# Asymptotic Gaussianity via coalescence probabilities in the Hammond-
Sheffield urn

Jan Lukas Igelbrink  Jan Lukas Igelbrink
Institut für Mathematik, Johannes Gutenberg-Universität Mainz and Goethe-
Universität Frankfurt, Germany<EMAIL_ADDRESS>and  Anton Wakolbinger
Anton Wakolbinger, Institut für Mathematik, Goethe-Universität Frankfurt,
Germany<EMAIL_ADDRESS>
###### Abstract.

For the renormalised sums of the random $\pm 1$-colouring of the connected
components of $\mathbb{Z}$ generated by the coalescing renewal processes in
the “power law Pólya’s urn” of Hammond and Sheffield [HS13] we prove
functional convergence towards fractional Brownian motion, closing a gap in
the tightness argument of their paper.

In addition, in the regime of the strong renewal theorem we gain insights into
the coalescing renewal processes in the Hammond-Sheffield urn (such as the
asymptotic depth of most recent common ancestors) and are able to control the
coalescence probabilities of two, three and four individuals that are randomly
sampled from $[n]$. This allows us to obtain a new, conceptual proof of the
asymptotic Gaussianity (including the functional convergence) of the
renormalised sums of more general colourings, which can be seen as an
invariance principle beyond the main result of [HS13].

In this proof, a key ingredient of independent interest is a sufficient
criterion for the asymptotic Gaussianity of the renormalised sums in randomly
coloured random partitions of $[n]$, based on Stein’s method.

Along the way we also prove a statement on the asymptotics of the coalescence
probabilities in the long-range seedbank model of Blath, González Casanova,
Kurt, and Spanò, see [BGKS13].

###### Key words and phrases:

power law Pólya’s urn, coalescing renewal processes, randomly coloured random
partitions, asymptotic normality, Stein’s method, seedbank coalescent.

###### 2020 Mathematics Subject Classification:

Primary 60G22; Secondary 60K05, 60F17, 60J90

###### Contents

  1. 1 Introduction
  2. 2 Coalescence probabilities in the Hammond-Sheffield urn
  3. 3 Asymptotic Gaussianity in randomly coloured random partitions
  4. 4 Pair coalescence probabilities: Proof of Proposition 2.1
  5. 5 Depth of most recent common ancestor: Proof of Proposition 2.4
  6. 6 Triplet and quartet coalescence probabilities: Proof of Proposition 2.3
    1. 6.1 Triplet coalescence probabilities
    2. 6.2 Quartet coalescence probabilities
  7. 7 A covariance estimate: Proof of Lemma 2.5
  8. 8 Asymptotic Gaussianity in the Hammond-Sheffield urn: Proof of Theorem 1.11.12
  9. 9 Tightness: Proof of Proposition 1.3
  10. 10 Coalescence probabilities in long-range seedbanks

##  1\. Introduction

We start with a brief description of the model of Hammond and Sheffield [HS13]
and then state our main results together with a short outline of the paper.

For $0<\alpha<\tfrac{1}{2}$ and a slowly varying function
$L\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}\rightarrow\mathbb{R}^{+}$
let $\mu\mathrel{\mathop{\mathchar 58\relax}}=\mu_{\alpha,L}$ be a probability
measure on $\mathbb{N}=\\{1,2,\ldots\\}$ having the power law tails

$\mu\left(\left\\{n,n+1,\ldots\right\\}\right)\sim n^{-\alpha}L(n)\mbox{ as
}n\to\infty,$ (1.1)

with the usual convention that for two sequences $f(n)$, $g(n)$ of real
numbers

$f(n)\sim g(n)\mbox{ as }n\rightarrow\infty$

means that $\lim_{n\to\infty}f(n)/g(n)=1$. Throughout it will be assumed that

$\mbox{the greatest common divisor of
}\left\\{n\in\mathbb{N}\mathrel{\mathop{\mathchar
58\relax}}\mu(n)>0\right\\}\mbox{ is one.}$ (1.2)

Let $R$ be an $\mathbb{N}$-valued random variable with distribution $\mu$. A
random directed graph $G_{\mu}$ with vertex set $\mathbb{Z}$ is generated in
the following way: Let $\left(R_{i}\right)_{i\in\mathbb{Z}}$ be a family of
independent copies of $R$. The random set of edges
$E\left(\mathcal{G}_{\mu}\right)$ is then given by

$E\left(\mathcal{G}_{\mu}\right)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\left(i,i-R_{i}\right)\mathrel{\mathop{\mathchar
58\relax}}i\in\mathbb{Z}\right\\}.$

This induces the random equivalence relation

$i\sim j\mathrel{\mathop{\mathchar 58\relax}}\Longleftrightarrow i\mbox{ and
}j\mbox{ belong to the same connected component of }\mathcal{G}_{\mu}.$ (1.3)

Note that the symbol $\sim$ is used in (1.1) and (1.3) in two different
meanings; this will cause no risk of confusion.

For $i\in\mathbb{Z}$ the connected component containing $i$ is denoted by
$\mathscr{C}_{i}$. The random variables $(R_{i})_{i\in\mathbb{Z}}$ give rise
to coalescing renewal processes starting from the integers; see Section 10 for
an interpretation (and extension) in terms of the long-range seedbank model of
Blath et al. [BGKS13]. In this terminology $\mathcal{G}_{\mu}$ is the graph of
ancestral lineages of the individuals $i\in\mathbb{Z}$, and the component
$\mathscr{C}_{i}$ consists of all $j\in\mathbb{Z}$ that are related to $i$,
see Figure 1 for an illustration.

Figure 1. A realisation of the ancestral lineages of the individuals
$\\{0,\ldots,100\\}$ traced back till $-200$. Each of the arcs corresponds to
an edge of $G_{\mu}$. All the outgoing edges from $i=0,\ldots,100$ which map
to an individual in $\\{0,\ldots,100\\}$ are drawn (in red), whereas for $i$
between $-200$ and $-1$ only those outgoing edges are drawn (in blue) that
belong to an ancestral lineage of some $j\in\\{0,\ldots,100\\}$. Here the
exponent $\alpha$ in (1.1) was chosen as $0.39$.

The probability that $0$ belongs to the ancestral lineage of $n$ is thus given
by the weight assigned to $n$ by the renewal measure,

$q_{n}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{\mathbf{P}}\left(\tilde{R}_{1}+\ldots+\tilde{R}_{j}=n\mbox{
for some }j\geq 0\right)$ (1.4)

with $\tilde{R}_{1},\tilde{R}_{2},\ldots$ being independent copies of $R$.
(Note that $\operatorname{\mathbf{P}}(0\sim n)$ is in general larger than
$q_{n}$ because $0$ and $n$ may be related to each other even if $0$ is not an
ancestor of $n$.)

Hammond and Sheffield suggest the picture of an urn in which the types of the
individuals $i$ are determined recursively: each individual $i\in\mathbb{Z}$
inherits the type (or “colour”) of its parent $i-R_{i}$. With $\\{+1,-1\\}$ as
the set of colours, they show that the set of random colourings of
$\mathbb{Z}$ that are consistent with $\mathcal{G}_{\mu}$ has a Gibbs
structure, with the extremal elements being given by i.i.d. assignments of
colours to the connected components of $\mathcal{G}_{\mu}$. The main result of
[HS13] concerns the asymptotics of the rescaled sum over the types of the
individuals $1,\ldots,\lfloor tn\rfloor$, $t\geq 0$, which as $n\to\infty$
turns out to converge to fractional Brownian motion. The individuals’ types
arise as follows:

Assume that each component of $\mathcal{G}_{\mu}$ gets coloured by an
independent copy of a real-valued random variable $Y$. In the situation of
[HS13], $Y$ is a centered Rademacher$(p)$ variable, i.e.

$Y=\xi-(2p-1)\mbox{ with
}\operatorname{\mathbf{P}}(\xi=+1)=p,\operatorname{\mathbf{P}}(\xi=-1)=1-p.$
(1.5)

For $i\in\mathbb{Z}$ the colour of the component $\mathscr{C}_{i}$ will be
denoted by $Y_{i}$. Define the “random walk” (with dependent increments)

$S_{n}\mathrel{\mathop{\mathchar 58\relax}}=\sum_{i=1}^{n}Y_{i},\quad
n=0,1,\ldots.$ (1.6)

By construction,

$\sigma_{n}^{2}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{\mathbf{Var}}[S_{n}]=\sum_{i,j\in[n]}\operatorname{\mathbf{Cov}}[Y_{i},Y_{j}]=\operatorname{\mathbf{E}}[Y^{2}]\,\sum_{i,j\in[n]}\operatorname{\mathbf{P}}(i\sim
j).$ (1.7)

[HS13, Lemma 3.1] show by Fourier and Tauberian arguments that

$\sum\limits_{i,j\in[n]}\operatorname{\mathbf{P}}(i\sim
j)\sim\frac{C_{\alpha}}{\alpha(2\alpha+1)}\frac{n^{2\alpha+1}}{L(n)^{2}}\quad\mbox{
as }n\to\infty,$ (1.8)

with

$C_{\alpha}\mathrel{\mathop{\mathchar 58\relax}}=\frac{1}{\sum_{m\geq
0}q_{m}^{2}}\frac{\Gamma(1-2\alpha)}{\Gamma(\alpha)\Gamma(1-\alpha)^{3}}.$
(1.9)

We will obtain (1.8) as a corollary of Proposition 2.1 below, which requires
the additional condition

$q_{n}\sim\frac{1}{\Gamma(\alpha)\Gamma(1-\alpha)}\frac{n^{\alpha-1}}{L(n)}\quad\mbox{as
}n\rightarrow\infty.$ (1.10)

This condition, which also appears in our Theorem 1.1, is equivalent to the
validity of the Strong Renewal Theorem for the renewal process with an
increment distribution $\mu$ satisfying (1.1) and (1.2), see Caravenna and
Doney [CD19], whose Theorem 1.4 gives necessary and sufficient conditions in
terms of $\mu$ for the validity of (1.10). A well-known sufficient condition
for (1.10) is the criterion of Doney [Don97]

$\sup_{n\geq
1}\frac{n\operatorname{\mathbf{P}}(R=n)}{\operatorname{\mathbf{P}}(R>n)}<\infty.$
(1.11)

For $\frac{i-1}{n}\leq t\leq\frac{i}{n}$, $i,n\in\mathbb{N}$, let $S^{(n)}(t)$
be the linear interpolation of $S_{i}/\sigma_{n}$ and $S_{i+1}/\sigma_{n}$.
Because of (1.7) and (1.8), for all $t\geq 0$,

$\operatorname{\mathbf{Var}}\left[S^{(n)}(t)\right]\to
t^{2\alpha+1}\quad\mbox{as }n\to\infty.$

Since $(S_{n})_{n\in\mathbb{N}_{0}}$ has stationary increments by
construction, this implies the convergence

$\operatorname{\mathbf{Cov}}\left(S_{s}^{(n)},S_{t}^{(n)}\right)\stackrel{{\scriptstyle[}}{{n}}\to\infty]{}\longrightarrow\frac{1}{2}\left(s^{2\alpha+1}+t^{2\alpha+1}-|t-s|^{2\alpha+1}\right),\quad
s,t\geq 0.$

The right-hand side is the covariance function of fractional Brownian motion
with Hurst parameter $H=\tfrac{1}{2}+\alpha$, which is the unique centered
Gaussian process with variance function $t^{2H}$, $t\geq 0$, stationary
increments and a.s. continuous paths. The processes $S^{(n)}$ are centered as
well. Thus, in order to prove that $S^{(n)}$ converges as $n\to\infty$ (in the
sense of finite dimensional distributions) to fractional Brownian motion with
Hurst parameter $H$, it only remains to show that the finite dimensional
distributions of $S^{(n)}$ are asymptotically Gaussian. This is provided by

######  Theorem 1.1.

Let $\mu$ be a probability measure on $\mathbb{N}$ satisfying (1.1) and (1.2).
Assume one of the following conditions (A) or 1.12:

  1. (A)

The colouring $Y$ is given by (1.5).

  2. (B)

The weights $q_{n}$ of the renewal measure specified in (1.4) satisfy the
asymptotics (1.10), and the colouring $Y$ obeys

$\operatorname{\mathbf{E}}\left[Y\right]=0\mbox{ and
}0<\operatorname{\mathbf{E}}\left[Y^{4}\right]<\infty.$ (1.12)

Then, for any fixed $d\in\mathbb{N}$ and fixed $0<t_{1}<\cdots<t_{d}<\infty$,
the sequence $\left(S_{\lfloor t_{1}n\rfloor},\ldots,S_{\lfloor
t_{d}n\rfloor}\right)_{n\in\mathbb{N}}$ is asymptotically Gaussian as
$n\to\infty$.

Under assumption (A) of Theorem 1.1, for each fixed $t>0$ asymptotic
Gaussianity of $S_{\lfloor tn\rfloor}$ as $n\to\infty$ is proved in [HS13] via
a martingale central limit theorem. The computations which ensure the
applicability of the martingale CLT are quite subtle and involved; from the
very beginning they make substantial use of the specific form (1.5) of the
colouring of the random graph $\mathcal{G}_{\mu}$. In [HS13] it is not
explicitly discussed whether these arguments also carry over to $S_{\lfloor
t_{1}n\rfloor},\ldots,S_{\lfloor t_{m}n\rfloor}$. However, again thanks to the
specific assumption (1.5) on $Y$ one can check that this is indeed the case,
thus rendering the asserted asymptotic Gaussianity of the finite dimensional
distributions of $S^{(n)}$.

Under assumption 1.12 we give a new, conceptual proof of the asymptotic
Gaussianity of the finite dimensional distributions of $S^{(n)}$. This proof,
which is completed in Section 8, is based on insights into the structure of
$\mathcal{G}_{\mu}$ which are stated in Section 2 and proved in Sections 4-7.
A key ingredient in the new proof is Theorem 3.1, which provides a criterion
for the asymptotic Gaussianity in randomly coloured random partitions also in
a more general setting. Proposition 3.3, which is instrumental in the proof of
Theorem 3.1, is based on Stein’s method and yields the closeness of the
distribution of $S_{n}/\sigma_{n}$ to the standard normal distribution in
terms of a bound that involves
$\operatorname{\mathbf{Var}}\left[Y^{2}\right]$; this explains the finiteness
condition of $\operatorname{\mathbf{E}}\left[Y^{4}\right]$ in (1.12).

Let us also mention that the loss of ground which comes with assuming the
“strong renewal” condition (1.10) in addition to (1.1) and (1.2) seems rather
minor. Indeed it becomes clear from the examples in [CD19, Section 10] that
the class of measures $\mu$ which satisfy (1.1) and (1.2) but fail to satisfy
(1.10) is rather special.

On the other hand, the benefit of assuming (1.10) is twofold. Firstly, it
allows a direct analysis of asymptotic properties of the genealogy of the
coalescing renewal processes in the Hammond-Sheffield urn, see Propositions
2.1, 2.3 and 2.4. Secondly, this opens the way to a two-step analysis (first
of the random partition of $\mathbb{Z}$, then of its random colouring) which
allows to derive the “invariance principle” stated in Theorem 1.11.12.

The following implication of Theorem 1.1 is immediate from its introductory
discussion.

######  Corollary 1.2.

Under assumptions (A) or 1.12, $S^{(n)}$ converges as $n\to\infty$ in the
sense of finite dimensional distributions to fractional Brownian motion with
Hurst parameter $H=\tfrac{1}{2}+\alpha$.

The next result, which will be proved in Section 9, amends the proof of [HS13,
Lemma 4.1], see Remark 9.2. Here, for each $n\in\mathbb{N}$ and $T>0$,
$\left(S^{(n)}(t)\right)_{0\leq t\leq T}$ is viewed as a random variable
taking its values in $C\left([0,T];\mathbb{R}\right)$, the space of continuous
functions from $[0,T]$ to $\mathbb{R}$, equipped with the $\sup$-norm.

######  Proposition 1.3.

Under the assumptions of Theorem 1.1, for all $T>0$ the sequence of random
variables $\left(S^{(n)}(t)\right)_{0\leq t\leq T}$ is tight.

A direct consequence of Corollary 1.2 and Proposition 1.3 is

######  Corollary 1.4.

Under the assumptions of Theorem 1.1, $S^{(n)}$ converges in distribution
(with respect to the topology of locally uniform convergence) to fractional
Brownian motion with Hurst parameter $H=\tfrac{1}{2}+\alpha$.

##  2\. Coalescence probabilities in the Hammond-Sheffield urn

In this section we will assume that the weights $q_{n}$ of the renewal measure
defined in (1.4) obey the asymptotics (1.10), see the discussion of this
condition in Section 1.

######  Proposition 2.1.

The coalescence probabilities for the ancestral lineages obey the asymptotics

$\operatorname{\mathbf{P}}(0\sim i)\sim
C_{\alpha}\frac{i^{2\alpha-1}}{L(i)^{2}}\quad\mbox{ as }i\to\infty,$ (2.1)

with $C_{\alpha}$ as in (1.9).

######  Remark 2.2.

  1. (a)

The asymptotics (1.8) is a direct consequence of (2.1). Indeed, the latter
implies

$\sum_{i\in[n]}(n-i)\operatorname{\mathbf{P}}(i\sim 0)\sim
n^{2\alpha+1}\frac{C_{\alpha}}{L(n)^{2}}\frac{1}{n}\sum_{i=1}^{n}\left(1-\frac{i}{n}\right)\left(\frac{i}{n}\right)^{2\alpha-1}\quad\mbox{as
}n\to\infty,$ (2.2)

with the limit of the Riemann sums being

$\int_{0}^{1}(1-x)x^{2\alpha-1}\mathrm{d}x=\frac{1}{2\alpha(2\alpha+1)}.$

Since the left-hand sides of (2.2) and (1.8) are equal, this shows the
asserted implication.

  2. (b)

In the light of the proof of Proposition 2.1 (carried out in Section 4) we
conjecture that increment distributions $\mu$ that satisfy (1.1) and violate
(1.10), generically also do not admit the asymptotics (2.1).

The next result, Proposition 2.3, will be instrumental in the proof of Theorem
1.1 under Assumption 1.12. This proposition will consider the probability that
three (respectively four) individuals that are randomly chosen from $[n]$
belong to the same component of $\mathcal{G}_{\mu}$, and will bound these
probabilities asymptotically as $n\to\infty$ by powers of the “pair
coalescence probability”. Note that for sufficiently small $\delta$ the powers
guaranteed by Proposition 2.3 are strictly larger than those required in
Theorem 3.2.

######  Proposition 2.3.

Let $\mathscr{I}^{(n)},\mathscr{J}^{(n)},\mathscr{K}^{(n)}$ and
$\mathscr{L}^{(n)}$ be independent and uniformly distributed on $[n]$, and
independent of the random graph $\mathcal{G}_{\mu}$. Then for all $\delta>0$,
as $n\to\infty$,

$\phantom{AAA}\operatorname{\mathbf{P}}\left(\mathscr{I}^{(n)}\sim\mathscr{J}^{(n)}\sim\mathscr{K}^{(n)}\right)=O\left(n^{4\alpha-2+\delta}\right)=O\left(\left(\operatorname{\mathbf{P}}\left(\mathscr{I}^{(n)}\sim\mathscr{J}^{(n)}\right)\right)^{2-\frac{\delta}{1-2\alpha}}\right),$
(2.3)
$\operatorname{\mathbf{P}}\left(\mathscr{I}^{(n)}\sim\mathscr{J}^{(n)}\sim\mathscr{K}^{(n)}\sim\mathscr{L}^{(n)}\right)=O\left(n^{6\alpha-3+\delta}\right)=O\left(\left(\operatorname{\mathbf{P}}\left(\mathscr{I}^{(n)}\sim\mathscr{J}^{(n)}\right)\right)^{3-\frac{\delta}{1-2\alpha}}\right).$
(2.4)

Proposition 2.3 will be proved in Section 6. Although the next result,
Proposition 2.4, will not be used explicitly in the proof of Theorem 1.1, it
seems interesting in its own right and also gives an intuition why the
estimates in Proposition 2.3 should hold. Qualitatively, Proposition 2.4 says
that for large $n$ the ancestral lineages of $0$ and $n$ with high probability
either coalesce quickly (i.e. _on the scale $n$_) or never. This makes it
believable that, as asserted in Proposition 2.3, the triplet coalescence
probability should asymptotically be comparable to the square of the pair
coalescence probability, and that the quartet coalescence probability should
roughly be equal to the third power of the pair coalescence probability.

######  Proposition 2.4.

Let $\mathscr{M}(0,n)\mathrel{\mathop{\mathchar 58\relax}}=\max\left\\{j\leq
0\mathrel{\mathop{\mathchar 58\relax}}j\sim 0\mbox{ and }j\sim n\right\\}$
(with $\max\emptyset\mathrel{\mathop{\mathchar 58\relax}}=-\infty$) be the
most recent common ancestor of $0$ and $n$, and put
$D_{n}\mathrel{\mathop{\mathchar 58\relax}}=-\mathscr{M}(0,n)$. Then, as
$n\to\infty$, the sequence of random variables $\frac{D_{n}}{n}$, conditioned
under $\\{0\sim n\\}$, converges in distribution to the random variable $D$
with density $B(\alpha,1-2\alpha)^{-1}x^{\alpha-1}(1+x)^{\alpha-1}\,dx$,
$x>0$.

Proposition 2.4 will be proved in Section 5. The distribution of the random
variable $D$ appearing in Proposition 2.4 is known as Beta prime distribution
with parameters $\alpha$ and $1-2\alpha$; it arises as the distribution of
$B/(1-B)$ where $B$ is Beta($\alpha,1-2\alpha$) distributed. See Figure 2 for
simulations of the ancestral lineages, which also illustrate the depths of the
most recent common ancestors.

I

II

Figure 2. This is a simulation of the ancestral lineages of the individuals
$0,\ldots,n$, with $n=100$ in panel (2I) and $n=1000$ in panel (2II).
Direction of time is vertical, and horizontal lines mark coalescence events.
The two panels give an impression of how the genealogical forest of the
individuals $i\in[n]$ scales with $n$, see e.g. Proposition 2.4. Like in
Figure 1, the parameter $\alpha$ was chosen as 0.39.

The following lemma will also be important in the proof of Theorem 1.1 under
Assumption 1.12.

######  Lemma 2.5.

Let $\sim$ be the random equivalence relation defined in (1.3). For
$i,j,k,\ell\in\mathbb{Z}$,

$\operatorname{\mathbf{Cov}}\left[I_{\\{i\sim
j\\}},I_{\\{k\sim\ell\\}}\right]\leq\operatorname{\mathbf{P}}\left(i\sim j\sim
k\sim\ell\right).$ (2.5)

Here and below, $I_{E}$ denotes the indicator variable of an event $E$.

######  Remark 2.6.

The proof of Lemma 2.5 (given in Section 7) shows that (2.5) holds for general
increment distributions $\mu$, without the assumptions (1.1) and (1.2).

##  3\. Asymptotic Gaussianity in randomly coloured random partitions

In this section we consider a situation that is more general than the one
described in Section 1. For $m\in\mathbb{N}$ let $\mathscr{P}^{(m)}$ be a
random partition of $[m]$. The (random) equivalence relation on $[m]$ induced
by $\mathscr{P}^{(m)}$ will be denoted by $\stackrel{{\scriptstyle
m}}{{\sim}}$, i.e.

$i\stackrel{{\scriptstyle m}}{{\sim}}j\mathrel{\mathop{\mathchar
58\relax}}\Longleftrightarrow i\mbox{ and }j\mbox{ belong to the same
partition element of }\mathscr{P}^{(m)}.$ (3.1)

The situation described in Section 1 fits into this framework, by choosing
$\stackrel{{\scriptstyle m}}{{\sim}}$ as the restriction to the set $[m]$ of
the equivalence relation $\sim$ defined in (1.3). Note, however, that this
kind of consistency of the relations $\stackrel{{\scriptstyle m}}{{\sim}}$ is
not required in the present section.

Let $Y$ be a real valued random variable with
$\operatorname{\mathbf{E}}[Y]~{}=~{}0$ and
$0<\operatorname{\mathbf{E}}[Y^{4}]<\infty$. Thinking of each partition
element being “coloured” by an independent copy of $Y$, we write $Y_{i}^{(m)}$
for the colour of the partition element in $\mathscr{P}^{(m)}$ to which
$i\in[m]$ belongs. We then define

$Z_{k}^{(m)}\mathrel{\mathop{\mathchar 58\relax}}=\sum_{i=1}^{k}Y^{(m)}_{i}.$

In the sequel we fix a natural number $d$ and real numbers
$0=\rho_{0}<\rho_{1}<\cdots<\rho_{d}=1$. The following theorem presents a
sufficient criterion for the asymptotic normality of the sequence of
$\mathbb{R}^{d}$-valued random variables

$\mathcal{Z}^{(m)}\mathrel{\mathop{\mathchar
58\relax}}=\left(Z^{(m)}_{\lfloor\rho_{1}m\rfloor},\ldots,Z^{(m)}_{\lfloor\rho_{d}m\rfloor}\right)$
(3.2)

as $m\to\infty$. To prepare for this, let for all $m\in\mathbb{N}$ the random
variables $\mathscr{I}^{(m)},\mathscr{J}^{(m)},\mathscr{K}^{(m)}$ and
$\mathscr{L}^{(m)}$ be independent and uniformly distributed on $[m]$, and
independent of $\mathscr{P}^{(m)}$ and of
$\left(Y^{(m)}_{i}\right)_{i\in[m]}$.

######  Theorem 3.1.

The sequence of $\mathbb{R}^{d}$-valued random variables $\mathcal{Z}^{(m)}$
defined in (3.2) is asymptotically Gaussian as $m\to\infty$ provided the
following conditions are satisfied:

$\displaystyle\operatorname{\mathbf{P}}\left(\mathscr{I}^{(m)}\stackrel{{\scriptstyle
m}}{{\sim}}\mathscr{J}^{(m)}\stackrel{{\scriptstyle
m}}{{\sim}}\mathscr{K}^{(m)}\right)\qquad$ $\displaystyle=$ $\displaystyle
o\left(\left(\operatorname{\mathbf{P}}\left(\mathscr{I}^{(m)}\stackrel{{\scriptstyle
m}}{{\sim}}\mathscr{J}^{(m)}\right)\right)^{3/2}\right)\mbox{ as }m\to\infty,$
(3.3)
$\displaystyle\operatorname{\mathbf{P}}\left(\mathscr{I}^{(m)}\stackrel{{\scriptstyle
m}}{{\sim}}\mathscr{J}^{(m)}\stackrel{{\scriptstyle
m}}{{\sim}}\mathscr{K}^{(m)}\stackrel{{\scriptstyle
m}}{{\sim}}\mathscr{L}^{(m)}\right)$ $\displaystyle=$ $\displaystyle
o\left(\left(\operatorname{\mathbf{P}}\left(\mathscr{I}^{(m)}\stackrel{{\scriptstyle
m}}{{\sim}}\mathscr{J}^{(m)}\right)\right)^{2}\right)\quad\mbox{ as
}m\to\infty,$ (3.4)
$\operatorname{\mathbf{Cov}}\left[I_{\\{i\stackrel{{\scriptstyle
m}}{{\sim}}j\\}},I_{\\{k\stackrel{{\scriptstyle
m}}{{\sim}}\ell\\}}\right]\leq\operatorname{\mathbf{P}}\left(i\stackrel{{\scriptstyle
m}}{{\sim}}j\stackrel{{\scriptstyle m}}{{\sim}}k\stackrel{{\scriptstyle
m}}{{\sim}}\ell\right)\quad\mbox{\rm for all }m\in\mathbb{N}\mbox{ \rm and
}i,j,k,\ell\in[m],$ (3.5)

and for all
$\left(\alpha_{1},\ldots,\alpha_{d}\right)\in\mathbb{R}^{d}\setminus\\{(0,\ldots,0)\\}$
and

$a_{i}^{(m)}\mathrel{\mathop{\mathchar 58\relax}}=\alpha_{g}\mbox{ \rm if
}\lfloor\rho_{g-1}m\rfloor<i\leq\lfloor\rho_{g}m\rfloor,\quad i=1,\ldots
m;\quad g=1,\ldots,d$

there exists a constant $\widetilde{C}>0$ (not depending on $m$) such that

$\sum_{i,j=1}^{m}a_{i}^{(m)}a_{j}^{(m)}\operatorname{\mathbf{P}}\left(i\stackrel{{\scriptstyle
m}}{{\sim}}j\right)\geq\widetilde{C}\sum_{i,j=1}^{m}\operatorname{\mathbf{P}}\left(i\stackrel{{\scriptstyle
m}}{{\sim}}j\right),\quad m\in\mathbb{N}.$ (3.6)

######  Remark 3.2.

  1. (a)

Asymptotic Gaussianity of the univariate sequence
$(Z_{m}^{(m)})_{m\in\mathbb{N}}$ is implied by the first three of the four
conditions in Theorem 3.1. Indeed, for $d=1$, condition (3.6) is automatically
satisfied with $\widetilde{C}\mathrel{\mathop{\mathchar
58\relax}}=\alpha_{1}$, since in that case $a_{i}^{(m)}=\alpha_{1}$ for
$i=1,\ldots,m$.

  2. (b)

If the partitions $\mathscr{P}^{(m)}$ are induced by a Hammond-Sheffield urn
as described in Section 1, then all conditions of Theorem 3.1 (including
condition (3.6) for all $d\in\mathbb{N}$) are implied by the assumption
(1.10), see Section 8.

Theorem 3.1 will be proved using the following proposition which, in turn,
will be deduced from a theorem of Charles Stein, see [Ste86, Lecture X,
Theorem 1]. Since $m$ will be fixed in this proposition, we will write $\sim$
instead of $\stackrel{{\scriptstyle m}}{{\sim}}$ for notational convenience
and without any risk of confusion.

######  Proposition 3.3.

For fixed $m\in\mathbb{N}$ and $a_{1},\ldots,a_{m}\in\mathbb{R}$ let

$\bar{S}_{m}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{i=1}^{m}a_{i}Y_{i},\quad\bar{\sigma}_{m}^{2}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{\mathbf{Var}}[\bar{S}_{m}],$ (3.7)

with $Y_{i}\mathrel{\mathop{\mathchar 58\relax}}=Y_{i}^{(m)}$ as defined at
the beginning of this section. Let $\mathcal{N}$ be a standard normal random
variable. Then for all continuously differentiable functions
$h\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}\rightarrow\mathbb{R}$ with
compact support

$\displaystyle\begin{split}&\left|\operatorname{\mathbf{E}}\left[h\left(\frac{\bar{S}_{m}}{\bar{\sigma}_{m}}\right)\right]-\operatorname{\mathbf{E}}[h(\mathcal{N})]\right|\\\
&\leq\frac{c_{1}(h)}{\bar{\sigma}_{m}^{2}}\sqrt{\operatorname{\mathbf{Var}}\left[\sum\limits_{i,j=1}^{m}Y_{i}^{2}a_{i}a_{j}I_{\\{i\sim
j\\}}\right]}+\frac{c_{2}(h)}{\bar{\sigma}_{m}^{3}}\operatorname{\mathbf{E}}\left[\sum\limits_{i,j,k=1}^{m}|Y_{i}|^{3}|a_{i}|a_{j}a_{k}I_{\\{i\sim
j\sim k\\}}\right].\end{split}$ (3.8)

where the finite numbers $c_{1}(h)$ and $c_{2}(h)$ are defined as

$c_{1}(h)\mathrel{\mathop{\mathchar
58\relax}}=2\sup|h-\operatorname{\mathbf{E}}[h(\mathcal{N})]|,\quad
c_{2}(h)\mathrel{\mathop{\mathchar 58\relax}}=2\sup|h^{\prime}|.$

###### Proof.

For $i\in[m]$ we put

$X_{i}\mathrel{\mathop{\mathchar
58\relax}}=\frac{a_{i}Y_{i}}{\bar{\sigma}_{m}},\qquad
M_{i}\mathrel{\mathop{\mathchar
58\relax}}=\\{j\in[m]\mathrel{\mathop{\mathchar 58\relax}}j\sim i\\},$ (3.9)

i.e., $M_{i}$ is that element of the partition $\mathscr{P}^{(m)}$ which
contains $i$. With $\mathscr{I}$ being a uniform pick from $[m]$ that is
independent of $\mathscr{P}^{(m)}$, we write

$M\mathrel{\mathop{\mathchar 58\relax}}=(M_{1},\ldots,M_{m}),\quad
W\mathrel{\mathop{\mathchar 58\relax}}=\sum_{i=1}^{m}X_{i},\quad
W^{\ast}\mathrel{\mathop{\mathchar 58\relax}}=W-\sum_{j\in
M_{\mathscr{I}}}X_{j},\quad G\mathrel{\mathop{\mathchar
58\relax}}=mX_{\mathscr{I}},$ $\mathcal{B}\mathrel{\mathop{\mathchar
58\relax}}=\sigma\left(M,X_{1},\ldots,X_{m}\right),\qquad\mathcal{C}\mathrel{\mathop{\mathchar
58\relax}}=\sigma\left(M,\mathscr{I},(X_{j})_{j\nsim\mathscr{I}}\right)$

and note that $W=\operatorname{\mathbf{E}}[G|\mathcal{B}]$ a.s. Now [Ste86,
Lecture X, Theorem 1] asserts that

$\displaystyle\left|\operatorname{\mathbf{E}}\left[h\left(W\right)\right]-\operatorname{\mathbf{E}}[h(\mathcal{N})]\right|$
$\displaystyle\leq$ $\displaystyle
c_{1}(h)\left(\sqrt{\operatorname{\mathbf{E}}\left[\left(1-\operatorname{\mathbf{E}}\left[G(W-W^{\ast})|\mathcal{B}\right]\right)^{2}\right]}+c\operatorname{\mathbf{E}}\left[\left|\operatorname{\mathbf{E}}\left[G\left|\mathcal{C}\right.\right]\right|\right]\right)$
$\displaystyle+c_{2}(h)\operatorname{\mathbf{E}}\left[|G|(W-W^{\ast})^{2}\right],$

with $N$, $c_{1}(h)$ and $c_{2}(h)$ as in (3.8) and a constant $c>0$. Let us
first turn to the term under the square root on the right-hand side of (3) and
observe that

$\displaystyle\operatorname{\mathbf{E}}\left[G(W-W^{*})|\mathcal{B}\right]=\operatorname{\mathbf{E}}\left[\left.mX_{\mathscr{I}}\sum_{j\in
M_{\mathscr{I}}}X_{j}\,\right|\,\mathcal{B}\right]=\sum_{i=1}^{m}X_{i}\sum_{j\in
M_{i}}X_{j}=\frac{1}{\bar{\sigma}_{m}^{2}}\sum_{i=1}^{m}a_{i}Y_{i}\sum_{j\in
M_{i}}a_{j}Y_{j}.$

The expectation of this random variable is 1, since

$\operatorname{\mathbf{E}}\left[\operatorname{\mathbf{E}}\left[\frac{1}{\bar{\sigma}_{m}^{2}}\sum_{i=1}^{m}a_{i}Y_{i}\sum_{j=1}^{m}a_{j}Y_{j}\bigg{|}\mathscr{P}^{(m)}\right]\right]=\frac{1}{\bar{\sigma}_{m}^{2}}\operatorname{\mathbf{Var}}[\bar{S}_{m}]=1.$

Hence the term under the square root in (3) equals

$\operatorname{\mathbf{Var}}\left[\sum_{i=1}^{m}X_{i}\sum_{j\in
M_{i}}X_{j}\right].$

The term
$\operatorname{\mathbf{E}}\left[\left|\operatorname{\mathbf{E}}\left[G\left|\mathcal{C}\right.\right]\right|\right]$
in the right-hand side of (3) vanishes, since the assumed independence of the
colouring and the partitions together with the assumption
$\operatorname{\mathbf{E}}[X_{i}]=0$ implies

$\operatorname{\mathbf{E}}\left[\left|\operatorname{\mathbf{E}}\left[G\left|\mathcal{C}\right.\right]\right|\right]=\operatorname{\mathbf{E}}\left[\left|\operatorname{\mathbf{E}}\left[\left.mX_{\mathscr{I}}\right|M,\mathscr{I}\right]\right|\right]=0.$

Finally, the rightmost term in (3) equals

$\displaystyle\operatorname{\mathbf{E}}\left[|G|\left(W-W^{*}\right)^{2}\right]=\operatorname{\mathbf{E}}\left[|G|\left(\sum_{j\in
M_{\mathscr{I}}}X_{j}\right)^{2}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{m}\sum_{i=1}^{m}\operatorname{\mathbf{E}}\left[I_{\\{\mathscr{I}=i\\}}|mX_{i}|\left(\sum_{j\in
M_{i}}X_{j}\right)^{2}\right]=\sum_{i=1}^{m}\operatorname{\mathbf{E}}\left[|X_{i}|\left(\sum_{j\in
M_{i}}X_{j}\right)^{2}\right].$

In summary we have shown that the right-hand side of (3) equals

$\displaystyle\phantom{AAAA}c_{1}(h)\sqrt{\operatorname{\mathbf{Var}}\left[\sum_{i=1}^{m}X_{i}\sum_{j\in
M_{i}}X_{j}\right]}+c_{2}(h)\operatorname{\mathbf{E}}\left[\sum_{i=1}^{m}|X_{i}|\left(\sum_{j\in
M_{i}}X_{j}\right)^{2}\right],$ (3.11)

which in turn is equal to the right-hand side of (3.8). This concludes the
proof of Proposition 3.3. ∎

###### Proof of Theorem 3.1.

It suffices to show that for all $d\in\mathbb{N}$ and
$(\alpha_{1},\ldots,\alpha_{d})\neq(0,\ldots,0)$, the linear combination

$\bar{Z}_{m}\mathrel{\mathop{\mathchar
58\relax}}=\alpha_{0}Z^{(m)}_{\lfloor\rho_{1}m\rfloor}+\alpha_{1}\left(Z^{(m)}_{\lfloor\rho_{2}m\rfloor}-Z^{(m)}_{\lfloor\rho_{1}m\rfloor}\right)+\cdots+\alpha_{d}\left(Z^{(m)}_{m}-Z^{(m)}_{\lfloor\rho_{d-1}m\rfloor}\right)$
(3.12)

is asymptotically Gaussian as $m\to\infty$. For $m\in\mathbb{N}$ we put

$a_{i}^{(m)}\mathrel{\mathop{\mathchar 58\relax}}=\alpha_{g}\mbox{ \rm if
}\lfloor\rho_{g-1}m\rfloor<i\leq\lfloor\rho_{g}m\rfloor,\quad i=1,\ldots
m;\quad g=1,\ldots,d.$ (3.13)

It is then readily checked that $\bar{Z}_{m}$ defined in (3.12) satisfies

$\bar{Z}_{m}=\sum_{i=1}^{m}a_{i}^{(m)}Y_{i}^{(m)},\quad m\in\mathbb{N},$
(3.14)

and thus fits into the frame of Proposition 3.3, with
$\bar{S}_{m}\mathrel{\mathop{\mathchar 58\relax}}=\bar{Z}_{m}$. To use this
proposition we will show that under the assumptions (3.3), (3.4), (3.5) and
(3.6) and with $a_{i}=a_{i}^{(m)}$ from (3.13), both summands in the right-
hand side of (3.8) converge to $0$ as $m\to\infty$. For notational convenience
we will for the rest of this proof suppress the superscript $m$ in the
equivalence relation $\stackrel{{\scriptstyle m}}{{\sim}}$, in the
coefficients $a_{i}^{(m)}$ and in the random variables $Y_{i}^{(m)}$,
$\mathscr{I}^{(m)}$, $\mathscr{J}^{(m)}$, $\mathscr{K}^{(m)}$,
$\mathscr{L}^{(m)}$.

For a constant $C$ not depending on $m$ we have

$\operatorname{\mathbf{Var}}\left[\sum\limits_{i,j=1}^{m}Y_{i}^{2}a_{i}a_{j}I_{\\{i\sim
j\\}}\right]\leq
C\operatorname{\mathbf{Var}}\left[\sum\limits_{i,j=1}^{m}Y_{i}^{2}I_{\\{i\sim
j\\}}\right],$ (3.15)
$\operatorname{\mathbf{E}}\left[\sum\limits_{i,j,k=1}^{m}|Y_{i}|^{3}|a_{i}|a_{j}a_{k}I_{\\{i\sim
j\sim k\\}}\right]\leq
C\operatorname{\mathbf{E}}\left[|Y|^{3}\right]\sum\limits_{i,j,k=1}^{m}\operatorname{\mathbf{P}}(i\sim
j\sim k).$ (3.16)

In order to bound
$\operatorname{\mathbf{Var}}\left[\sum\limits_{i,j=1}^{m}Y_{i}^{2}I_{\\{i\sim
j\\}}\right]$ from above, we decompose the variance with respect to
$\mathscr{P}^{(m)}$ and first note that

$\operatorname{\mathbf{E}}\left[\left.\sum\limits_{i,j=1}^{m}Y_{i}^{2}I_{\\{i\sim
j\\}}\right|\mathscr{P}^{(m)}\right]=\operatorname{\mathbf{E}}\left[Y^{2}\right]\sum\limits_{i,j=1}^{m}I_{\\{i\sim
j\\}}.$

The variance of the latter is

$\operatorname{\mathbf{E}}\left[Y^{2}\right]^{2}\sum_{i,j,k,\ell\in[m]}\,\operatorname{\mathbf{Cov}}\left[I_{\\{i\sim
j\\}},I_{\\{k\sim\ell\\}}\right]$

which by assumption (3.5) is not larger than

$\operatorname{\mathbf{E}}\left[Y^{2}\right]^{2}\sum_{i,j,k,\ell\in[m]}\operatorname{\mathbf{P}}(i\sim
j\sim k\sim\ell).$ (3.17)

Next we note that

$\displaystyle\begin{split}\operatorname{\mathbf{Var}}\left[\left.\sum\limits_{i,j=1}^{m}Y_{i}^{2}I_{\\{i\sim
j\\}}\right|\mathscr{P}^{(n)}\right]&=&&\sum\limits_{i,j,k,\ell=1}^{m}\operatorname{\mathbf{Cov}}\left[\left.Y_{i}^{2}I_{\\{i\sim
j\\}},Y_{k}^{2}I_{\\{k\sim\ell\\}}\right|\mathscr{P}^{(m)}\right]\\\
&=&&\operatorname{\mathbf{Var}}\left[Y^{2}\right]\sum\limits_{i,j,k,\ell=1}^{m}I_{\\{i\sim
j\sim k\sim\ell\\}}.\end{split}$

Taking expectation of the latter and adding this to (3.17) we obtain

$\operatorname{\mathbf{Var}}\left[\sum\limits_{i,j=1}^{m}Y_{i}^{2}I_{\\{i\sim
j\\}}\right]\leq\operatorname{\mathbf{E}}\left[Y^{4}\right]\sum_{i,j,k,\ell\in[m]}\operatorname{\mathbf{P}}(i\sim
j\sim
k\sim\ell)=O\left(m^{4}\operatorname{\mathbf{P}}\left(\mathscr{I}\sim\mathscr{J}\sim\mathscr{K}\sim\mathscr{L}\right)\right),$

which in view of (3.4) is
$o\left(m^{2}\operatorname{\mathbf{P}}\left(\mathscr{I}\sim\mathscr{J}\right)\right)^{2}$.

Likewise the right-hand side of (3.16) is
$o\left(m^{2}\left(\operatorname{\mathbf{P}}\left(\mathscr{I}\sim\mathscr{J}\right)\right)^{3/2}\right)$.
From (3.7) and (3.14) we get

$\bar{\sigma}_{m}^{2}=\sum_{i,j\in[m]}a_{i}^{(m)}a_{j}^{(m)}\operatorname{\mathbf{Cov}}\left[Y_{i},Y_{j}\right]=\operatorname{\mathbf{Var}}[Y]\sum_{i,j\in[m]}a_{i}^{(m)}a_{j}^{(m)}\operatorname{\mathbf{P}}(i\sim
j),$ (3.18)

which due to assumption (3.6) is bounded from below by
$\widetilde{C}m^{2}\operatorname{\mathbf{P}}(\mathscr{I}\sim\mathscr{J})$.
Thus under the assumptions of Theorem 3.1 the right-hand side of (3.8)
converges to 0 as $m\to\infty$, which shows that Theorem 3.1 is a consequence
of Proposition 3.3. ∎

##  4\. Pair coalescence probabilities: Proof of Proposition 2.1

Let $R^{(i)}_{k}$, $i\in\mathbb{Z}$, $k\in\mathbb{N}$ be independent copies of
$R$, and define

$A_{i}\mathrel{\mathop{\mathchar
58\relax}}=\left\\{n\in\mathbb{Z}\mathrel{\mathop{\mathchar
58\relax}}i-R_{1}^{(i)}-\cdots-R_{j}^{(i)}=n\mbox{ for some }j\geq
0\right\\}.$ (4.1)

Note that the $A_{i}$ are independent and thus can be seen as decoupled
versions of the ancestral lineages of the individuals $i\in\mathbb{Z}$. In
particular they do not coalesce if they meet. Decomposing with respect to the
most recent collision time one obtains immediately (cf. [HS13, p. 711]) that
for $i>0$

$\operatorname{\mathbf{P}}\left(A_{0}\cap A_{i}\neq\emptyset\right)\sum_{m\geq
0}q_{m}^{2}=\operatorname{\mathbf{E}}\left[\left|A_{0}\cap
A_{i}\right|\right]=\sum_{m\geq 0}q_{m}q_{m+i},$ (4.2)

hence

$\operatorname{\mathbf{P}}(0\sim i)=\frac{\sum_{m\geq
0}q_{m}q_{m+i}}{\sum_{m\geq 0}q_{m}^{2}},\quad i\geq 0.$ (4.3)

We will now assume (in accordance with the assumptions in Proposition 2.1)
that the weights $q_{n}$ of the renewal measure defined in (1.4) have the
property (1.10). Under this condition we will prove

######  Proposition 4.1.

As $m\to\infty$,

$\sum_{j\geq
1}q_{j}q_{m+j}\sim\frac{1}{\Gamma(\alpha)^{2}\Gamma(1-\alpha)^{2}}\frac{m^{2\alpha-1}}{L(m)^{2}}\int_{0}^{\infty}(1+x)^{\alpha-1}x^{\alpha-1}\mathrm{d}x.$
(4.4)

The asymptotics (2.1) claimed in Proposition 2.1 is immediate from (4.3)
combined with (4.4).

The remainder of this section is devoted to the proof of Proposition 4.1. We
will prove (4.4) first under a special assumption on the Karamata
representation of the slowly varying function $L$.

######  Lemma 4.2.

Let $\alpha\in(0,\frac{1}{2})$, and consider

$r_{n}\mathrel{\mathop{\mathchar 58\relax}}=n^{\alpha-1}K(n)$ (4.5)

where $K(n)$ is of the form

$K(n)=\exp\left(\int_{B}^{n}\frac{l(t)}{t}\,\mathrm{d}t\right)$ (4.6)

with $B$ a positive constant and $l(t)$, $t\geq B$, a bounded measurable
function converging to $0$ as $t\to\infty$. Then $(r_{n})$ is ultimately
decreasing, and

$\sum_{j\geq 1}r_{j}r_{i+j}\sim
K(i)^{2}i^{2\alpha-1}\int_{0}^{\infty}(1+x)^{\alpha-1}x^{\alpha-1}\mathrm{d}x\quad\mbox{as
}i\to\infty,$ (4.7)

with

$\int_{0}^{\infty}(1+x)^{\alpha-1}x^{\alpha-1}dx=B(\alpha,1-2\alpha)=\frac{\Gamma(\alpha)\Gamma(1-2\alpha)}{\Gamma(1-\alpha)}.$
(4.8)

###### Proof.

a) The equality (4.8) is readily checked by substituting $y=\frac{x}{1+x}$.

b) The fact that $(r_{n})$ is ultimately decreasing follows from (4.5)
together with the Karamata representation (4.6) of $K(n)$. To see this, we
argue as follows, putting $\beta\mathrel{\mathop{\mathchar
58\relax}}=1-\alpha$. Since $l(t)$ tends to zero for $t\rightarrow\infty$ we
know that there exists $n_{0}\in\mathbb{N}$ such that for all $t\geq n_{0}$
one has $l(t)<\beta$. This implies

$\displaystyle\frac{r_{n}}{r_{n+1}}$ $\displaystyle=$
$\displaystyle\frac{n^{-\beta}}{(n+1)^{-\beta}}\cdot\frac{K(n)}{K(n+1)}$
$\displaystyle=$
$\displaystyle\frac{n^{-\beta}}{(n+1)^{-\beta}}\exp\left(-\int_{n}^{n+1}\frac{l(t)}{t}\mathrm{d}t\right)$
$\displaystyle=$
$\displaystyle\exp\left(\beta\left(\ln(n+1)-\ln(n)\right)-\int_{n}^{n+1}\frac{l(t)}{t}\mathrm{d}t\right)$
$\displaystyle=$
$\displaystyle\exp\left(\int_{n}^{n+1}\left(\frac{\beta}{t}-\frac{l(t)}{t}\right)\mathrm{d}t\right).$

Since by assumption the integrand on the right-hand side is strictly positive
for $n\geq n_{0}$, we obtain that $(r_{n})_{n}$ is decreasing for $n\geq
n_{0}$.

c) In view of (4.5), the claimed asymptotics (4.7) is equivalent to

$\lim_{i\to\infty}\frac{1}{i}\sum_{j\geq
1}\frac{r_{j}}{r_{i}}\frac{r_{i+j}}{r_{i}}=\int_{0}^{\infty}(1+x)^{\alpha-1}x^{\alpha-1}\mathrm{d}x.$
(4.9)

We now set out to prove (4.9). To this purpose we show first that for all
$\varepsilon>0$ there exists an $N\in\mathbb{N}$ such that for all
sufficiently large $i$

$\frac{1}{i}\sum_{j\geq
Ni}\frac{r_{j}}{r_{i}}\frac{r_{i+j}}{r_{i}}<\varepsilon.$ (4.10)

Since $(r_{n})$ is ultimately decreasing, (4.10) will follow if we can show
that exists an $N\in\mathbb{N}$ such that for all sufficiently large $i$

$\frac{1}{i}\sum_{j\geq Ni}\left(\frac{r_{j}}{r_{i}}\right)^{2}<\varepsilon.$
(4.11)

Again because of the ultimate monotonicity of $(r_{n})$, the left-hand side.
of (4.11) is for sufficiently large $i$ bounded from above by

$\sum_{m=N}^{\infty}\left(\frac{r_{mi}}{r_{i}}\right)^{2}=\sum_{m=N}^{\infty}m^{2\alpha-2}\left(\frac{K(mi)}{K(i)}\right)^{2}.$
(4.12)

Using (4.6) one obtains that for any $\delta>0$ and $i$ so large that
$l(t)<\delta$ for all $t\geq i$,

$\frac{K(mi)}{K(i)}=\exp\left(\int_{i}^{mi}\frac{l(t)}{t}\mathrm{d}t\right)\leq\exp\left(\delta\left(\ln(mi)-\ln(i)\right)\right)\leq
m^{\delta},$ (4.13)

which implies by dominated convergence that for sufficiently large $N$ the
right-hand side of (4.12) is smaller than $\varepsilon$ for all sufficiently
large $i$. We have thus proved (4.10).

Next we show that for all $\varepsilon>0$ there exists an $\eta\in\mathbb{N}$
such that for all sufficiently large $i$

$\frac{1}{i}\sum_{j\leq\eta
i}\frac{r_{j}}{r_{i}}\frac{r_{i+j}}{r_{i}}<\varepsilon.$ (4.14)

Again by ultimate monotonicity of $(r_{n})$ which gives us
$\frac{r_{i+j}}{r_{i}}\leq 1$ for $i$ large enough, for this it suffices to
show that for all $\varepsilon>0$ there exists an $\eta>0\in\mathbb{N}$ such
that for all sufficiently large $i$

$\frac{1}{i}\sum_{j\leq\eta i}\frac{r_{j}}{r_{i}}<\varepsilon.$ (4.15)

From [Fel71, Theorem 5 on p. 447] we obtain that

$\sum_{j\leq\eta i}r_{j}\sim\frac{1}{\alpha}(\eta i)^{\alpha}K(\lfloor\eta
i\rfloor)\quad\mbox{as }i\to\infty,$ (4.16)

and hence

$\frac{1}{i}\sum_{j\leq\eta
i}\frac{r_{j}}{r_{i}}\sim\frac{1}{\alpha}\eta^{\alpha}\frac{K(\lfloor\eta
i\rfloor)}{K(i)}\sim\frac{\eta^{\alpha}}{\alpha}\quad\mbox{as }i\to\infty,$
(4.17)

which proves (4.15), and hence also (4.14). (The last asymptotic is by the
fact that $K$ is slowly varying.)

In view of (4.5), (4.10) and (4.14), for proving (4.7) it remains to show that

$\lim_{i\to\infty}\frac{1}{i}\sum_{\eta i\leq j\leq
Ni}\frac{K(j)}{K(i)}\frac{K(i+j)}{K(i)}\left(1+\frac{j}{i}\right)^{\alpha-1}\left(\frac{j}{i}\right)^{\alpha-1}=\int_{\eta}^{N}(1+x)^{\alpha-1}x^{\alpha-1}\mathrm{d}x.$
(4.18)

From (4.6) one derives that

$\lim_{i\to\infty}\sup_{\eta i\leq
j\leq(N+1)i}\left|\frac{K(j)}{K(i)}-1\right|=0.$

Hence (4.18) boils down to a convergence of Riemann sums to its integral
limit, and the proof of Lemma 4.2 is done. ∎

Let us now complete the proof of Proposition 4.1.

###### Proof.

The asymptotics (1.10) can be rewritten as

$q_{n}=C_{n}n^{\alpha-1}\tilde{L}(n)$

where $\tilde{L}(n)$ is a slowly varying function and $C_{n}\to C>0$.

As in the proof of Lemma 4.2 it suffices to show that

$\frac{1}{i}\sum_{j\geq
1}\frac{q_{j}}{q_{i}}\frac{q_{i+j}}{q_{i}}\to\int_{0}^{\infty}(1+x)^{\alpha-1}x^{\alpha-1}\mathrm{d}x.$
(4.19)

Because of the Karamata representation theorem (see e.g. Theorem 1.3.1 in
[BGT87]) there exists a $K(n)$ satisfying (4.6) and a sequence $D_{n}$
converging to a positive constant $D$ such that

$\tilde{L}(n)=D_{n}K(n),\quad n=1,2,\ldots$ (4.20)

Defining $r_{n}$ as in (4.5) we have

$q_{n}=D_{n}C_{n}r_{n},\quad n=1,2,\ldots$

Since the asymptotics of neither the left-hand side of (4.9) nor that of the
left-hand side of (4.19) reacts to the omission of a fixed finite number of
summands, we see that (4.9) carries over to (4.19). ∎

##  5\. Depth of most recent common ancestor: Proof of Proposition 2.4

In this section we will assume that the weights $q_{n}$ of the renewal measure
defined in (1.4) obey the asymptotics (1.10), see the discussion after
equation (1.10). For $i,m\in\mathbb{N}$ we set

$\displaystyle
f_{i}(m)=\operatorname{\mathbf{P}}\left(\mathscr{M}(0,i)=-m\right),\quad
F_{i}(m)=\operatorname{\mathbf{P}}\left(\mathscr{M}(0,i)\geq-m\right).$

For the independent couplings $A_{i}$, $i\in\mathbb{Z}$, of the ancestral
lines of $0$ and $i$ as defined in (4.1) we have for all $r>0$ and
$i\in\mathbb{Z}$

$\displaystyle\sum_{k=0}^{ri}q_{k}q_{k+i}$ $\displaystyle=$
$\displaystyle\operatorname{\mathbf{E}}\left[|A_{0}\cap
A_{i}\cap\\{0,\ldots,-ri\\}|\right]$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{ri}\left(f_{i}(k)\sum_{l=0}^{ri-k}q_{l}^{2}\right)\quad\leq\quad\left(\sum_{l=0}^{ri}q_{l}^{2}\right)F_{i}(ri),$

and consequently

$\displaystyle\operatorname{\mathbf{P}}(D_{i}\leq
ri)=F_{i}(ri)\geq\frac{\sum_{k=0}^{ri}q_{k}q_{k+i}}{\sum_{l=0}^{ri}q_{l}^{2}}.$

As in the proof of Proposition 4.1 we obtain

$\sum_{k=0}^{ri}q_{k}q_{k+i}\sim
i^{2\alpha-1}\frac{1}{\Gamma(1-\alpha)^{2}\Gamma(\alpha)^{2}}\int_{0}^{r}x^{\alpha-1}(1+x)^{\alpha-1}\mathrm{d}x\quad\mbox{
as }i\to\infty.$

Together with the asymptotics (2.1) this gives

$\liminf_{i\to\infty}\operatorname{\mathbf{P}}\left(D_{i}\leq ri\,|\,0\sim
i\right)\geq
B(\alpha,1-2\alpha)^{-1}\int_{0}^{r}x^{\alpha-1}(1+x)^{\alpha-1}\mathrm{d}x.$
(5.1)

For the upper estimate we choose some arbitrary $\varepsilon>0$ and observe

$\displaystyle\sum_{k=0}^{(r+\varepsilon)i}q_{k}q_{k+i}$ $\displaystyle=$
$\displaystyle\operatorname{\mathbf{E}}\left[|A_{0}\cap
A_{i}\cap\\{0,\ldots,-(r+\varepsilon)i\\}|\right]$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{(r+\varepsilon)i}\left(f_{i}(k)\sum_{l=0}^{(r+\varepsilon)i-k}q_{l}^{2}\right)\quad\geq\quad\sum_{k=0}^{ri}\left(f_{i}(k)\sum_{l=0}^{(r+\varepsilon)i-k}q_{l}^{2}\right)$
$\displaystyle\geq$ $\displaystyle\left(\sum_{l=0}^{\varepsilon
i}q_{l}^{2}\right)\sum_{k=0}^{ri}f_{i}(k)\qquad=\quad\left(\sum_{l=0}^{\varepsilon
i}q_{l}^{2}\right)F_{i}(ri).$

Using (5), now with $r+\varepsilon$ instead of $r$, we get

$\limsup_{i\to\infty}\operatorname{\mathbf{P}}\left(D_{i}\leq ri\,|\,0\sim
i\right)\leq
B(\alpha,1-2\alpha)^{-1}\int_{0}^{r+\varepsilon}x^{\alpha-1}(1+x)^{\alpha-1}\mathrm{d}x.$

Since $\varepsilon$ was arbitrary, this together with (5.1) gives the
assertion of Proposition 2.4.

##  6\. Triplet and quartet coalescence probabilities: Proof of Proposition
2.3

In this section we will assume that the weights $q_{n}$ of the renewal measure
defined in (1.4) obey the asymptotics (1.10). We now turn to the asymptotic
analysis of triplet and quartet coalescence probabilities.

###  6.1. Triplet coalescence probabilities

######  Lemma 6.1.

For all $r>0$ and $i\in\mathbb{N}$ we have for a slowly varying function
$\tilde{L}$ not depending on $r$

$\operatorname{\mathbf{P}}\left(0\sim
i\sim\lfloor(1+r)i\rfloor\right)\leq\left(r^{\alpha-1}+r^{2\alpha-1}\right)\tilde{L}(i)\,i^{4\alpha-2}.$
(6.1)

###### Proof.

Let $(A_{k})$ be the independent (non-merging) ancestral lineages defined in
(4.1). We set

$A_{k,\ell}\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}A_{\max(A_{k}\cap A_{\ell})}&\mbox{ if }A_{k}\cap
A_{\ell}\neq\emptyset,\\\ \emptyset&\mbox{ if }A_{k}\cap
A_{\ell}=\emptyset,\end{cases}$ $B_{k,\ell}\mathrel{\mathop{\mathchar
58\relax}}=\bigcup_{g\in A_{k}\cap A_{\ell}}A_{g}.$

In words, $B_{k,\ell}$ is the union of the (non-merging) ancestral lineages
starting at all the points of intersection of $A_{k}$ and $A_{\ell}$.
Distinguishing the 3 shapes of the ancestral tree of the individuals $0$, $i$
and $(1+r)i$ on the event $E\mathrel{\mathop{\mathchar 58\relax}}=\\{0\sim
i\sim(1+r)i\\}$ (and omitting the Gauss brackets in $\lfloor(1+r)i\rfloor$
etc. for the sake of readability) we have by subadditivity

$\displaystyle\operatorname{\mathbf{P}}\left(0\sim i\sim(1+r)i\right)$
$\displaystyle\leq$
$\displaystyle\operatorname{\mathbf{P}}\left(E;\mathscr{M}(0,i)\geq\mathscr{M}(0,(1+r)i\right)+\operatorname{\mathbf{P}}\left(E;\mathscr{M}(0,(1+r)i)\geq\mathscr{M}(0,i)\right)$
$\displaystyle+\operatorname{\mathbf{P}}\left(E;\mathscr{M}(i,(1+r)i)\geq\mathscr{M}(0,i)\right)$
$\displaystyle\leq$ $\displaystyle\operatorname{\mathbf{P}}\left(A_{0,i}\cap
A_{(1+r)i}\neq\emptyset\right)+\operatorname{\mathbf{P}}\left(A_{0,(1+r)i}\cap
A_{i}\neq\emptyset\right)+\operatorname{\mathbf{P}}\left(A_{i,(1+r)i}\cap
A_{0}\neq\emptyset\right)$ $\displaystyle\leq$
$\displaystyle\operatorname{\mathbf{E}}\left[|A_{0,i}\cap
A_{(1+r)i}|\right]+\operatorname{\mathbf{E}}\left[|A_{0,(1+r)i}\cap
A_{i}|\right]+\operatorname{\mathbf{E}}\left[|A_{i,(1+r)i}\cap A_{0}|\right]$
$\displaystyle\leq$ $\displaystyle\operatorname{\mathbf{E}}\left[|B_{0,i}\cap
A_{(1+r)i}|\right]+\operatorname{\mathbf{E}}\left[|B_{0,(1+r)i}\cap
A_{i}|\right]+\operatorname{\mathbf{E}}\left[|B_{i,(1+r)i}\cap A_{0}|\right].$
$\displaystyle=$ $\displaystyle\sum_{m\geq 0}q_{m}q_{ri+m}\sum_{n\geq
0}q_{n}q_{n+i-m}+\sum_{m\geq 0}q_{m}q_{m+(1+r)i}\sum_{n\geq 0}q_{n}q_{n+i+m}$
$\displaystyle+\sum_{m\geq 0}q_{m}q_{ri+m}\sum_{n\geq 0}q_{n}q_{n+i-m}$

An inspection of the third summand on the right-hand side leads to

$\displaystyle\sum_{m\geq 0}q_{m}q_{ri+m}\sum_{n\geq 0}q_{n}q_{n+i-m}$
$\displaystyle=$ $\displaystyle q_{m}q_{ri+i}\sum_{n\geq
0}q_{n}q_{n}+\sum_{m\neq i}q_{m}q_{ri+m}\sum_{n\geq 0}q_{n}q_{n+i-m}$
$\displaystyle\leq$ $\displaystyle C\left(q_{i}q_{(1+r)i}+\sum_{m\neq
i}\frac{m^{\alpha-1}}{L(m)}\frac{(m+ri)^{\alpha-1}}{L(m+ri)}\frac{|i-m|^{2\alpha-1}}{L(|i-m|)^{2}}\right)$
$\displaystyle\sim$ $\displaystyle
C\left(\frac{i^{2\alpha-2}}{L(i)^{2}}+\frac{i^{4\alpha-2}}{L(i)^{4}}\int_{0}^{\infty}x^{\alpha-1}(x+r)^{\alpha-1}|1-x|^{2\alpha-1}\mathrm{d}x\right)$
$\displaystyle\leq$ $\displaystyle\tilde{L}(i)i^{4\alpha-2}r^{\alpha-1},$

where $\tilde{L}$ is a slowly varying function that dominates
$C\left(\tfrac{1}{L^{2}}+\tfrac{1}{L^{4}}\right)$, and where the asymptotics
is justified in the same way as (4.18) was derived first for slowly varying
functions satisfying (4.6) and then, using the Karamata representation, for
general slowly varying functions obeying (4.20). The first and the second
summand on the r.h.s of (6.1) can be analysed in an analogous manner, leading
to the bounds $\tilde{L}(i)i^{4\alpha-2}r^{2\alpha-1}$ and
$\tilde{L}(i)i^{4\alpha-2}r^{\alpha-1}$, respectively. ∎

###### Proof of Proposition 2.3 Part 1.

We set out to show (2.3), and first observe that

$\sum_{i=0}^{n}\sum_{j=0}^{n}\operatorname{\mathbf{P}}\left(0\sim i\sim
j\right)\\\ \leq\sum_{i=0}^{n}\operatorname{\mathbf{P}}\left(0\sim
i\right)+2\sum_{i=1}^{n}\sum_{j=1}^{n-i}\operatorname{\mathbf{P}}\left(0\sim
i\sim i+j\right)$ (6.3)

By Lemma 6.1 we get, noting that $r^{\alpha-1}+r^{2\alpha-1}\leq
2r^{\alpha-1}+1$,

$\displaystyle\begin{split}\sum_{i=1}^{n}\sum_{j=1}^{n-i}\operatorname{\mathbf{P}}\left(0\sim
i\sim
i+j\right)&\leq\sum_{i=1}^{n}\sum_{j=1}^{n-i}\tilde{L}(i)i^{4\alpha-1}\left(\left(\frac{j}{i}\right)^{\alpha-1}+1\right)\\\
&\leq\tilde{L}(n)\left(\sum_{i=1}^{n}i^{3\alpha-1}(n-i)^{\alpha}+\sum_{i=1}^{n}i^{4\alpha-1}\right)=O\left(n^{4\alpha+\delta}\right)\end{split}$
(6.4)

for each $\delta>0$. From (6.3) combined with (2.1) and (6.4) we obtain for
each $\delta>0$ the estimate

$\sum_{i=0}^{n}\sum_{j=0}^{n}\operatorname{\mathbf{P}}\left(0\sim i\sim
j\right)\leq Cn^{4\alpha+\delta},$ (6.5)

where the constant $C$ depends on $\delta$ but not on $n$. An analogous
calculation for $k$ in place of $0$ shows that also

$\sum_{i=0}^{n}\sum_{j=0}^{n}\operatorname{\mathbf{P}}\left(k\sim i\sim
j\right)\leq Cn^{4\alpha+\delta},\quad k\in[n],$ (6.6)

where $C$ can be chosen uniformly in $n$ and in $k\in[n]$. (An intuitive
reason for this uniformity comes from the fact that for each $k\in[n]$ and
small $\varepsilon>0$, the big majority of the pairs $(i,j)\in[n]^{2}$ leads
to pairwise distances $|i-j|$, $|i-k|$ and $|j-k|$ that are all between
$\varepsilon n$ and $n$.)

The first equality in (2.3) now follows directly from (6.6), and the second
equality in (2.3) is immediate from (1.8). ∎

###  6.2. Quartet coalescence probabilities

######  Lemma 6.2.

For all $r_{2}>r_{1}>0$ and $i\in\mathbb{N}$ we have for a slowly varying
function $\bar{L}$ not depending on $r$

$\displaystyle\operatorname{\mathbf{P}}\left(0\sim
i\sim\lfloor(1+r_{1})i\rfloor\sim\lfloor(1+r_{2})i\rfloor\right)$ (6.7)
$\displaystyle\leq$
$\displaystyle\left(r_{1}^{\alpha-1}+r_{1}^{2\alpha-1}+r_{2}^{\alpha-1}+r_{2}^{2\alpha-1}\right)\bar{L}(i)i^{6\alpha-3}.$

###### Proof.

Again let $(A_{i})$ be the independent (non-merging) ancestral lineages
defined in (4.1). Let $B_{k,\ell}$ be as in (6.1) and set

$B_{[k,\ell]j}\mathrel{\mathop{\mathchar 58\relax}}=\bigcup_{m\in
B_{k,\ell}}\bigcup_{g\in A_{m}\cap A_{j}}A_{g}.$

We fix $i\in\mathbb{N}$ and set with $j_{1}\mathrel{\mathop{\mathchar
58\relax}}=(1+r_{1})i$, $j_{2}\mathrel{\mathop{\mathchar
58\relax}}=(1+r_{2})i$,

$\displaystyle Q\mathrel{\mathop{\mathchar 58\relax}}=$
$\displaystyle\\{(0,i,j_{1},j_{2}),(0,i,j_{2},j_{1}),(0,j_{1},i,j_{2}),(0,j_{1},j_{2},i),$
$\displaystyle(0,j_{2},i,j_{1}),(0,j_{2},j_{1},i),(i,j_{1},0,j_{2}),(i,j_{1},j_{2},0),$
$\displaystyle(i,j_{2},0,j_{1}),(i,j_{2},j_{1},0),(j_{1},j_{2},0,i),(j_{1},j_{2},i,0)\\}$

We now argue in a similar way as in the proof of Proposition 6.1. By
subadditivity we get

$\displaystyle\operatorname{\mathbf{P}}\left(0\sim
i\sim(1+r_{1})i\sim(1+r_{2})\right)\leq\sum_{(k_{1},k_{2},k_{3},k_{4})\in
Q}\operatorname{\mathbf{E}}\left[|B_{[k_{1},k_{2}]k_{3}}\cup
A_{k_{4}}|\right].$

By the very same arguments as in the proof of Lemma 6.1 one checks that each
of the twelve summands is bounded by the right-hand side of (6.7). ∎

###### Proof of Proposition 2.3 Part 2.

We are now going to prove (2.4), and first set out to show that for all
$\delta>0$

$\sum_{i=1}^{n}\sum_{j=1}^{n-i}\sum_{\ell=1}^{n-i-j}\operatorname{\mathbf{P}}\left(0\sim
i\sim i+j\sim i+j+\ell\right)=O\left(n^{6\alpha+\delta}\right).$ (6.8)

By Lemma 6.2, and since
$r_{1}^{\alpha-1}+r_{1}^{2\alpha-1}+r_{2}^{\alpha-1}+r_{2}^{2\alpha-1}\leq
4(r_{1}^{\alpha-1}+1)$, the left-hand side. of (6.8) is bounded from above by

$C\sum_{i=1}^{n}\bar{L}(i)i^{6\alpha-3}\sum_{j=1}^{n-i}\sum_{\ell=1}^{n-i-j}\left[\left(\frac{j}{i}\right)^{\alpha-1}+1\right].$

By arguments analogous to those leading to (6.3) in the proof of Part 1, this
implies (6.8). As in the proof of Part 1 we can argue that, as $n\to\infty$
the terms

$\sum_{i=0}^{n}\sum_{j=0}^{n}\sum_{\ell=0}^{n}\operatorname{\mathbf{P}}\left(k\sim
i\sim j\sim\ell\right)$

are of the same order uniformly in for all $k\in\\{0,1,\ldots,n\\}$, so it is
enough to look at the case $k=0$. Also, from (2.1) and (2.3) it is clear that
we may then restrict to pairwise distinct $i,j,\ell$. The first equality in
(2.4) thus follows from (6.8), and (like in Part 1) the second equality in
(2.4) is clear from (1.8). ∎

##  7\. A covariance estimate: Proof of Lemma 2.5

For $i=k$ or $j=\ell$ the assertion of Lemma 2.5 is clearly true because then
the left-hand side. of (2.5) vanishes. For $i=k$ we have

$\operatorname{\mathbf{Cov}}[I_{\\{i\sim
j\\}},I_{\\{i\sim\ell\\}}]=\operatorname{\mathbf{P}}(i\sim
j\sim\ell)-\operatorname{\mathbf{P}}(i\sim
j)\operatorname{\mathbf{P}}(i\sim\ell)\leq\operatorname{\mathbf{P}}(i\sim
j\sim\ell).$

Hence we may assume without loss of generality that $i,j,k,\ell$ are pairwise
distinct. We then have

$\operatorname{\mathbf{Cov}}\left[I_{i\sim
k},I_{j\sim\ell}\right]=\operatorname{\mathbf{P}}\left(i\sim k\cap
j\sim\ell\right)-\operatorname{\mathbf{P}}\left(i\sim
k\right)\operatorname{\mathbf{P}}\left(j\sim\ell\right)$ (7.1)

and

$\operatorname{\mathbf{P}}\left(i\sim k\cap
j\sim\ell\right)=\operatorname{\mathbf{P}}\left(i\sim k\not\sim
j\sim\ell\right)+\operatorname{\mathbf{P}}\left(i\sim k\sim j\sim\ell\right).$
(7.2)

By (7.1) and (7.2), the inequality (2.5) is immediate from the following

######  Lemma 7.1.

For pairwise distinct $i,j,k,\ell$ we have

$\operatorname{\mathbf{P}}\left(i\sim k\not\sim
j\sim\ell\right)\leq\operatorname{\mathbf{P}}\left(i\sim
k\right)\operatorname{\mathbf{P}}\left(j\sim\ell\right)$

###### Proof.

Let $A_{g}$, $g\in\mathbb{Z}$, be defined as in (4.1). For $m\in\mathbb{N}$
and $i_{1}<\ldots<i_{m}\in\mathbb{Z}$ we define an
$(A_{i_{1}},\ldots,A_{i_{m}})$-measurable random graph
$G^{\\{i_{1},\ldots,i_{m}\\}}$ which is equal in distribution to the subgraph
of $\mathcal{G}_{\mu}$ that is formed by the (possibly coalescing) ancestral
lineages of $i_{1},i_{2},\ldots,i_{m}$. The construction of
$G^{\\{i_{1},\ldots,i_{m}\\}}$ is done inductively in the following “lookdown”
manner: the ancestral lineage of $i_{1}$ is taken as $A_{i_{1}}$,
correspondingly, we put $G^{\\{i_{1}\\}}\mathrel{\mathop{\mathchar
58\relax}}=A_{i_{1}}$. The ancestral lineage of $i_{h+1}$ is given by
$A_{i_{h+1}}$ as long as the latter did not meet
$G^{\\{i_{1},\ldots,i_{h}\\}}$. At the time of the first (seen in backward
time direction) collision of $A_{i_{h+1}}$ with
$G^{\\{i_{1},\ldots,i_{h}\\}}$, the ancestral lineage of $i_{h+1}$ is
continued by the lineage in $G^{\\{i_{1},\ldots,i_{h}\\}}$ that starts in the
meeting point (and the continuation of $A_{i_{h+1}}$ from there on is erased).
An inspection of $G^{\\{i,j,k,\ell\\}}$ reveals that

$\displaystyle\operatorname{\mathbf{P}}\left(\left\\{i\sim k\not\sim
j\sim\ell\right\\}\right)$ $\displaystyle=$
$\displaystyle\operatorname{\mathbf{P}}\left(\left\\{A_{i}\cap
A_{k}\not=\emptyset\right\\}\cap\left\\{A_{j}\cap
A_{\ell}\not=\emptyset\right\\}\cap\left\\{G^{\\{i,k\\}}\cap
G^{\\{j,\ell\\}}=\emptyset\right\\}\right)$ $\displaystyle\leq$
$\displaystyle\operatorname{\mathbf{P}}\left(\left\\{A_{i}\cap
A_{k}\not=\emptyset\right\\}\cap\left\\{A_{j}\cap
A_{\ell}\not=\emptyset\right\\}\right),$

which because of mutual independence of the $A_{g}$ gives the assertion of the
lemma. ∎

##  8\. Asymptotic Gaussianity in the Hammond-Sheffield urn:
Proof of Theorem 1.11.12

We are going to apply Theorem 3.1, with $\mathscr{P}^{(m)}$ being the
partition on $[m]$ that is generated by the Hammond-Sheffield urn, i.e. by the
equivalence class $\sim$ defined in (1.3). For $d\in\mathbb{N}$ and
$t_{1}<\cdots<t_{d}$ as prescribed in Theorem 1.1 we apply Theorem 3.1 with
$m=m(n)=\lfloor t_{d}n\rfloor$ and $\rho_{g}\mathrel{\mathop{\mathchar
58\relax}}=\lfloor t_{g}n\rfloor/\lfloor t_{d}n\rfloor$, $g=1,\ldots,d$. Under
condition 1.12 of Theorem 1.1, Proposition 2.3 ensures the validity of
assumptions (3.3) and (3.4), and Lemma 2.5 guarantees that (3.5) is fulfilled.
It remains to check the assumption (3.6). Indeed, with

$a(x)\mathrel{\mathop{\mathchar 58\relax}}=\alpha_{g}\mbox{ if
}\rho_{g-1}<x\leq\rho_{g},\quad x\in(0,1],\quad g=1,\ldots,d,$

because of $\operatorname{\mathbf{P}}(i\sim
j)=\operatorname{\mathbf{P}}(0\sim|i-j|)$ and in view of Proposition 2.1 the
left-hand side. of (3.6) has the asymptotics

$m^{-2}\sum_{i,j\in[m]}a^{(m)}_{i}a^{(m)}_{j}\operatorname{\mathbf{P}}(i\sim
j)\sim\frac{Cm^{2\alpha-1}}{L(m)^{2}}\int_{0}^{1}\int_{0}^{1}a(x)a(y)|x-y|^{2\alpha-1}\mathrm{d}x\,\mathrm{d}y\quad\mbox{as
}m\to\infty$ (8.1)

The Riesz kernel $|x-y|^{2\alpha-1}$ is positive definite (see e.g. [Dos98]),
hence the integral term in (8.1) is strictly positive, and consequently the
left-hand side of (8.1) is of the order $m^{2\alpha+1}$ as $m\to\infty$.
Because of (1.8), this is also the order of the right-hand side of (3.6).

##  9\. Tightness: Proof of Proposition 1.3

Inspired by the proof of Theorem 1 in [Sot01], which shows tightness of a
different approximation scheme for fractional Brownian motion, we will make
use of the following

######  Lemma 9.1 ([Bil68, Theorem 13.5]).

Let $T>0$ and $(\zeta^{(n)}(t))_{0\leq t\leq T}$ be continuous processes that
converge to a continuous process $(\zeta(t))_{0\leq t\leq T}$ in the sense of
finite dimensional distributions. Assume that for a nondecreasing continuous
function $F$ on $[0,T]$, for some $\gamma>1$, for all $0\leq s\leq t\leq u\leq
T$, for some $N\in\mathbb{N}$ and all $n\geq N$

$\operatorname{\mathbf{E}}\left[\left|\zeta^{(n)}(t)-\zeta^{(n)}(s)\right|\cdot\left|\zeta^{(n)}(u)-\zeta^{(n)}(t)\right|\right]\leq\left[F(u)-F(s)\right]^{\gamma}.$
(9.1)

Then $(\zeta^{n}(t))_{0\leq t\leq T}$ converges in distribution in
$\left(C\left([0,T]\right),||\cdot||_{L^{\infty}([0,T])}\right)$ to
$(\zeta(t))_{0\leq t\leq T}$.

###### Proof of Proposition 1.3.

We first note the following immediate consequence of (1.7) and (1.8): There
exist constants $c,c^{\prime}>0$ such that

$cn^{2\alpha+1}L(n)^{-2}\leq\sigma_{n}^{2}\leq
c^{\prime}n^{2\alpha+1}L(n)^{-2},\quad\forall n\in\mathbb{N}.$ (9.2)

Next we fix $0\leq s\leq t\leq u\leq T$ and $j,k,l\in\mathbb{N}$ satisfying

$\frac{j}{n}\leq s<\frac{j+1}{n},\quad\frac{k}{n}\leq
t<\frac{k+1}{n},\quad\frac{l}{n}\leq u<\frac{l+1}{n}.$

The definition of $S^{(n)}$ as a linear interpolation gives

$\sigma_{n}(S^{(n)}(t)-S^{(n)}(s))=\sum_{m=j+1}^{k}Y_{m}+\left[1-\left(sn-j\right)\right]Y_{j+1}+\left[tn-k\right]Y_{k+1}.$

We note that $0\leq\left[1-\left(sn-j\right)\right]\leq 1$,
$0\leq\left[tn-k\right]\leq 1$ and get that for some constants
$c_{1},c_{2},c_{3},c_{4}$

$\displaystyle\operatorname{\mathbf{Var}}\left[S^{(n)}(t)-{S}^{(n)}(s)\right]$
$\displaystyle\leq$
$\displaystyle\left[2\operatorname{\mathbf{Var}}\left[S^{(k-j)}\right]+2c_{1}\operatorname{\mathbf{Var}}\left[Y_{j+1}+Y_{k+1}\right]\right]\sigma_{n}^{-2}$
(9.3) $\displaystyle\leq$
$\displaystyle\left[2c^{\prime}(k-j)^{2\alpha+1}L(k-j)^{-2}+c_{2}\right]\
\sigma_{n}^{-2}$ $\displaystyle\leq$
$\displaystyle\left[c_{3}(k-j)^{2\alpha+1}L(k-j)^{-2}\right]\sigma_{n}^{-2}$
$\displaystyle\leq$ $\displaystyle
c_{4}\left(\frac{k-j}{n}\right)^{2\alpha+1}\left(\frac{L(n)}{L(k-j)}\right)^{2}.$

Here the first inequality holds because for any two square-integrable random
variables $G_{1}$, $G_{2}$ one has
$\operatorname{\mathbf{Var}}\left[G_{1}+G_{2}\right]\leq
2\operatorname{\mathbf{Var}}[G_{1}]+2\operatorname{\mathbf{Var}}[G_{2}]$, the
second and the last inequality hold because of (9.2), and the third one holds
for some $c_{3}>0$, some $N_{1}\in\mathbb{N}$ and all $n\geq N_{1}$ because
$L$ is a slowly varying function. (Remember that $k,j,l$ depend on $n$ for
fixed time points $s,t,u$.) Analogously,

$\operatorname{\mathbf{Var}}\left[S^{(n)}(u)-S^{(n)}(t)\right]\leq
c_{3}\left(\frac{l-k}{n}\right)^{2\alpha+1}\left(\frac{L(n)}{L(l-k)}\right)^{2}.$
(9.4)

To use Lemma 9.1 we have to bound the expectation

$\operatorname{\mathbf{E}}\left[\left|S^{(n)}(t)-S^{(n)}(s)\right|\cdot\left|S^{(n)}(u)-S^{(n)}(t)\right|\right].$

For $l\leq j+1$ we get that $|u-t|\leq\frac{2}{n}$ and $|t-s|\leq\frac{2}{n}$,
so that by basic calculus $(u-t)(t-s)\leq(u-s)^{1+\alpha}$. By the definition
of $\mu$ and the fact that $L$ is slowly varying one has $L(n)\leq
n^{-\alpha}$ for $n$ large enough. So linear interpolation gives:

$\displaystyle\operatorname{\mathbf{E}}\left[|S^{(n)}(t)-S^{(n)}(s)|\cdot|S^{(n)}(u)-S^{(n)}(s)|\right]$
$\displaystyle\leq$
$\displaystyle\left(\tilde{c}n^{-\frac{1}{2}-\alpha}L(n)\right)^{2}\left((t-s)\cdot(u-s)\right)$
$\displaystyle\leq$
$\displaystyle\left(\tilde{c}n^{-\frac{1}{2}}\right)^{2}(u-s)^{1+2\alpha}\,=\,\tilde{c}^{2}\frac{1}{n}(u-s)^{1+2\alpha}$
$\displaystyle\leq$
$\displaystyle\tilde{c}^{2}(u-s)^{1+2\alpha}\,=\,\left[c_{0}u-c_{0}s\right]^{1+2\alpha}$

for some $c_{0}>0$. Now assume $l>j+1$. Cauchy-Schwarz and the estimates
(9.3),(9.4) yield

$\displaystyle\operatorname{\mathbf{E}}\left[\left|S^{(n)}(t)-S^{(n)}(s)\right|\cdot\left|S^{(n)}(u)-S^{(n)}(t)\right|\right]$
$\displaystyle\leq$
$\displaystyle\sqrt{\operatorname{\mathbf{Var}}\left[S^{(n)}(t)-S^{(n)}(s)\right]}\sqrt{\operatorname{\mathbf{Var}}\left[S^{(n)}(u)-S^{(n)}(t)\right]}$
$\displaystyle\leq$
$\displaystyle\sqrt{c_{3}\left(\frac{k-j}{n}\right)^{2\alpha+1}\left(\frac{L(n)}{L(k-j)}\right)^{2}}\sqrt{c_{3}\left(\frac{l-k}{n}\right)^{2\alpha+1}\left(\frac{L(n)}{L(l-k)}\right)^{2}}$
$\displaystyle\leq$ $\displaystyle
c_{3}\left(\frac{l-j}{n}\right)^{2\alpha+1}\left(\frac{L(n)}{L(k-j)L(l-k)}\right)^{2}$
$\displaystyle\leq$ $\displaystyle
2c_{3}\left(\frac{l-j}{n}\right)^{2\alpha+1}\leq\quad\left(2c_{3}\cdot
2(u-s)\right)^{2\alpha+1}.$

(The third inequality holds because of $4|k-j|\cdot|l-k|\leq|l-j|^{2}$, and
the fourth one holds for $n\geq N_{2}$ for some $N_{2}\in\mathbb{N}$ because
$L$ is a slowly varying function.)

Thus, $\zeta^{(n)}\mathrel{\mathop{\mathchar 58\relax}}=S^{(n)}$ fulfills
condition (9.1) with with $N\mathrel{\mathop{\mathchar
58\relax}}=\max\\{N_{1},N_{2}\\}$, $\gamma\mathrel{\mathop{\mathchar
58\relax}}=\alpha+1/2$ and $F(x)\mathrel{\mathop{\mathchar
58\relax}}=4\max\\{c_{0},c_{3}\\}x$. In view of Corollary 1.2 we can thus
apply Lemma 9.1 and conclude the assertion of Proposition 1.3. ∎

######  Remark 9.2.

In [HS13] the functional convergence of $S^{(n)}$ was deduced from a tail
estimate (uniform in $n$) on $\max_{t\in[0,1]}S^{(n)}_{t}$, stated in [HS13,
Lemma 4.1]. The proof of this lemma given there relies on the statement that
for a certain sequence $(r_{n})$ the inequalities (4.11) in [HS13] imply
boundedness of $(r_{n})$. There are, however, examples of unbounded sequences
which fulfill these inequalities. Still, things clear up nicely because the
assertion of [HS13, Lemma 4.1] is a quick consequence of Corollary 1.4 and the
Borell-TIS inequality.

##  10\. Coalescence probabilities in long-range seedbanks

In this section we will assume that the weights $q_{n}$ of the renewal measure
defined in (1.4) obey the asymptotics (1.10), see the discussion after
equation (1.10). Following [BGKS13] we extend the model described in Section 1
as follows. For fixed $N\in\mathbb{N}$, the set of vertices of
$\mathcal{G}_{\mu}$ is now $\mathbb{Z}\times[N]$. The set of those vertices
whose first component is $i$ constitutes the population of individuals living
at time $i$. The parent of the individual $(i,k)$ is $(i-R_{i,k},H_{i,k})$,
where the random variables $R_{i,k}$ are independent copies of $R$ and the
random variables $H_{i,k}$ are i.i.d. picks from $[N]$. In words, each
individual chooses its parent uniformly from a previous time with delay (or
dormancy) distribution $\mu$. The corresponding urn model, which goes back to
Kaj, Krone and Lascoux ([KKL01]), thus specialises to the Hammond-Sheffield
urn for $N=1$.

Again we write $(i,k)\sim(j,\ell)$ if the two individuals $(i,k)$ and
$(j,\ell)$ belong to the same connected component of $\mathcal{G}_{\mu}$.
Thanks to Proposition 4.1, which also provides a proof of [BGKS13, Lemma 3.1
(c)], we arrive at the following analogue of Proposition 2.1 (see also
[BGKS13, Theorem 3(c))]:

######  Proposition 10.1.

For all $k,\ell\in[N]$

$\operatorname{\mathbf{P}}\left((0,k)\sim(i,\ell)\right)\sim
C_{\alpha,N}\frac{i^{2\alpha-1}}{L(i)^{2}}\quad\mbox{ as }i\to\infty,$ (10.1)

where now

$C_{\alpha,N}\mathrel{\mathop{\mathchar 58\relax}}=\frac{1}{N+\sum_{m\geq
1}q_{m}^{2}}\,\frac{\Gamma(1-2\alpha)}{\Gamma(\alpha)\Gamma(1-\alpha)^{3}}.$

###### Proof.

Denote by $\tilde{A}_{0,k}$ and $\tilde{A}_{i,\ell}$ the decoupled ancestral
lineages of the individuals $(0,k)$ and $(i,\ell)$ constructed in analogy to
(4.1). Like in (4.2) we observe

$\operatorname{\mathbf{E}}\left[|\tilde{A}_{0,k}\cap\tilde{A}_{i,\ell}|\right]=\sum_{m\geq
0}\frac{1}{N}q_{m}q_{m+i}.$

Decomposing at the most recent collision time of $\tilde{A}_{0,k}$ and
$\tilde{A}_{i,\ell}$ we get

$\displaystyle\operatorname{\mathbf{E}}\left[\tilde{A}_{0,k}\cap\tilde{A}_{0,\ell}\right]=\operatorname{\mathbf{P}}\left(\tilde{A}_{0,k}\cap\tilde{A}_{i,\ell}\not=\emptyset\right)\cdot\left(1+\sum_{m\geq
1}\frac{1}{N}q_{m}^{2}\right).$

The last two equalities combine to

$\displaystyle\operatorname{\mathbf{P}}\left(|A_{0,k}\cap
A_{i,\ell}|\not=\emptyset\right)=\frac{\ \sum_{m\geq
0}q_{m}q_{m+i}}{N+\sum_{m\geq 1}q_{m}^{2}}.$

The claimed asymptotics (10.1) is now immediate from Proposition 4.1. ∎

######  Remark 10.2.

For given natural numbers $n$ and $N$ let
$\mathscr{I},\mathscr{J},\mathscr{K}$ and $\mathscr{L}$ be independent and
uniformly distributed on $[n]\times[N]$. In complete analogy to Proposition
2.3 one can derive that for all $\delta>0$ and for a constant $C$ not
depending on $n$ and $N$

$\displaystyle\operatorname{\mathbf{P}}\left(\mathscr{I}\sim\mathscr{J}\sim\mathscr{K}\right)\qquad$
$\displaystyle\leq$ $\displaystyle\frac{C}{N^{2}}n^{4\alpha-2+\delta},$
$\displaystyle\operatorname{\mathbf{P}}\left(\mathscr{I}\sim\mathscr{J}\sim\mathscr{K}\sim\mathscr{L}\right)$
$\displaystyle\leq$ $\displaystyle\frac{C}{N^{3}}n^{6\alpha-3+\delta}.$

Consequently, along the lines of the proof of Theorem 1.1 one obtains the
convergence of an analogue of $(S^{(n)})$ towards fractional Brownian motion
also in the long-range seedbank model of [BGKS13].

#### Acknowledgment.

We thank Matthias Birkner, Florin Boenkost, Adrian González Casanova, Alan
Hammond and Nicola Kistler for stimulating discussions and valuable hints. We
also thank the _Allianz für Hochleistungsrechnen Rheinland-Pfalz_ for granting
us access to the High Performance Computing Elwetritsch, on which simulations
have been performed which inspired results in this work. We are also grateful
to two anonymous referees whose careful reading and thoughtful comments led to
a substantial improvement of the presentation.

## References

  * [BGKS13] J. Blath, A. González Casanova, N. Kurt, and D. Spanò.  The ancestral process of long-range seed bank models.  Journal of Applied Probability, 50(3):741–759, 2013.  URL http://www.jstor.org/stable/43283498.
  * [BGT87] N. H. Bingham, C. M. Goldie, and J. L. Teugels.  Regular variation.  Cambridge University Press Cambridge [Cambridgeshire] ; New York, 1987\.  URL http://www.loc.gov/catdir/toc/cam027/86028422.html.
  * [Bil68] P. Billingsley.  Convergence of probability measures 2nd edition.  John Wiley & Sons, Inc., New York-London-Sydney, 1968.
  * [CD19] F. Caravenna and R. Doney.  Local large deviations and the strong renewal theorem.  Electronic Journal of Probability, 24:1 – 48, 2019.  doi:10.1214/19-EJP319.
  * [Don97] R. Doney.  One-sided local large deviation and renewal theorems in the case of infinite mean.  Probability Theory and Related Fields, 107:451–465, 04 1997.  doi:10.1007/s004400050093.
  * [Dos98] M. R. Dostanić.  Spectral properties of the operator of riesz potential type.  Proceedings of the American Mathematical Society, 126(8):2291–2297, 1998.  URL http://www.jstor.org/stable/118743.
  * [Fel71] W. Feller.  An introduction to probability theory and its applications. Vol. II. Second edition. John Wiley & Sons Inc., New York, 1971.
  * [HS13] A. Hammond and S. Sheffield.  Power law Pólya’s urn and fractional brownian motion.  Probability Theory and Related Fields, 157(3):691–719, 2013.  doi:10.1007/s00440-012-0468-6.
  * [KKL01] I. Kaj, S. Krone, and M. Lascoux.  Coalescent theory for seed bank models.  Journal of Applied Probability, 38:285–300, 06 2001.  doi:10.1017/S0021900200019860.
  * [Sot01] T. Sottinen.  Fractional Brownian motion, random walks and binary market models.  Finance Stoch., 5(3):343–355, 2001.  doi:10.1007/PL00013536.
  * [Ste86] C. Stein.  Approximate computation of expectations.  Lecture Notes-Monograph Series, 7:i–164, 1986.  URL http://www.jstor.org/stable/4355512.