A}}h(x,n)$
$\displaystyle=\eta\int_{\mathbb{R}_{+}}\mathrm{L}_{\gamma}(d\sigma)\sum_{j\in\mathbb{N}}\mathbb{P}(K_{\sigma}=j)\int_{\Delta}D_{j}(\zeta)\mathbb{P}(\sum_{i\in\mathbb{N}}\zeta_{i}B_{i}^{x}\in
dy)\left(h(y,n)-h(x,n)\right)$ $\displaystyle=\mathcal{\hat{G}}h(x,n),$

where $\mathcal{\hat{G}}$ is the generator of $\\{\hat{N}_{t}\\}_{t\geq 0}$
applied to $h$, seen as a function of $n$. ∎

## Acknowledgements

The authors thank three anonymous referees for very improving commentaries.
ACG thanks Jochen Blath for helpful discussions. AGC was supported by CONACyT
Grant A1-S-14615, VMP by DGAPA-UNAM postdoctoral program and ASJ by CONACyT
Grant CB-2014/243068.

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