## Scaling limits for the block counting process and the fixation line of a
class of $\Lambda$-coalescents

Martin Möhle and Benedict Vetter111Mathematisches Institut, Eberhard Karls
Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany, E-mail
addresses<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We provide scaling limits for the block counting process and the fixation line
of $\Lambda$-coalescents as the initial state $n$ tends to infinity under the
assumption that the measure $\Lambda$ on $[0,1]$ satisfies
$\int_{[0,1]}u^{-1}(\Lambda-b\lambda)({\rm d}u)<\infty$ for some $b>0$. Here
$\lambda$ denotes the Lebesgue measure. The main result states that the block
counting process, properly logarithmically scaled, converges in the Skorohod
space to an Ornstein–Uhlenbeck type process as $n$ tends to infinity. The
result is applied to beta coalescents with parameters $1$ and $b>0$. We split
the generators into two parts by additively decomposing $\Lambda$ and then
prove the uniform convergence of both parts separately.

Keywords: coalescent process; block counting process; fixation line;
Bolthausen–Sznitman coalescent; Ornstein–Uhlenbeck type process; self-
decomposability; generalized Mehler semigroup; time-inhomogeneous process;
weak convergence.

2020 Mathematics Subject Classification: Primary 60J90; Secondary 60J27.

###  1 Introduction

The $\Lambda$-coalescent, independently introduced by Pitman [16] and Sagitov
[18], is a Markov process $\Pi=(\Pi_{t})_{t\geq 0}$ with càdlàg paths, values
in the space of partitions of ${\mathbb{N}}:=\\{1,2,\ldots\\}$, starting at
time $t=0$ from the partition $\\{\\{1\\},\\{2\\},\ldots\\}$ of ${\mathbb{N}}$
into singletons, whose behavior is fully determined by a finite measure
$\Lambda$ on the Borel subsets of $[0,1]$. If the process is in a state with
$k\geq 2$ blocks, any particular $j\in\\{2,\ldots,k\\}$ blocks merge at the
rate

$\displaystyle\lambda_{k,j}=\int_{[0,1]}u^{j-2}(1-u)^{k-j}\Lambda({\rm d}u).$

The reader is referred to [3] for a survey of $\Lambda$-coalescents. Unless
$\Lambda(\\{1\\})>0$, either $\Pi_{t}$ has infinitely many blocks for all
$t>0$ almost surely or finitely many blocks for all $t>0$ almost surely. The
$\Lambda$-coalescent is said to stay infinite in the first case and to come
down from infinity in the second. An atom of $\Lambda$ at $1$ corresponds to
the rate of jumping to the trivial (and absorbing) partition consisting only
of the block ${\mathbb{N}}$. For $t\geq 0$ let $N_{t}^{(n)}$ denote the number
of blocks of the restriction
$\Pi_{t}^{(n)}:=\\{B\cap[n]|B\in\Pi_{t},B\cap[n]\neq\emptyset\\}$ of $\Pi_{t}$
to $[n]:=\\{1,\ldots,n\\}$. The block counting process
$N^{(n)}:=(N_{t}^{(n)})_{t\geq 0}$ is a $[n]$-valued conservative Markov
process with càdlàg paths that jumps from $k\geq 2$ to
$j\in\\{1,\ldots,k-1\\}$ at the rate

$\displaystyle
q_{k,j}=\binom{k}{j-1}\int_{[0,1]}u^{k-j-1}(1-u)^{j-1}\Lambda({\rm d}u).$

Clearly, $N^{(n)}$ starts in $n$ at time $t=0$, has decreasing paths and
eventually reaches the absorbing state $1$. Our main objective is to analyze
the limiting behavior of the block counting process of $\Lambda$-coalescents
that stay infinite as the initial state $n$ tends to infinity by determining
suitable scaling constants. For $\Lambda$-coalescents with dust the scaling is
$n$. A block $B\in\Pi_{t}$ of size $|B|=1$ is called a singleton. The number
of singletons in $[n]$ divided by $n$ converges to the frequency of singletons
as $n$ tends to infinity, and, if $\int_{[0,1]}u^{-1}\Lambda({\rm d}u)<\infty$
and $\Lambda(\\{0\\})=0$, then the frequency of singletons is strictly
positive or the $\Lambda$-coalescent is said to have dust. It is known that
$N^{(n)}/n$ converges in the Skorohod space $D_{[0,1]}[0,\infty)$ to the
frequency of singletons process as $n$ tends to infinity [9, Theorem 2.13].

The Bolthausen–Sznitman coalescent, where $\Lambda$ is the uniform
distribution on $[0,1]$, is an example of a dust-free $\Lambda$-coalescent
that stays infinite [5]. In this case it was shown in [14] via moment
calculations that $(N_{t}^{(n)}/n^{\exp(-t)})_{t\geq 0}$ converges in
$D_{[0,\infty)}[0,\infty)$ to the Mittag–Leffler process as $n$ tends to
infinity.

We provide unified proofs for both limit theorems, stated later as Corollaries
3 and 4, and extend the convergence to $\Lambda$-coalescents, where the
measure $\Lambda$ is the sum of the uniform distribution on $[0,1]$ multiplied
by a constant $b\geq 0$ and a measure that corresponds to a coalescent with
dust. This includes the $\Lambda$-coalescent where $\Lambda=\beta(1,b)$ is the
beta distribution with parameters $1$ and $b>0$. Theorem 1 below states that
$(\log N_{t}^{(n)}-\exp(-bt)\log n)_{t\geq 0}$ converges in
$D_{{\mathbb{R}}}[0,\infty)$ as $n$ tends to infinity. The logarithmic version
of the convergence result has the advantage of putting the limiting process in
Theorem 1 to a class of processes, which has been studied in the literature.
The limiting process can be represented as the solution of the Langevin
equation with Lévy noise instead of a Brownian motion and is sometimes called
Ornstein–Uhlenbeck type process [19] or generalized Ornstein–Uhlenbeck
process. First Wolfe [25] studied Ornstein–Uhlenbeck type processes on
${\mathbb{R}}$, later Jurek and Vervaat [12] on Banach spaces and Sato and
Yamazato [19] on ${\mathbb{R}}^{d}$, then Applebaum [2] on Hilbert spaces.
Under the logarithmic moment condition (17) the marginal distributions of the
Ornstein–Uhlenbeck type process converge in distribution as time tends to
infinity to the unique stationary distribution. The stationary distribution is
self-decomposable. A real-valued random variable $S$ is self-decomposable if
for every $\alpha\in[0,1]$ there exists a random variable $S_{\alpha}$
independent of $S$ such that $S$ has the same distribution as $\alpha
S+S_{\alpha}$. If $\phi$ is the characteristic function of $S$ then $S$ is
self-decomposable if and only if $x\mapsto\phi(x)/\phi(\alpha x),$
$x\in{\mathbb{R}},$ is the characteristic function of a real-valued random
variable for every $\alpha\in[0,1]$. A distribution $\mu$ on ${\mathbb{R}}$ is
self-decomposable if there exists a self-decomposable random variable with
distribution $\mu$. Conversely, every self-decomposable distribution can be
obtained in this way. For the $\beta(1,b)$-coalescent (17) is satisfied.

The fixation line $L=(L_{t})_{t\geq 0}$ is a ${\mathbb{N}}$-valued Markov
process that jumps from $k\in{\mathbb{N}}$ to $j\in\\{k+1,\ldots\\}$ at the
rate

$\displaystyle\gamma_{k,j}=\binom{j}{j-k+1}\int_{[0,1]}u^{j-k-1}(1-u)^{k}\Lambda({\rm
d}u).$

The fixation line is the ’time-reversal’ of the block counting process, in the
sense that the hitting times $\inf\\{t\geq 0|N_{t}^{(n)}\leq m\\}$ and
$\inf\\{t\geq 0|L_{t}^{(m)}\geq n\\}$ share the same distribution [11, Lemma
2.1]. Equivalently, the process $L$ is Siegmund-dual [22] to the block
counting process, i.e., if $L^{(m)}=(L_{t}^{(m)})_{t\geq 0}$ denotes the
fixation line starting in $L_{0}^{(m)}=m,$ $m\in{\mathbb{N}},$ at time $t=0$
then (see [13])

$\displaystyle{\mathbb{P}}(L_{t}^{(m)}\geq n)={\mathbb{P}}(N_{t}^{(n)}\leq
m),\qquad m,n\in{\mathbb{N}},t\geq 0.$ (1)

For a thorough definition of the fixation line see [11] and the references
therein. Theorem 5 states that $(\log L_{t}^{(n)}-\exp(bt)\log n)_{t\geq 0}$
converges in $D_{{\mathbb{R}}}[0,\infty)$ as the initial value $L_{0}^{(n)}=n$
tends to infinity.

The article is organized as follows. In Section 2 the main convergence result
(Theorem 1) is stated and applied to the $\beta(1,b)$-coalescent with
parameter $b>0$. The limiting process is an Ornstein–Uhlenbeck type process.
Well-known results are applied to our setting in Section 3. In particular, the
generator of the limiting process is determined. The line of proof is in some
sense reversed. First we prove Corollaries 3 and 4 in Sections 4 and 5 by
showing the convergence of the generators of the (logarithm of the) scaled
block counting processes. The decomposition of $\Lambda$ into the uniform
distribution multiplied by a constant and a measure that corresponds to a
coalescent with dust is transferred to the generators. This enables us to use
relations obtained in Sections 4 and 5 to prove Theorem 1 in Section 6. The
proof of Theorem 5 is conducted in Section 7 with similar methods.

Notation. Let $E$ be a complete separable metric space. The Banach space
$B(E)$ of bounded measurable functions $f:E\to{\mathbb{R}}$ is equipped with
the usual supremum norm $\|f\|:=\sup_{x\in E}|f(x)|$ and the Banach subspace
$\widehat{C}(E)\subset B(E)$ consists of continuous functions vanishing at
infinity. If $E\subseteq{\mathbb{R}}^{d}$ for some $d\in{\mathbb{N}}$ then
$C_{k}(E)$ denotes the space of $k$-times continuously differentiable
functions. The Borel-$\sigma$-field on ${\mathbb{R}}$ is denoted by
$\mathcal{B}$, $\Lambda$ is a (non-zero) finite measure on
$\mathcal{B}\cap[0,1]$ with $\Lambda(\\{0\\})=\Lambda(\\{1\\})=0$ and
$\lambda$ denotes the Lebesgue measure on $([0,1],\mathcal{B}\cap[0,1])$. The
generators, usually denoted by $A$, are understood to be defined on (a
subspace of) $\widehat{C}(E)$. For a measure space $(\Omega,\mathcal{F},\mu)$
and $p>0$ the space of measurable functions $f:\Omega\to{\mathbb{R}}$ with
$\int|f|^{p}{\rm d}\mu<\infty$ is denoted by $L^{p}(\mu)$ or in short,
$L^{p}$.

###  2 Results

Let $\Lambda$ be a finite measure on $([0,1],\mathcal{B}\cap[0,1])$ with no
mass at $0$ and $1$. For $b\geq 0$ the map
$B\mapsto(\Lambda-b\lambda)(B)=\Lambda(B)-b\lambda(B),$
$B\in\mathcal{B}\cap[0,1],$ might possibly be a signed measure. Hahn’s
decomposition theorem states the existence of some $A\in\mathcal{B}\cap[0,1]$
such that $(\Lambda-b\lambda)^{+}(B):=(\Lambda-b\lambda)(B\cap A),$
$B\in\mathcal{B}\cap[0,1],$ and
$(\Lambda-b\lambda)^{-}(B):=-(\Lambda-b\lambda)(B\cap A^{c}),$
$B\in\mathcal{B}\cap[0,1],$ define nonnegative measures. The two nonnegative
measures $(\Lambda-b\lambda)^{+}$ and $(\Lambda-b\lambda)^{-}$ constitute the
Jordan decomposition of $\Lambda-b\lambda$. Using this decomposition one can
integrate with respect to a signed measure by defining $\int f{\rm
d}(\Lambda-b\lambda):=\int f{\rm d}(\Lambda-b\lambda)^{+}-\int f{\rm
d}(\Lambda-b\lambda)^{-}$ for $f\in L^{1}((\Lambda-b\lambda)^{+})\cap
L^{1}((\Lambda-b\lambda)^{-})$. The assumption of Theorem 1 below is the
following.

Assumption A. There exists $b\geq 0$ such that
$\int_{[0,1]}u^{-1}(\Lambda-b\lambda)^{+}({\rm d}u)<\infty$ and
$\int_{[0,1]}u^{-1}(\Lambda-b\lambda)^{-}({\rm d}u)<\infty$.

Note that the constant $b\geq 0$ is uniquely determined by $\Lambda$, if it
exists. Schweinsberg’s criterion [21] shows that the $\Lambda$-coalescent does
not come down from infinity under Assumption A, see Lemma 8 in the appendix.
Moreover, the $\Lambda$-coalescent is dust-free if $b>0$. Assumption A is for
example satisfied, if $\Lambda$ has density $f\in C_{1}([0,1])$ with respect
to $\lambda$ for which $\lim_{u\searrow 0}f^{\prime}(u)$ exists and is finite.
In this case $b=\lim_{u\searrow 0}f(u)$.

Suppose that $\Lambda$ satisfies Assumption A. Let
$\Gamma(z):=\int_{0}^{\infty}u^{z-1}e^{-u}{\rm d}u,$ ${\rm Re}(z)>0,$ denote
the gamma function and
$\Psi(z):=(\log\Gamma)^{\prime}(z)=\Gamma^{\prime}(z)/\Gamma(z),$ ${\rm
Re}(z)>0,$ the digamma function. Define

$\displaystyle a:=b(1+\Psi(1))-\int_{[0,1]}u^{-1}(\Lambda-b\lambda)({\rm d}u)$
(2)

and $\psi:{\mathbb{R}}\to{\mathbb{C}}$ via

$\displaystyle\psi(x):=iax+\int_{[0,1]}(e^{ix\log(1-u)}-1+ixu)u^{-2}\Lambda({\rm
d}u),\qquad x\in{\mathbb{R}}.$ (3)

Substituting $g:(0,1)\to{\mathbb{R}},$ $g(u):=\log(1-u),$ $u\in(0,1)$, shows
that

$\displaystyle\psi(x)=iax+\int_{(-\infty,0)}(e^{ixu}-1+ix(1-e^{u}))\varrho({\rm
d}u),\qquad x\in{\mathbb{R}},$

where the measure $\varrho$, defined via

$\displaystyle\varrho(A):=\int_{g^{-1}(A)}u^{-2}\Lambda({\rm
d}u)=\int_{A}(1-e^{u})^{-2}\Lambda_{g}({\rm d}u),\qquad A\in\mathcal{B},$ (4)

satisfies $\int_{{\mathbb{R}}}(u^{2}\wedge 1)\varrho({\rm d}u)<\infty$ and
$\varrho(\\{0\\})=0$. Hence $\varrho$ is a Lévy measure and $e^{\psi(x)},$
$x\in{\mathbb{R}},$ is the characteristic function of an infinitely divisible
distribution.

###### Theorem 1

Suppose that $\Lambda$ satisfies Assumption A. Then the possibly time-
inhomogeneous Markov process $X^{(n)}:=(X_{t}^{(n)})_{t\geq 0}:=(\log
N_{t}^{(n)}-e^{-bt}\log n)_{t\geq 0}$ converges in
$D_{{\mathbb{R}}}[0,\infty)$ as $n\to\infty$ to the time-homogeneous Markov
process $X=(X_{t})_{t\geq 0}$ with initial value $X_{0}=0$ and semigroup
$(T_{t})_{t\geq 0}$ given by

$\displaystyle
T_{t}f(x):={\mathbb{E}}(f(X_{s+t})|X_{s}=x)={\mathbb{E}}(f(e^{-bt}x+X_{t})),\qquad
x\in{\mathbb{R}},f\in B({\mathbb{R}}),s,t\geq 0,$ (5)

where $X_{t}$ has characteristic function $\phi_{t}$ given by

$\displaystyle\phi_{t}(x)=\exp\Big{(}\int_{0}^{t}\psi(e^{-bs}x){\rm
d}s\Big{)},\qquad x\in{\mathbb{R}},t\geq 0,$ (6)

and $\psi$ is given by (3).

Distributions with characteristic functions of the form (6), where $\psi$ is
more generally the characteristic exponent of an infinitely divisible
distribution, have been studied in [25] on ${\mathbb{R}}$ and on more general
state spaces in [2], [12] and [19]. Since
$\phi_{t+s}(x)=\phi_{t}(e^{-bs}x)\phi_{s}(x),$ $x\in{\mathbb{R}},$ for
$s,t\geq 0$, the semigroup $(T_{t})_{t\geq 0}$ belongs to the class of
generalized Mehler semigroups [4]. In particular, the semigroup
$(T_{t})_{t\geq 0}$ is Feller, i.e.,
$T_{t}(\widehat{C}({\mathbb{R}}))\subseteq\widehat{C}({\mathbb{R}})$ for each
$t\geq 0$ and $(T_{t})_{t\geq 0}$ is strongly continuous on
$\widehat{C}({\mathbb{R}})$. Hence the Markov process $X$ in Theorem 1 exists
and has paths in $D_{{\mathbb{R}}}[0,\infty)$.

The beta distribution $\beta(a,b)$ with parameters $a,b>0$ has density
$u\mapsto\Gamma(a+b)/(\Gamma(a)\Gamma(b))\linebreak u^{a-1}(1-u)^{b-1},$
$u\in(0,1),$ with respect to Lebesgue measure on $(0,1)$. The class of beta
coalescents are the processes for which $\Lambda=\beta(a,b)$ for some $a,b>0$.
They have been extensively studied in the literature due to the easy
computability of the jump rates

$\displaystyle
q_{k,j}=\frac{\Gamma(a+b)\Gamma(k+1)\Gamma(j-1+b)\Gamma(k-j-1+a)}{\Gamma(a)\Gamma(b)\Gamma(k-2+a+b)\Gamma(j)\Gamma(k-j+2)},\qquad
j\in\\{1,\ldots,k-1\\},k\geq 2.$ (7)

The beta coalescent comes down from infinity if and only if $0<a<1$ [21,
Example 15]. If $a>1$ then the beta coalescent has dust. For $a=1$ the beta
coalescent is dust-free and does not come down from infinity.

###### Example 2

Suppose that $\Lambda=\beta(1,b)$ with $b>0$. From the observation stated
below Assumption A we conclude that Assumption A is satisfied with the same
constant $b$. The ’dust-part’ $\Lambda-b\lambda$ has possibly negative density
$u\mapsto b((1-u)^{b-1}-1),$ $u\in(0,1),$ with respect to Lebesgue measure on
$(0,1)$. The underlying Lévy measure $\varrho$ has density $f$ with respect to
Lebesgue measure on ${\mathbb{R}}\setminus\\{0\\}$ given by
$f(u):=be^{bu}(1-e^{u})^{-2}$ for $u<0$ and $f(u):=0$ for $u>0$. Calculations
involving Gauß’ representation [24, p. 247] for the digamma function $\Psi$
(see Proposition 11 in the appendix) show that $a=b(1+\Psi(b))$ and

$\displaystyle\psi(x)=b((1-b)\Psi(b)-(1-b-ix)\Psi(b+ix)),\qquad
x\in{\mathbb{R}}.$ (8)

According to Theorem 1 the process $(\log N_{t}^{(n)}-e^{-bt}\log n)_{t\geq
0}$ converges in $D_{{\mathbb{R}}}[0,\infty)$ as $n\to\infty$ to a Markov
process $X=(X_{t})_{t\geq 0}$ with initial value $X_{0}=0$ and semigroup
$(T_{t})_{t\geq 0}$ given by

$\displaystyle
T_{t}f(x):={\mathbb{E}}(f(X_{s+t})|X_{s}=x)={\mathbb{E}}(f(e^{-bt}x+X_{t})),\qquad
x\in{\mathbb{R}},f\in B({\mathbb{R}}),s,t\geq 0,$

where $X_{t}$ has characteristic function $\phi_{t}$ given by (6). Since
$\int_{(1-e^{-1},1)}\log\log(1-u)^{-1}\Lambda({\rm
d}u)=\int_{1-e^{-1}}^{1}\log\log(1-u)^{-1}b(1-u)^{b-1}{\rm d}u<\infty$, the
logarithmic moment condition of Lemma 7 is satisfied and $X_{t}$ converges in
distribution as $t\to\infty$ to the unique stationary distribution $\mu$ of
$X$. The distribution $\mu$ is self-decomposable with characteristic function
$\phi$ given by

$\displaystyle\phi(x)=\exp\Big{(}\int_{0}^{\infty}\psi(e^{-bs}x){\rm
d}s\Big{)}=\exp\Big{(}(1-b)\int_{0}^{x}\frac{\Psi(b)-\Psi(b+iu)}{u}{\rm
d}u\Big{)}\frac{\Gamma(b+ix)}{\Gamma(b)},\quad x\in{\mathbb{R}}.$ (9)

In the last step equation (8) and the fact that
$\Psi(z)=(\log\Gamma(z))^{\prime},$ ${\rm Re}(z)>0,$ has been used. The
characteristic function $\phi_{t}$ of $X_{t}$ is hence given by

$\displaystyle\phi_{t}(x)=\frac{\phi(x)}{\phi(e^{-bt}x)}=\exp\Big{(}(1-b)\int_{e^{-bt}x}^{x}\frac{\Psi(b)-\Psi(b+iu)}{u}{\rm
d}u\Big{)}\frac{\Gamma(b+ix)}{\Gamma(b+ie^{-bt}x)},\quad
x\in{\mathbb{R}},t\geq 0.$

If $Z$ has a gamma distribution with parameter $b$ and $1$, i.e., $Z$ has
density $u\mapsto u^{b-1}e^{-u}(\Gamma(b))^{-1},$ $u>0$, with respect to
Lebesgue measure on $(0,\infty)$ then $\log Z$ has a self-decomposable
distribution and characteristic function $\Gamma(b+ix)/\Gamma(b),$
$x\in{\mathbb{R}},$ see [23, V, Example 9.18]. If $b<1$ then the first factor
on the right-hand side of (9) is the characteristic function of a self-
decomposable distribution (see [23, V, Theorem 6.7] and the proof of
Proposition 11). The underlying characteristic exponent
$(1-b)(\Psi(b)-\Psi(b+iu)),$ $u\in{\mathbb{R}},$ corresponds to the negative
of a drift-free subordinator. Similarly to the convergence above,
$(N_{t}^{(n)}/n^{e^{-bt}})_{t\geq 0}$ converges in $D_{[0,\infty)}[0,\infty)$
to $(\exp(X_{t}))_{t\geq 0}$ as $n\to\infty$.

The two cases mentioned in the introduction arise from Assumption A as
follows. If $\int_{[0,1]}u^{-1}\Lambda({\rm d}u)<\infty$, then the
$\Lambda$-coalescent has dust and Assumption A is satisfied with $b=0$.
Corollary 3 below has been proven in [9] and [15]. In both articles the blocks
of the coalescent are allowed to merge simultaneously. In [15] the convergence
of the generators has been proven and even a rate of convergence has been
determined. In this article the uniform convergence of the generators is going
to be proven as well, but with different techniques. In [9] the convergence of
the corresponding semigroups has been shown, which is equivalent to the
convergence of the generators on a core. We carry out the proof since parts
are used to verify Theorem 1.

######  Corollary 3 (dust case)

Suppose $\int_{[0,1]}u^{-1}\Lambda({\rm d}u)<\infty$. Then the time-
homogeneous Markov process $X^{(n)}:=(X_{t}^{(n)})_{t\geq
0}:=(N_{t}^{(n)}-\log n)_{t\geq 0}$ converges in $D_{{\mathbb{R}}}[0,\infty)$
as $n\to\infty$ to a limiting process $X=(X_{t})_{t\geq 0}$ with initial value
$X_{0}=0$ and semigroup $(T_{t})_{t\geq 0}$ given by

$\displaystyle
T_{t}f(x):={\mathbb{E}}(f(X_{s+t})|X_{s}=x)={\mathbb{E}}(f(x+X_{t})),\qquad
x\in{\mathbb{R}},f\in B({\mathbb{R}}),s,t\geq 0,$ (10)

where $X_{t}$ has characteristic function
${\mathbb{E}}(\exp(ixX_{t}))=\exp(t\psi(x)),$ $x\in{\mathbb{R}},t\geq 0,$ with

$\displaystyle\psi(x)=\int_{[0,1]}(e^{ix\log(1-u)}-1)u^{-2}\Lambda({\rm
d}u),\qquad x\in{\mathbb{R}}.$ (11)

Observe that $-X$ is a pure-jump subordinator with characteristic exponent
$x\mapsto\psi(-x),$ $x\in{\mathbb{R}}$.

Note that here $b=0$, $a=\int_{[0,1]}u^{-1}\Lambda({\rm d}u)$ and the
definitions (3) and (11) for $\psi$ coincide. Hence Theorem 1 and Corollary 3
describe the same limiting process.

For $\Lambda=\lambda$ Assumption A is satisfied with $b=1$. The block counting
process of the Bolthausen–Sznitman coalescent has been treated in [13] and
[14]. [13] and [14] have proven that the semigroup of
$(N_{t}^{(n)}/n^{e^{-t}})_{t\geq 0}$ converges on a dense subset of
$B([0,\infty))$ to the semigroup of a Feller process as $n$ tends to infinity,
hence the processes converge in $D_{[0,\infty)}[0,\infty)$. Taking logarithms
does not spoil the convergence. If $f\in\widehat{C}({\mathbb{R}})$ then
$g:=f\circ\log\in\widehat{C}([0,\infty))$, and the semigroup and hence the
generator $A^{(n)}$ of the logarithm of the scaled block counting process
$X^{(n)}=(X_{t}^{(n)})_{t\geq 0}=(\log N_{t}^{(n)}-e^{-t}\log n)_{t\geq 0}$
converge as well. We prove the convergence of $A^{(n)}$ in Section 5 directly.
Since the scaling depends on $t$, the process $X^{(n)}$ is time-inhomogeneous,
and [13] introduces the time-space process in order to transfer the question
of convergence to time-homogeneous Markov processes. The time-space process is
revisited in Section 5. Since $\lambda=\beta(1,1)$, the following result is
the particular case $b=1$ of Example 2.

######  Corollary 4 (Bolthausen–Sznitman case)

Suppose $\Lambda=\lambda$. Then the time-inhomogeneous Markov process
$X^{(n)}:=(X_{t}^{(n)})_{t\geq 0}:=(N_{t}^{(n)}-e^{-t}\log n)_{t\geq 0}$
converges in $D_{{\mathbb{R}}}[0,\infty)$ as $n\to\infty$ to the time-
homogeneous Markov process $X=(X_{t})_{t\geq 0}$ with initial value $X_{0}=0$
and semigroup $(T_{t})_{t\geq 0}$ given by

$\displaystyle
T_{t}f(x):={\mathbb{E}}(f(X_{s+t})|X_{s}=x)={\mathbb{E}}(f(e^{-t}x+X_{t})),\qquad
x\in{\mathbb{R}},f\in B({\mathbb{R}}),s,t\geq 0,$

where $X_{t}$ has characteristic function
$\phi_{t}(x):={\mathbb{E}}(\exp(ixX_{t}))=\Gamma(1+ix)/\Gamma(1+ie^{-t}x),$
$x\in{\mathbb{R}},t\geq 0$.

Note that here $b=1,$ $a=1+\Psi(1)$ and the underlying Lévy measure $\varrho$
has density $f$ with respect to the Lebesgue measure on
${\mathbb{R}}\setminus\\{0\\}$ given by $f(u):=e^{u}(1-e^{u})^{-2}$ for $u<0$
and $f(u):=0$ for $u>0$. Example 2 with $b=1$ states that
$\psi(x)=ix\Psi(1+ix),$ $x\in{\mathbb{R}},$ and that $X_{t}$ converges in
distribution as $t\to\infty$ to the unique stationary distribution $\mu$ of
$X$ with characteristic function $\phi(x)=\Gamma(1+ix),$ $x\in{\mathbb{R}}$.
Let $Z$ have an exponential distribution with parameter $1$. Then (see e.g.
[23, V, Example 9.15]) $\log Z$ is the negative of a Gumbel distributed random
variable and ${\mathbb{E}}(e^{ix\log Z})=\Gamma(1+ix),$ $x\in{\mathbb{R}}$.
Hence $-X_{t}$ converges in distribution as $t\to\infty$ to the Gumbel
distribution.

A convergence result for the fixation line can be stated analogously to
Theorem 1.

###### Theorem 5

Suppose that $\Lambda$ satisfies Assumption A. Then the possibly time-
inhomogeneous Markov process $Y^{(n)}:=(Y_{t}^{(n)})_{t\geq 0}:=(\log
L_{t}^{(n)}-e^{bt}\log n)_{t\geq 0}$ converges in $D_{{\mathbb{R}}}[0,\infty)$
as $n\to\infty$ to the time-homogeneous Markov process $Y=(Y_{t})_{t\geq 0}$
with initial value $Y_{0}=0$ and semigroup $(T_{t})_{t\geq 0}$ given by

$\displaystyle
T_{t}f(y):={\mathbb{E}}(f(Y_{s+t})|Y_{s}=y)={\mathbb{E}}(f(e^{bt}y+Y_{t})),\qquad
y\in{\mathbb{R}},f\in B({\mathbb{R}}),s,t\geq 0,$ (12)

where $Y_{t}$ has characteristic function $\chi_{t}$ given by

$\displaystyle\chi_{t}(y)=\exp\Big{(}\int_{0}^{t}\psi(-e^{bs}y){\rm
d}s\Big{)},\qquad y\in{\mathbb{R}},t\geq 0,$ (13)

and $\psi$ is given by (3).

  * Remark.

    1. 1.

The process defined by (12) and (13) is an Ornstein–Uhlenbeck type process
with underlying characteristic exponent $y\mapsto\psi(-y),$
$y\in{\mathbb{R}}.$ The semigroup defined by (12) belongs to the class of
generalized Mehler semigroups, since
$\chi_{t+s}(y)=\chi_{t}(e^{bs}y)\chi_{s}(y),$ $y\in{\mathbb{R}},$ for $s,t\geq
0$ [4].

    2. 2.

Let the random variable $S_{t}$ have characteristic function $\phi_{t}$, given
by (6), for $t\geq 0$. Conditional on $X_{s}=x$, $X_{t+s}$ is distributed as
$e^{-bt}x+S_{t}$ for all $x\in{\mathbb{R}}$. Note that
$Y_{t}\overset{\text{d}}{=}-e^{bt}X_{t}\overset{\text{d}}{=}-e^{bt}S_{t}$ and
that conditional on $Y_{s}=y$, $Y_{t+s}$ is distributed as
$e^{bt}y-e^{bt}S_{t}$ such that

$\displaystyle{\mathbb{P}}(e^{Y_{t+s}}\geq
x|e^{Y_{s}}=y)={\mathbb{P}}(y^{e^{bt}}e^{-e^{bt}S_{t}}\geq
x)={\mathbb{P}}(x^{e^{-bt}}e^{S_{t}}\leq y)={\mathbb{P}}(e^{X_{t+s}}\leq
y|e^{X_{s}}=x)$

for all $x,y,s,t\geq 0$, i.e., $e^{Y}$ is Siegmund-dual to $e^{X}$ (see [22]).

    3. 3.

The Bolthausen–Sznitman case $\Lambda=\lambda$ is stated in [13, Theorem 3.1
b)] in non-logarithmic form. The fixation line in the Bolthausen–Sznitman
coalescent is a continuous-time discrete state space branching process in
which the offspring distribution has probability generating function
$f(s)=s+(1-s)\log(1-s),$ $s\in[0,1].$ The limiting process described in
Theorem 5 is the logarithm of Neveu’s continuous-state branching process. By
Corollary 4, the characteristic functions $\chi_{t}$ of the marginal
distributions are given by (see [13, Eq. (19)])

$\displaystyle\chi_{t}(y)=\phi_{t}(-e^{t}y)=\Gamma(1-ie^{bt}y)/\Gamma(1-iy),\qquad
y\in{\mathbb{R}},t\geq 0.$

###  3 The limiting process

Standard computations (see [20, Lemma 17.1]) show that $\phi_{t}$, given by
(6), is the characteristic function of an infinitely divisible distribution
for each $t\geq 0$ without Gaussian component and Lévy measure $\varrho_{t}$
given by

$\displaystyle\varrho_{t}(A)=\int_{{\mathbb{R}}\setminus\\{0\\}}\int_{0}^{t}1_{A}(e^{-bs}u){\rm
d}s\varrho({\rm d}u),\qquad A\in\mathcal{B},t\geq 0.$

Sato and Yamazato [19, Theorem 3.1] provide a formula for the generator
corresponding to the semigroup $(T_{t})_{t\geq 0}$ given by (5).

###### Lemma 6

Suppose that $\Lambda$ satisfies Assumption A. (Let $\psi$ be given by (3),
$\phi_{t}$ be defined by (6) and let the random variable $X_{t}$ have
characteristic function $\phi_{t}$ for each for $t\geq 0$.) The family of
operators $(T_{t})_{t\geq 0}$ defined by (5) is a Feller semigroup. Let $D$
denote the space of twice differentiable functions
$f:{\mathbb{R}}\to{\mathbb{R}}$ such that
$f,f^{\prime},f^{\prime\prime}\in\widehat{C}({\mathbb{R}})$ and such that the
map $x\mapsto xf^{\prime}(x),$ $x\in{\mathbb{R}},$ belongs to
$\widehat{C}({\mathbb{R}})$. Then $D$ is a core for the generator $A$
corresponding to $(T_{t})_{t\geq 0}$ and

$\displaystyle
Af(x)=f^{\prime}(x)(a-bx)+\int_{[0,1]}(f(x+\log(1-u))-f(x)+uf^{\prime}(x))u^{-2}\Lambda({\rm
d}u)$ (14)

for $x\in{\mathbb{R}}$ and $f\in D$, where $a$ is given by (2).

  * Proof. Substituting $g:(0,1)\to{\mathbb{R}},$ $g(u):=\log(1-u),$ $u\in(0,1)$ shows that (14) is an integro-differential operator of the form (1.1) of Sato and Yamazato [19] with dimension $d=1$. In [19], operators of this form are initially considered as acting on the space $C_{c}^{2}$ of twice differentiable functions with compact support (see the explanations after Eq. (1.2) in [19]), but Step 3 of the proof of [19, Theorem 3.1] shows that (14) even holds for functions $f\in D$ ($\supset C_{c}^{2}$). Note that the space $D$ is denoted by $F_{1}$ in [19]. The fact that $D$ is a core for $A$ is only a different phrasing of the claim in Step 5 of the proof of [19, Theorem 3.1]. $\Box$

  * Remark. The limiting process in Theorem 1 arises as the solution of a certain stochastic differential equation. Let the Lévy process $L=(L_{t})_{t\geq 0}$ with characteristic functions ${\mathbb{E}}(e^{ixL_{t}})=e^{t\psi(x)},$ $x\in{\mathbb{R}},t\geq 0,$ be adapted to the filtration $(\mathcal{F}_{t})_{t\geq 0}$ which satisfies the usual hypotheses such that $L_{t+s}-L_{s}$ is independent of $\mathcal{F}_{s}$ for all $s,t\geq 0$. In this remark $\psi$ is allowed to be the characteristic exponent of an arbitrary infinitely divisible distribution on ${\mathbb{R}}$ and $b>0$ is fixed. The Langevin equation with Lévy noise instead of a Brownian motion

$\displaystyle{\rm d}X_{t}=-bX_{t}{\rm d}t+{\rm d}L_{t},\qquad t\geq 0,$ (15)

with initial value $X_{0}=0$ has an unique $(\mathcal{F}_{t})_{t\geq
0}$-adapted solution $X=(X_{t})_{t\geq 0}$ with càdlàg paths. The solution of
(15) or the corresponding semigroup are hence sometimes called
Ornstein–Uhlenbeck type or generalized Ornstein–Uhlenbeck process or
semigroup. It holds that

$\displaystyle X_{t}=\int_{0}^{t}e^{-b(t-s)}{\rm d}L_{s},\qquad t\geq 0.$ (16)

Various constructions for the integral (16) are possible. In Applebaum [1,
Sections 6.3 and 6.2] the stochastic integral is the Itô-integral with respect
to semimartingales. Wolfe [25] constructed the integral as a random Bochner
integral, which exists in the sense of convergence in probability. Jurek and
Vervaat [12] constructed the stochastic integral as a pathwise Laplace-
Stieltjes integral (and using integration by parts). The process $X$ is a
stochastically continuous Markov process and the corresponding semigroup is
given by (5), where the characteristic functions of $X_{t}$ are given by (6)
with underlying infinitely divisible characteristic exponent $\psi$ for $t\geq
0$.

Suppose that $b>0$ and that the Lévy measure $\varrho$ of the characteristic
exponent $\psi$ satisfies

$\displaystyle\int_{\\{|x|>1\\}}\log(1+|u|)\varrho({\rm d}u)<\infty.$ (17)

According to [19, Theorems 4.1 and 4.2], $X_{t}$ converges in distribution as
$t\to\infty$ to the unique stationary distribution $\mu$ of $X$. The
distribution $\mu$ is self-decomposable and conversely every self-decomposable
distribution can be obtained as the stationary distribution of an
Ornstein–Uhlenbeck type process. If (17) does not hold, then there exists no
stationary distribution. The following Lemma is an application to our setting.

###### Lemma 7

Suppose that $\Lambda$ satisfies Assumption A. Let $X=(X_{t})_{t\geq 0}$ be as
in Theorem 1. If further $\int_{(\varepsilon,1)}\log\log(1-u)^{-1}\Lambda({\rm
d}u)<\infty$ for some $1-e^{-1}<\varepsilon<1$ then $X_{t}$ converges in
distribution as $t\to\infty$ to the unique stationary distribution $\mu$ of
$X$. The distribution $\mu$ is self-decomposable with characteristic function
$\phi$ given by

$\displaystyle\phi(x)=\exp\Big{(}\int_{0}^{\infty}\psi(e^{-bs}x){\rm
d}s\Big{)},\qquad x\in{\mathbb{R}}.$ (18)

The characteristic function $\phi_{t}$ of $X_{t}$ satisfies
$\phi_{t}(x)=\phi(x)/\phi(e^{-bt}x),$ $x\in{\mathbb{R}}$.

If $\int_{(\varepsilon,1)}\log\log(1-u)^{-1}(u^{-2})\Lambda({\rm d}u)=\infty$
for $0<\varepsilon<1$, then, for every $l$,

$\displaystyle\lim_{t\to\infty}\sup_{x\in{\mathbb{R}}}\sup_{y\in{\mathbb{R}}}{\mathbb{P}}(|e^{-bt}x+X_{t}-y|\leq
l)=0.$ (19)

The process has no stationary distribution.

###  4 Proof of Corollary 3

In this section $\Lambda$ satisfies the dust condition
$\int_{[0,1]}u^{-1}\Lambda({\rm d}u)<\infty$. Let
$E_{n}:=\\{x\in{\mathbb{R}}|e^{x}n\in[n]\\}$ denote the state space of
$X^{(n)}=(X_{t}^{(n)})_{t\geq 0}=(\log N_{t}^{(n)}-\log n)_{t\geq 0}$ for each
$n\in{\mathbb{N}}$. Define $k:=k(x,n):=e^{x}n\in[n]$ for $x\in E_{n}$ and
$n\in{\mathbb{N}}$ such that the generator $A^{(n)}$ of $X^{(n)}$ can be
represented as

$\displaystyle
A^{(n)}f(x)=\sum_{j=1}^{k-1}(f(x+\log\tfrac{j}{k})-f(x))q_{k,j},\qquad x\in
E_{n},f\in\widehat{C}({\mathbb{R}}),n\in{\mathbb{N}}.$ (20)

The process $X=(X_{t})_{t\geq 0}$ defined by (10) and (11) is a Feller process
in $\widehat{C}({\mathbb{R}})$. Let $A$ denote the generator. From [20,
Theorem 31.5] it follows that the space $\widehat{C}_{2}({\mathbb{R}})$ of
twice differentiable functions $f\in C_{2}({\mathbb{R}})$ with
$f,f^{\prime},f^{\prime\prime}\in\widehat{C}({\mathbb{R}})$ is a core for $A$
and

$\displaystyle Af(x)=\int_{[0,1]}(f(x+\log(1-u))-f(x))u^{-2}\Lambda({\rm
d}u),\qquad x\in{\mathbb{R}},f\in\widehat{C}_{2}({\mathbb{R}}).$ (21)

The idea to prove the uniform convergence of the generators is the following:
write the jump rates as values of a distribution depending on $k$ (with some
minor rectifications) whose limiting behavior as $k\to\infty$ can be
determined. The generator can then be written as the mean of a random variable
and classical weak convergence results can be applied.

  * Proof. [of Corollary 3] Let $f\in\widehat{C}_{2}({\mathbb{R}})$. Define $h:[0,1]\times{\mathbb{R}}\to{\mathbb{R}}$ via $h(u,x):=u^{-1}(f(x+\log(1-u))-f(x)),$ $u\in(0,1),$ $h(0,x):=\lim_{u\searrow 0}h(u,x)=-f^{\prime}(x)$ and $h(1,x):=\lim_{u\nearrow 1}h(u,x)=-f(x)$ for $x\in{\mathbb{R}}$. Differentiating $s\mapsto f(x+\log(1-us)),$ $s\in(0,1),$ leads to

$\displaystyle
f(x+\log(1-u))-f(x)=-u\int_{0}^{1}\frac{f^{\prime}(x+\log(1-us))}{1-us}{\rm
d}s,\qquad u\in[0,1),x\in{\mathbb{R}},$

such that

$\displaystyle h(u,x)=-\int_{0}^{1}\frac{f^{\prime}(x+\log(1-us))}{1-us}{\rm
d}s,\qquad u\in[0,1),x\in{\mathbb{R}},$

and $h$ stays bounded even as $u$ tends to $0$. Define

$\displaystyle
S(k,x):=\sum_{j=1}^{k-1}(f(x+\log\tfrac{j}{k})-f(x))q_{k,j},\quad
I(x):=\int_{[0,1]}h(u,x)u^{-1}\Lambda({\rm d}u),\qquad
k\in{\mathbb{N}},x\in{\mathbb{R}},$ (22)

such that $A^{(n)}f(x)=S(k,x)$ for $x\in E_{n}$ and $n\in{\mathbb{N}}$ and
$I(x)=Af(x)$ for $x\in{\mathbb{R}}$. Substituting $k-j$ for $j$ and the
definition of $h$ yield

$\displaystyle S(k,x)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{k-1}(f(x+\log(1-\tfrac{j}{k}))-f(x))q_{k,k-j}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{k-1}h(\tfrac{j}{k},x)\frac{j}{k}\binom{k}{j+1}\int_{[0,1]}u^{j-1}(1-u)^{k-j-1}\Lambda({\rm
d}u)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{k-1}h(\tfrac{j}{k},x)\frac{j}{j+1}\binom{k-1}{j}\int_{[0,1]}u^{j-1}(1-u)^{k-j-1}\Lambda({\rm
d}u).$

Set $c:=\int_{[0,1]}u^{-1}\Lambda({\rm d}u)>0$ and define the probability
measure $\mathrm{Q}$ on $([0,1],\mathcal{B}\cap[0,1])$ via
$\mathrm{Q}(A):=c^{-1}\int_{A}u^{-1}\Lambda({\rm d}u),$
$A\in\mathcal{B}\cap[0,1].$ Let the random variables $Z_{k},$
$k\in{\mathbb{N}},$ have distribution given by

$\displaystyle{\mathbb{P}}(Z_{k}=j)=\binom{k-1}{j}\int_{[0,1]}u^{j}(1-u)^{k-1-j}\mathrm{Q}({\rm
d}u),\qquad j\in\\{0,\ldots,k-1\\},$

i.e., $Z_{k}$ has a mixed binomial distribution with sample size $k-1$ and
random success rate $\mathrm{Q}$. Let the random variable $Z$ have
distribution $\mathrm{Q}$. Then

$\displaystyle S(k,x)=c{\mathbb{E}}((1-(Z_{k}+1)^{-1})h(Z_{k}/k,x)),\qquad
k\in{\mathbb{N}},x\in{\mathbb{R}},$

and $I(x)=c{\mathbb{E}}(h(Z,x)),$ $x\in{\mathbb{R}}.$ It is straightforward to
check that $Z_{k}/k\to Z$ in distribution as $k\to\infty$, e.g., by verifying
the convergence of the cumulative distribution functions (cdf) on the set of
continuity points of the cdf of $Z$. In particular,
$\lim_{k\to\infty}{\mathbb{P}}(Z_{k}\leq C)=Q(0)=0$ for every $C>0$ such that
$\lim_{k\to\infty}{\mathbb{E}}((Z_{k}+1)^{-1})=0$. Since $h$ is bounded and
$f,f^{\prime}\in\widehat{C}({\mathbb{R}})$ are uniformly continuous, the
family of functions $\\{h(\cdot,x)|x\in{\mathbb{R}}\\}$ is equicontinuous on
$[\delta,1-\delta]$ for every $0<\delta<1/2$ and uniformly bounded on $[0,1]$.
From Lemma 9 it follows that
${\mathbb{E}}(h(Z_{k}/k,x))\to{\mathbb{E}}(h(Z,x))$ uniformly in
$x\in{\mathbb{R}}$ as $k\to\infty$, thus

$\displaystyle\lim_{k\to\infty}\sup_{x\in{\mathbb{R}}}|S(k,x)-I(x)|=0.$ (23)

From $\lim_{x\to-\infty}h(Z,x)=0$ a.s., the fact that $h$ is bounded and the
dominated convergence theorem it follows that

$\displaystyle\lim_{x\to-\infty}|I(x)|=c\lim_{x\to-\infty}|{\mathbb{E}}(h(Z,x))|=0.$
(24)

Since $f\in\widehat{C}({\mathbb{R}})$, $\lim_{x\to-\infty}S(k,x)=0$ for any
$k\in{\mathbb{N}}$. Due to (23) and (24),

$\displaystyle\lim_{x\to-\infty}\sup_{k\in{\mathbb{N}}}|S(k,x)|=0.$ (25)

As $n\to\infty$, $k=k(x,n)=e^{x}n\to\infty$ or $x\to-\infty$. For example, for
$n\in{\mathbb{N}}$ and $x\in E_{n}$, either $k\geq n^{1/2}$ or
$x<-\tfrac{1}{2}\log n$. Distinguishing the two cases leads to

$\displaystyle\lim_{n\to\infty}\sup_{x\in E_{n}}|A^{(n)}f(x)-Af(x)|$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leq\lim_{k\to\infty}\sup_{x\in{\mathbb{R}}}|S(k,x)-I(x)|+\lim_{x\to-\infty}\sup_{k\in{\mathbb{N}}}|S(k,x)|+\lim_{x\to-\infty}|I(x)|=0.$
(26)

By [8, I, Theorem 6.1 and IV, Theorem 2.5], $X^{(n)}\to X$ in
$D_{{\mathbb{R}}}[0,\infty)$ as $n\to\infty$. $\Box$

  * Remark.

    1. 1.

The generator $A^{(n)}$ converges even if $\Lambda(\\{1\\})>0$. In this case
the atom at $1$ can be split off from $\Lambda$ such that
$q_{k,j}=\binom{k}{j-1}\int_{[0,1)}u^{k-j-1}(1-u)^{j-1}\Lambda|_{[0,1)}({\rm
d}u)+\Lambda(\\{1\\})1_{\\{1\\}}(j),$ $j\in\\{1,\ldots,k-1\\},k\geq 2,$ where
the first summand are the jump rates of the block counting process
corresponding to the restriction $\Lambda|_{[0,1)}$ of $\Lambda$ to $[0,1)$,
i.e., a measure with no atom at $1$. Thus,

$\displaystyle A^{(n)}f(x)=S(k,x)+(f(\log n^{-1})-f(x))\Lambda(\\{1\\}),\qquad
x\in E_{n},f\in\widehat{C}({\mathbb{R}}),n\in{\mathbb{N}},$

where the jump rates in $S(k,x)$ correspond to $\Lambda|_{[0,1)}$, and

$\displaystyle
Af(x)=I(x)+h(1,x)\Lambda(\\{1\\})=I(x)-f(x)\Lambda(\\{1\\}),\qquad
x\in(-\infty,0],$

where $I(x)=\int h(u,x)\Lambda|_{[0,1)}({\rm d}u),$ $x\in{\mathbb{R}}$. The
additional term corresponds to the killing of the subordinator $-X$ at the
rate $\Lambda(\\{1\\})$. Since $f\in\widehat{C}({\mathbb{R}})$,
$\lim_{n\to\infty}\sup_{x\in E_{n}}|(f(\log
n^{-1})-f(x))\Lambda(\\{1\\})+f(x)\Lambda(\\{1\\})|=\Lambda(\\{1\\})\lim_{n\to\infty}|f(\log
n^{-1})|=0$, i.e., the additional term converges, and again (26) holds true.

    2. 2.

The approach to the convergence of the generators is related to Bernstein
polynomials. The $(k-1)$-th Bernstein polynomial

$\displaystyle\sum_{j=0}^{k-1}h(\tfrac{j}{k-1},x)\binom{k-1}{j}u^{j}(1-u)^{k-1-j}$

of $h(\cdot,x)$ converges uniformly in $u\in[0,1]$ to $h(u,x)$ as
$k\to\infty$, if $x\in{\mathbb{R}}$ is fixed.

###  5 Proofs concerning the Bolthausen–Sznitman coalescent

In this section $\Lambda=\lambda$ is the Lebesgue measure on $[0,1]$. Define
$\alpha:=\alpha(t):=e^{-t},$ $t\geq 0$. The process
$X^{(n)}=(X_{t}^{(n)})_{t\geq 0}=(\log N_{t}^{(n)}-\alpha\log n)_{t\geq 0}$ is
a time-inhomogeneous Markov process. In order to prove convergence in
$D_{{\mathbb{R}}}[0,\infty)$ to $X$ we want to show the uniform convergence of
the generators. Typical convergence results are stated for time-homogeneous
Markov processes and in order to use these we are going to introduce the time-
space process.

####  5.1 Time-space process: semigroup and generator

Define the time-space processes $\widetilde{X}^{(n)}:=(t,X_{t}^{(n)})_{t\geq
0},$ $n\in{\mathbb{N}}$, and $\widetilde{X}:=(t,X_{t})_{t\geq 0}$. It has been
proven in [6] that $\widetilde{X}^{(n)}$ and $\widetilde{X}$ are time-
homogeneous Markov processes (and exist on a new probability space). In the
following the tilde symbol indicates the time-space setting. Let
$\widetilde{E}_{n}:=\\{(s,x)\in[0,\infty)\times{\mathbb{R}}|e^{x}n^{\alpha(s)}\in[n]\\}$
denote the state space of $\widetilde{X}^{(n)}$,
$\widetilde{E}:=[0,\infty)\times{\mathbb{R}}$ denote the state space of
$\widetilde{X}$ and define $k:=k(s,x,n):=e^{x}n^{\alpha(s)}\in{\mathbb{N}}$
for $(s,x)\in\widetilde{E}_{n}$ and $n\in{\mathbb{N}}$. Given $f\in
B(\widetilde{E})$ and $s\geq 0$, denote the function $x\mapsto f(s,x),$
$x\in{\mathbb{R}}$, by $\pi f(s,x)$. The limiting process $X$ already is time-
homogeneous. Recall that $D$, the space of twice differentiable functions
$f:{\mathbb{R}}\to{\mathbb{R}}$ such that
$f,f^{\prime},f^{\prime\prime}\in\widehat{C}({\mathbb{R}})$ and such that the
map $x\mapsto xf^{\prime}(x),$ $x\in{\mathbb{R}},$ belongs to
$\widehat{C}({\mathbb{R}})$, is a core for the generator $A$ of the semigroup
$(T_{t})_{t\geq 0}$ corresponding to $X$. The semigroup
$(\widetilde{T}_{t})_{t\geq 0}$ of $\widetilde{X}$, given by

$\displaystyle\widetilde{T}_{t}f(s,x):={\mathbb{E}}(f(s+t,X_{s+t})|X_{s}=x)={\mathbb{E}}(f(s+t,\alpha(t)x+X_{t})),\quad(s,x)\in\widetilde{E},f\in
B(\widetilde{E}),t\geq 0,$

is a Feller semigroup. Let $\widetilde{D}$ denote the space of functions
$f\in\widehat{C}(\widetilde{E})$ of the form
$f(s,x)=\sum_{i=1}^{l}g_{i}(s)h_{i}(x)$ with $l\in{\mathbb{N}},h_{i}\in D$ and
$g_{i}\in C_{1}([0,\infty))$ such that
$g_{i},g_{i}^{\prime}\in\widehat{C}([0,\infty))$ for $i=1,\ldots,l$.
Proposition 10 states that $\widetilde{D}$ is a core for the generator
$\widetilde{A}$ of $(\widetilde{T}_{t})_{t\geq 0}$ and

$\displaystyle\widetilde{A}f(s,x)=\frac{\partial}{\partial s}f(s,x)+A\pi
f(s,x),\qquad(s,x)\in\widetilde{E},f\in\widetilde{D}.$ (27)

The ’semigroup’ $(T_{s,t}^{(n)})_{s,t\geq 0}$ of $X^{(n)}$ is given by

$\displaystyle T_{s,t}^{(n)}f(x)$ $\displaystyle:=$
$\displaystyle{\mathbb{E}}(f(X_{s+t}^{(n)})|X_{s}^{(n)}=x)={\mathbb{E}}(f(\log
N_{s+t}^{(n)}-\alpha(s+t)\log n|N_{s}^{(n)}=k)$ $\displaystyle=$
$\displaystyle{\mathbb{E}}(f(\log N_{t}^{(k)}-\alpha(s+t)\log
n)),\qquad(s,x)\in\widetilde{E}_{n},f\in B({\mathbb{R}}),t\geq 0.$

The ’generator’ $(A_{s}^{(n)})_{s\geq 0}$ of $(T_{s,t}^{(n)})_{s,t\geq 0}$ is
given by

$\displaystyle A_{s}^{(n)}f(x)$ $\displaystyle:=$ $\displaystyle\lim_{t\to
0}t^{-1}(T_{s,t}^{(n)}f(x)-f(x))$ (28) $\displaystyle=$
$\displaystyle\lim_{t\to 0}t^{-1}({\mathbb{E}}(f(\log
N_{t}^{(k)}-\alpha(s+t)\log n))-f(x))$ $\displaystyle=$
$\displaystyle-f^{\prime}(x)\alpha^{\prime}(s)\log
n+\sum_{j=1}^{k-1}(f(x+\log\tfrac{j}{k})-f(x))q_{k,j},\quad(s,x)\in\widetilde{E}_{n}.$

Here $f\in C_{1}({\mathbb{R}})$ such that
$f,f^{\prime}\in\widehat{C}({\mathbb{R}})$. The semigroup
$(\widetilde{T}_{t}^{(n)})_{t\geq 0}$ of $\widetilde{X}^{(n)}$, given by

$\displaystyle\widetilde{T}_{t}^{(n)}(s,x)$ $\displaystyle:=$
$\displaystyle{\mathbb{E}}(f(s+t,X_{s+t})|X_{s}^{(n)}=x)$ $\displaystyle=$
$\displaystyle{\mathbb{E}}(f(s+t,\log N_{t}^{(k)}-\alpha(s+t)\log
n)),\qquad(s,x)\in\widetilde{E}_{n},f\in B(\widetilde{E}_{n}),t\geq
0,n\in{\mathbb{N}},$

is a Feller semigroup on $\widehat{C}(\widetilde{E}_{n})$ for every
$n\in{\mathbb{N}}$. On $\widetilde{D}$, or more precisely, for the restriction
of $f\in\widetilde{D}$ to $\widetilde{E}_{n}$ the generator
$\widetilde{A}^{(n)}$ of $\widetilde{T}^{(n)}$ is given by

$\displaystyle\widetilde{A}^{(n)}f(s,x)=\frac{\partial}{\partial
s}f(s,x)+A_{s}^{(n)}\pi
f(s,x),\qquad(s,x)\in\widetilde{E}_{n},n\in{\mathbb{N}}.$ (29)

####  5.2 Proof of Corollary 4

  * Proof. [of Corollary 4] Let $f\in D$. The approach to the proof is the same as in Section 4, but the function $u\mapsto f(x+\log(1-u)),$ $u\in[0,1]$, demands second order approximation like in the integral part of the limiting generator (14). Define $h:[0,1]\times{\mathbb{R}}\to{\mathbb{R}}$ via $h(u,x):=u^{-2}(f(x+\log(1-u))-f(x)+uf^{\prime}(x)),$ $u\in(0,1),$ $h(0,x):=\lim_{u\searrow 0}h(u,x)=2^{-1}(f^{\prime\prime}(x)-f^{\prime}(x))$ and, since $f\in\widehat{C}({\mathbb{R}})$, $h(1,x):=\lim_{u\nearrow 1}h(u,x)=f^{\prime}(x)-f(x)$ for $x\in{\mathbb{R}}$. Taylor’s theorem applied to $u\mapsto f(x+\log(1-u)),$ $u<1$, with evaluation point $u=0$ and exact integral remainder yields

$\displaystyle h(u,x)$ $\displaystyle=$ $\displaystyle
u^{-2}\int_{0}^{u}\frac{u-s}{(1-s)^{2}}(f^{\prime\prime}(x+\log(1-s))-f^{\prime}(x+\log(1-s))){\rm
d}s$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{1-s}{(1-us)^{2}}(f^{\prime\prime}(x+\log(1-us))-f^{\prime}(x+\log(1-us))){\rm
d}s,\quad u\in[0,1),x\in{\mathbb{R}}.$

The latter formula of $h(u,x)$ shows that $h$ is bounded even as $u$ tends to
zero. Putting $k=k(s,x,n)=e^{x}n^{\alpha(s)}$ in (28) yields

$\displaystyle A_{s}^{(n)}f(x)$ $\displaystyle=$ $\displaystyle
f^{\prime}(x)R(k,x)+S(k,x),\qquad(s,x)\in\widetilde{E}_{n},n\in{\mathbb{N}},$

where

$\displaystyle R(k,x):=\log k-\sum_{j=1}^{k-1}\tfrac{k-j}{k}q_{k,j}-x,\qquad
k\in{\mathbb{N}},x\in{\mathbb{R}},$ (30)

and

$\displaystyle
S(k,x):=\sum_{j=1}^{k-1}(f(x+\log\tfrac{j}{k})-f(x)+\tfrac{k-j}{k}f^{\prime}(x))q_{k,j},\qquad
k\in{\mathbb{N}},x\in{\mathbb{R}}.$ (31)

Further define $I(x):=\int_{[0,1]}h(u,x)\Lambda({\rm d}u),$
$x\in{\mathbb{R}}$. By Eq. (7) with $a=b=1$,
$\tfrac{k-j}{k}q_{k,j}=(k-j+1)^{-1},$ $j\in\\{1,\ldots,k-1\\},k\geq 2,$ such
that $\sum_{j=1}^{k-1}\tfrac{k-j}{k}q_{k,j}=\sum_{j=2}^{k}j^{-1}$ for $k\geq
2$. As $n\to\infty$, $k=k(s,x,n)=e^{x}n^{\alpha(s)}\to\infty$ or
$x\to-\infty$. Fix $T>0$. E.g., if $s\in[0,T]$ then either $k\geq
n^{\alpha(T+\delta)}$ or $x<-\alpha(T)(1-\alpha(\delta))\log n$, where
$\delta>0$ is a constant. The well-known asymptotics of the harmonic numbers
states that $\sup_{x\in{\mathbb{R}}}|R(k,x)-(1+\Psi(1)-x)|=|\log
k-\sum_{j=1}^{k}j^{-1}-\Psi(1)|\rightarrow 0$ as $k\to\infty$. Clearly,
$\lim_{x\to-\infty}|f^{\prime}(x)|=0$. Dividing the state space as above
therefore implies

$\displaystyle\lim\limits_{n\to\infty}\sup_{(s,x)\in\widetilde{E}_{n},s\in[0,T]}|f^{\prime}(x)||R(k,x)-(1+\Psi(1)-x)|=0.$
(32)

Substituting $k-j-1$ for $j$ in (31) yields

$\displaystyle S(k,x)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{k-2}(f(x+\log(1-\tfrac{j+1}{k}))-f(x)+\tfrac{j+1}{k}f^{\prime}(x))q_{k,k-j-1}$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{k-2}h(\tfrac{j+1}{k},x)\frac{(j+1)^{2}}{k^{2}}\binom{k}{j+2}\int_{[0,1]}u^{j}(1-u)^{k-2-j}\Lambda({\rm
d}u)$ $\displaystyle=$
$\displaystyle\frac{k-1}{k}\sum_{j=0}^{k-2}h(\tfrac{j+1}{k},x)\frac{j+1}{j+2}\binom{k-2}{j}\int_{[0,1]}u^{j}(1-u)^{k-2-j}\Lambda({\rm
d}u),\qquad k\in{\mathbb{N}},x\in{\mathbb{R}}.$

Set $c:=\Lambda([0,1])>0$ and define the probability measure $\mathrm{Q}$ on
$([0,1],\mathcal{B}\cap[0,1])$ as $Q:=c^{-1}\Lambda$. Let the random variables
$Z_{k},$ $k\in{\mathbb{N}},$ have distribution given by

$\displaystyle{\mathbb{P}}(Z_{k}=j)=\binom{k-2}{j}\int_{[0,1]}u^{j}(1-u)^{k-2-j}\mathrm{Q}({\rm
d}u),\qquad j\in\\{0,\ldots,k-2\\},$

i.e., $Z_{k}$ has a mixed binomial distribution with sample size $k-2$ and
random success rate $\mathrm{Q}$. Let $Z$ have distribution $\mathrm{Q}$. Then

$\displaystyle
S(k,x)=c(1-k^{-1}){\mathbb{E}}((1-(Z_{k}+2)^{-1})h((Z_{k}+1)/k,x)),\qquad
k\in{\mathbb{N}},x\in{\mathbb{R}},$

and $I(x)=c{\mathbb{E}}(h(Z,x)),$ $x\in{\mathbb{R}}$. It is easy to check that
$(Z_{k}+1)/k\rightarrow Z$ in distribution as $k\to\infty$. The family of
functions $\\{h(\cdot,x)|x\in{\mathbb{R}}\\}$ is equicontinuous on
$[\delta,1-\delta]$ for every $0<\delta<1/2$ and uniformly bounded on $[0,1]$.
Due to $Q(\\{0\\})=c^{-1}\Lambda(\\{0\\})=0$, $Z_{k}\to\infty$ a.s. as
$k\to\infty$ and thus $\lim_{k\to\infty}{\mathbb{E}}(1/(Z_{k}+2))=0$ such that
the additional factor $1-(Z_{k}+2)^{-1}$ in the mean above can be omitted when
considering the limit of $S(k,x)$ as $k\to\infty$. From Lemma 9 it follows
that

$\displaystyle\lim_{k\to\infty}\sup_{x\in{\mathbb{R}}}|S(k,x)-I(x)|=0.$ (33)

From $\lim_{x\to-\infty}h(Z,x)=0$ a.s., the fact that the functions
$h(\cdot,x),$ $x\in{\mathbb{R}},$ are uniformly bounded and the dominated
convergence theorem it follows that

$\displaystyle\lim_{x\to-\infty}|I(x)|=c\lim_{x\to-\infty}|{\mathbb{E}}(h(Z,x))|=0.$
(34)

Since $f,f^{\prime}\in\widehat{C}({\mathbb{R}})$, $\lim_{x\to-\infty}S(k,x)=0$
for any $k\in{\mathbb{N}}$ and, in view of (33) and (34),

$\displaystyle\lim_{x\to-\infty}\sup_{k\in{\mathbb{N}}}|S(k,x)|=0.$ (35)

As seen in the proof of Corollary 3, Eqs. (33)-(35) imply

$\displaystyle\lim_{n\to\infty}\sup_{(s,x)\in\widetilde{E}_{n},s\in[0,T]}|S(k,x)-I(x)|=0.$
(36)

By (32),
$\lim_{n\to\infty}\sup_{(s,x)\in\widetilde{E}_{n},s\in[0,T]}|A_{s}^{(n)}f(x)-Af(x)|=0.$
Due to (27) and (29),
$\lim_{n\to\infty}\sup_{(s,x)\in\widetilde{E}_{n},s\in[0,T]}|\widetilde{A}^{(n)}f(s,x)-\widetilde{A}f(s,x)|=0$
for every function $f$ belonging to the core $\widetilde{D}$ and each $T>0$.
From [8, IV, Corollary 8.7] it follows that
$\widetilde{X}^{(n)}\to\widetilde{X}$ in $D_{\widetilde{E}}[0,\infty)$, hence
$X^{(n)}\to X$ in $D_{{\mathbb{R}}}[0,\infty)$ as $n\to\infty$. $\Box$

  * Remark.

    1. 1.

Note that if $\Lambda=\lambda$ then $Z_{k}$ has a discrete uniform
distribution on $\\{0,\ldots,k-2\\}$ and $Z$ has a continuous uniform
distribution on $(0,1)$.

    2. 2.

Put
$\gamma(k):=\sum_{j=1}^{k-1}\tfrac{k-j}{k}q_{k,j}=\sum_{j=2}^{k}(j-1)\binom{k}{j}\lambda_{k,j}$
for $k\geq 2$. Among dust-free $\Lambda$-coalescents that stay infinite the
proof works for the Bolthausen–Sznitman coalescent, because the precise
asymptotics of $\gamma(k)/k=\log k-1$ as $k\to\infty$ is known. Observe that
in the proof of Corollary 4 the fact that $\Lambda=\lambda$ is only used to
verify (32).

###  6 Proof of Theorem 1

In this section $\Lambda$ satisfies Assumption A. We continue to use the time-
space setting and the notation of Subsection 5.1 with $\alpha$ replaced by
$\alpha:=\alpha(t):=e^{-bt},$ $t\geq 0$. Define
$\Lambda_{D}:=\Lambda-b\lambda$ and let $\Lambda_{D}^{+},\Lambda_{D}^{-}$
denote the nonnegative measures constituting the Jordan decomposition
$\Lambda_{D}=\Lambda_{D}^{+}-\Lambda_{D}^{-}$ of $\Lambda_{D}$. The
decomposition of $\Lambda$ into a ’Bolthausen–Sznitman part’ $b\lambda$ and a
’dust part’ $\Lambda_{D}$ is transferred to the jump rates and the generator.
Proving Theorem 1 now only requires to suitable arrange equations already
obtained in Sections 4 and 5. To be precise, the results of Section 4 are
applied to the summands $\Lambda_{D}^{\pm}$ of $\Lambda_{D}$, but we omit this
detail in the following.

  * Proof. [of Theorem 1] Let $q_{k,j}^{\lambda},q_{k,j}^{D,+}$ and $q_{k,j}^{D,-}$ denote the rates of the block counting process corresponding to $\lambda,\Lambda_{D}^{+}$ and $\Lambda_{D}^{-}$, respectively, and define $q_{k,j}^{D}:=q_{k,j}^{D,+}-q_{k,j}^{D,-}$ for $j\in\\{1,\ldots,k\\}$ and $k\in{\mathbb{N}}$. Obviously, $q_{k,j}=bq_{k,j}^{\lambda}+q_{k,j}^{D}$. Recall that $k=k(s,x,n)=e^{x}n^{\alpha(s)}\in{\mathbb{N}}$ for $(s,x)\in\widetilde{E}_{n}$ and $n\in{\mathbb{N}}$. From (28) it follows that the ’generator’ $A_{s}^{(n)}$ of $X^{(n)}=(X_{t}^{(n)})_{t\geq 0}=(\log N_{t}^{(n)}-\alpha(t)\log n)_{t\geq 0}$ is given by

$\displaystyle
A_{s}^{(n)}f(x)=bR(k,x)f^{\prime}(x)+bS_{BS}(k,x)+S_{D}(k,x),\qquad(s,x)\in\widetilde{E}_{n},n\in{\mathbb{N}},$

where

$\displaystyle R(k,x)$ $\displaystyle:=$ $\displaystyle\log
k-\sum_{j=1}^{k-1}\tfrac{k-j}{k}q_{k,j}^{\lambda}-x,\qquad
k\in{\mathbb{N}},x\in{\mathbb{R}},$ $\displaystyle S_{BS}(k,x)$
$\displaystyle:=$
$\displaystyle\sum_{j=1}^{k-1}(f(x+\log\tfrac{j}{k})-f(x)+\tfrac{k-j}{k}f^{\prime}(x))q_{k,j}^{\lambda},\qquad
k\in{\mathbb{N}},x\in{\mathbb{R}},$ $\displaystyle S_{D}(k,x)$
$\displaystyle:=$
$\displaystyle\sum_{j=1}^{k-1}(f(x+\log\tfrac{j}{k})-f(x))q_{k,j}^{D},\qquad
k\in{\mathbb{N}},x\in{\mathbb{R}},$

are defined as in (30), (31) and (22), and $f\in C_{1}({\mathbb{R}})$ such
that $f,f^{\prime}\in\widehat{C}({\mathbb{R}})$. By Lemma 6 and (2), the
generator $A$ of $X=(X_{t})_{t\geq 0}$ can be written as

$\displaystyle Af(x)$ $\displaystyle=$ $\displaystyle
b(1+\Psi(1)-x)f^{\prime}(x)+b\int_{[0,1]}\frac{f(x+\log(1-u))-f(x)+uf^{\prime}(x)}{u^{2}}\lambda({\rm
d}u)$
$\displaystyle+\int_{[0,1]}\frac{f(x+\log(1-u))-f(x)}{u^{2}}\Lambda_{D}({\rm
d}u),\qquad x\in{\mathbb{R}},f\in D.$

From Eqs. (32), (36) and (23)-(25) it follows that
$\lim_{n\to\infty}\sup_{(s,x)\in\widetilde{E}_{n},s\in[0,T]}|A_{s}^{(n)}f(x)-Af(x)|=0$
for $f\in D$. Due to (27) and (29),

$\displaystyle\lim_{n\to\infty}\sup_{(s,x)\in\widetilde{E}_{n},s\in[0,T]}|\widetilde{A}^{(n)}f(s,x)-\widetilde{A}f(s,x)|=0$
(37)

for every $f\in\widetilde{D}$ and $T>0$. By Proposition 10, the space
$\widetilde{D}$ is a core for $\widetilde{A}$. Thus, it follows from [8, IV,
Corollary 8.7] that $\widetilde{X}^{(n)}\to\widetilde{X}$ in
$D_{\widetilde{E}}[0,\infty)$, hence $X^{(n)}\to X$ in
$D_{{\mathbb{R}}}[0,\infty)$ as $n\to\infty$. $\Box$

###  7 Proof of Theorem 5

In this section $\Lambda$ satisfies Assumption A. The process
$Y^{(n)}=(Y_{t}^{(n)})_{t\geq 0}=(\log L_{t}^{(n)}-e^{bt}\log n)_{t\geq 0}$ is
a possibly time-inhomogeneous Markov process, hence we set up the time-space
framework. We provide two proofs. Using Theorem 1 and Siegmund-duality, in the
first proof the convergence of the one-dimensional distributions and
subsequently the uniform convergence of the semigroups is shown. The second
proof, in which the uniform convergence of generators is shown, resembles
previous ones.

  * Proof. [First proof of Theorem 5] For $x\in{\mathbb{R}}$ and $t\geq 0$ define $m:=\lceil e^{y}n^{e^{bt}}\rceil\in{\mathbb{N}}$. If $\varrho_{t}((-\infty,0))=\int_{[0,1]}u^{-2}\Lambda({\rm d}u)=\infty$ then $X_{t}$ has a continuous distribution for every $t>0$. Eq. (1) and Theorem 1 imply that

$\displaystyle{\mathbb{P}}(Y_{t}^{(n)}\geq y)$ $\displaystyle=$
$\displaystyle{\mathbb{P}}(L_{t}^{(n)}\geq m)={\mathbb{P}}(N_{t}^{(m)}\leq
n)={\mathbb{P}}(X_{t}^{(m)}\leq\log n-e^{-bt}\log m)$ (38)
$\displaystyle\rightarrow$
$\displaystyle{\mathbb{P}}(X_{t}\leq-e^{-bt}y)={\mathbb{P}}(-e^{bt}X_{t}\geq
y),\qquad y\in{\mathbb{R}},t\geq 0,$

as $n\to\infty$. If $\int_{[0,1]}u^{-2}\Lambda({\rm d}u)<\infty$ then the dust
condition is satisfied such that $b=0$ and (38) holds true for $-y$ in the set
$C_{X_{t}}$ of continuity points of $X_{t}$. Since
$Y_{t}\overset{\text{d}}{=}-e^{bt}X_{t}$,
$\lim_{n\to\infty}{\mathbb{P}}(-Y_{t}^{(n)}\leq-y)={\mathbb{P}}(-Y_{t}\leq-y)$
for every $-y\in C_{X_{t}}=C_{-Y_{t}}$. Thus, $Y_{t}^{(n)}$ converges in
distribution to $Y_{t}$ as $n\to\infty$ for every $t\geq 0$.

Define the time-space processes $\widetilde{Y}^{(n)}:=(t,Y_{t}^{(n)})_{t\geq
0},$ $n\in{\mathbb{N}},$ and $\widetilde{Y}:=(t,Y_{t})_{t\geq 0}$. The
processes $\widetilde{Y}^{(n)}$ and $\widetilde{Y}$ are time-homogeneous
Markov processes with state spaces $\widetilde{E}_{n}=\\{(s,y)|s\geq
0,e^{y}n^{e^{bs}}\in\\{n,n+1,\ldots\\}\\}$ and
$\widetilde{E}=[0,\infty)\times{\mathbb{R}}$ and semigroups
$(\widetilde{T}_{t}^{(n)})_{t\geq 0}$ and $(\widetilde{T}_{t})_{t\geq 0}$.
Define $k:=k(s,y,n):=e^{y}n^{e^{bs}}\in\\{n,n+1,\ldots\\}$ for
$(s,y)\in\widetilde{E}_{n}$ and $n\in{\mathbb{N}}$. Then

$\displaystyle\widetilde{T}_{t}^{(n)}f(s,y)$ $\displaystyle=$
$\displaystyle{\mathbb{E}}(f(s+t,Y_{s+t}^{(n)})|Y_{s}^{(n)}=y)={\mathbb{E}}(f(s+t,\log
L_{t}^{(k)}-e^{b(t+s)}\log n))$ $\displaystyle=$
$\displaystyle{\mathbb{E}}(f(s+t,e^{bt}y+Y_{t}^{(k)})),\qquad(s,y)\in\widetilde{E}_{n},f\in
B(\widetilde{E}),t\geq 0,n\in{\mathbb{N}}.$

Fix $t>0$ and first let $f\in B(\widetilde{E})$ be of the form
$f(s,y)=g(s)h(y),$ $(s,y)\in\widetilde{E}$, where $g\in B([0,\infty))$ and
$h\in\widehat{C}({\mathbb{R}})$. Clearly,
$\widetilde{T}_{t}^{(n)}f(s,y)=g(s+t){\mathbb{E}}(h(e^{bt}y+Y_{t}^{(k)})),$
$(s,y)\in\widetilde{E}_{n},n\in{\mathbb{N}},$ and
$\widetilde{T}_{t}f(s,y)={\mathbb{E}}(f(s+t,Y_{s+t})|Y_{s}=y)=g(s+t)T_{t}h(y)=g(s+t){\mathbb{E}}(h(e^{bt}y+Y_{t})),$
$(s,y)\in\widetilde{E},$ where the distribution of $Y_{t}$ is defined by its
characteristic function $\chi_{t}$, given by (13). Note that $h$ is uniformly
continuous and bounded. For $y\in{\mathbb{R}}$ define the function
$h_{y}:{\mathbb{R}}\to{\mathbb{R}}$ via $h_{y}(x):=h(e^{bt}y+x),$
$x\in{\mathbb{R}}$. The family of functions $\\{h_{y}|y\in{\mathbb{R}}\\}$ is
equicontinuous and uniformly bounded. From the weak convergence of
$Y_{t}^{(k)}$ to $Y_{t}$ as $k\to\infty$ and [17, Theorem 3.1] it follows that
$\lim_{k\to\infty}\sup_{y\in{\mathbb{R}}}|{\mathbb{E}}(h(e^{bt}y+Y_{t}^{(k)}))-{\mathbb{E}}(h(e^{bt}y+Y_{t}))|=0.$
Since $k=e^{y}n^{e^{bs}}\geq n$ for $(s,y)\in\widetilde{E}_{n}$ and
$n\in{\mathbb{N}}$,
$\lim_{n\to\infty}\sup_{(s,y)\in\widetilde{E}_{n}}|{\mathbb{E}}(h(e^{bt}y+Y_{t}^{(k)}))-{\mathbb{E}}(h(e^{bt}y+Y_{t}))|=0$.
Thus,

$\displaystyle\lim_{n\to\infty}\sup_{(s,y)\in\widetilde{E}_{n}}|\widetilde{T}_{t}^{(n)}f(s,y)-\widetilde{T}_{t}f(s,y)|=0.$
(39)

The algebra of functions $f\in B(\widetilde{E})$ of the form
$f(s,y)=\sum_{i=1}^{l}g_{i}(s)h_{i}(y),$ $(s,y)\in\widetilde{E},$ where
$l\in{\mathbb{N}},g_{i}\in B([0,\infty))$ and
$h_{i}\in\widehat{C}({\mathbb{R}})$, separates points and vanishes nowhere.
According to the Stone–Weierstrass theorem for locally compact spaces (see
e.g. [7]) it is a dense subset of $B(\widetilde{E})$ such that (39) holds true
for $f\in B(\widetilde{E})$. [8, IV, Theorem 2.11] states that
$\widetilde{Y}^{(n)}\to\widetilde{Y}$ in $D_{\widetilde{E}}[0,\infty)$, hence
$Y^{(n)}\to Y$ in $D_{{\mathbb{R}}}[0,\infty)$ as $n\to\infty$. $\Box$

The process $Y$ defined by (12) and (13) is an Ornstein–Uhlenbeck type process
(with nonnegative linear drift) as in [19]. The underlying infinitely
divisible distribution has characteristic exponent $y\mapsto\psi(-y),$
$y\in{\mathbb{R}}.$ According to [19, Theorem 3.1], $D$ is a core for the
corresponding generator $A$ and

$\displaystyle
Af(y)=f^{\prime}(y)(-a+by)+\int_{[0,1]}(f(y-\log(1-u))-f(y)-uf^{\prime}(y))u^{-2}\Lambda({\rm
d}u)$ (40)

for $y\in{\mathbb{R}}$ and $f\in D$; comparatively see Lemma 6 and its proof.

  * Proof. [Second proof of Theorem 5] The ’generator’ $(A_{s}^{(n)})_{s\geq 0}$ of $Y^{(n)}$ is given by

$\displaystyle A_{s}^{(n)}f(y)=-f^{\prime}(y)be^{bs}\log
n+\sum_{j>e^{y}n^{e^{bs}}}(f(\log j-e^{bs}\log
n)-f(y))\gamma_{e^{y}n^{e^{bs}},j},\quad(s,y)\in\widetilde{E}_{n},n\in{\mathbb{N}}.$

Here $f\in C_{1}({\mathbb{R}})$ such that
$f,f^{\prime}\in\widehat{C}({\mathbb{R}})$. Putting
$k:=k(s,y,n):=e^{y}n^{e^{bs}}$ for $(s,y)\in\widetilde{E}_{n}$ and
$n\in{\mathbb{N}}$ yields

$\displaystyle A_{s}^{(n)}f(y)=bf^{\prime}(y)(-\log
k+y)+\sum_{j=1}^{\infty}(f(y+\log(1+\tfrac{j}{k}))-f(y))\gamma_{k,k+j},\quad(s,y)\in\widetilde{E}_{n},n\in{\mathbb{N}}.$

Define $\Lambda_{D}:=\Lambda-b\lambda$ and let
$\Lambda_{D}^{+},\Lambda_{D}^{-}$ denote the nonnegative measures constituting
the Jordan decomposition $\Lambda_{D}=\Lambda_{D}^{+}-\Lambda_{D}^{-}$ of
$\Lambda_{D}$. Let $\gamma_{k,j}^{\lambda},\gamma_{k,j}^{D,+}$ and
$\gamma_{k,j}^{D,-}$ denote the jump rates of the fixation line corresponding
to $\lambda,\Lambda_{D}^{+}$ and $\Lambda_{D}^{-}$, respectively, and define
$\gamma_{k,j}^{D}:=\gamma_{k,j}^{D,+}-\gamma_{k,j}^{D,-}$ for
$j\in\\{k,k+1,\ldots\\}$ and $k\in{\mathbb{N}}$. Then
$\gamma_{k,k+j}=b\gamma_{k,k+j}^{\lambda}+\gamma_{k,k+j}^{D},$
$k\in{\mathbb{N}},j\in{\mathbb{N}}_{0},$ and

$\displaystyle
A_{s}^{(n)}f(y)=bf^{\prime}(y)R(k,y)+bS_{BS}(k,y)+S_{D}(k,y),\qquad(s,y)\in\widetilde{E}_{n},n\in{\mathbb{N}},$
(41)

where

$\displaystyle R(k,y)$ $\displaystyle:=$ $\displaystyle-\log
k+y+\sum_{j=1}^{k}\tfrac{j}{k}\gamma_{k,k+j}^{\lambda},\qquad
k\in{\mathbb{N}},y\in{\mathbb{R}},$ $\displaystyle S_{BS}(k,y)$
$\displaystyle:=$
$\displaystyle\sum_{j=1}^{\infty}(f(y+\log(1+\tfrac{j}{k}))-f(y)-\tfrac{j}{k}1_{[0,1]}(\tfrac{j}{k})f^{\prime}(y))\gamma_{k,k+j}^{\lambda},\qquad
k\in{\mathbb{N}},y\in{\mathbb{R}},$ $\displaystyle S_{D}(k,y)$
$\displaystyle:=$
$\displaystyle\sum_{j=1}^{\infty}(f(y+\log(1+\tfrac{j}{k}))-f(y))\gamma_{k,k+j}^{D},\qquad
k\in{\mathbb{N}},y\in{\mathbb{R}},$

and $f\in C_{1}({\mathbb{R}})$ such that
$f,f^{\prime}\in\widehat{C}({\mathbb{R}})$. Using the decomposition of
$\Lambda$ on Eq. (40) yields

$\displaystyle Af(y)=bf^{\prime}(y)(-1-\Psi(1)+y)+bI_{BS}(y)+I_{D}(y),\qquad
y\in{\mathbb{R}},f\in D,$ (42)

where

$\displaystyle I_{BS}(y)$ $\displaystyle:=$
$\displaystyle\int_{[0,1]}(f(y-\log(1-u))-f(y)-uf^{\prime}(y))u^{-2}\lambda({\rm
d}u),\qquad y\in{\mathbb{R}},$ $\displaystyle I_{D}(y)$ $\displaystyle:=$
$\displaystyle\int_{[0,1]}(f(y-\log(1-u))-f(y))u^{-2}\Lambda_{D}({\rm
d}u),\qquad y\in{\mathbb{R}}.$

Let $f\in D$. In the Bolthausen–Sznitman coalescent
$\gamma_{k,k+j}^{\lambda}=k/(j(j+1))$ for $k,j\in{\mathbb{N}}$ such that
$\sum_{j=1}^{k}\tfrac{j}{k}\gamma_{k,k+j}^{\lambda}=\sum_{j=1}^{k}(j+1)^{-1}=H_{k+1}-1=\log
k-1-\Psi(1)+o(1)$ as $k\to\infty$. Here $H_{k}$ denotes the $k$-th harmonic
number for $k\in{\mathbb{N}}$. Thus,

$\displaystyle\lim_{k\to\infty}\sup_{y\in{\mathbb{R}}}|R(k,y)-(-1-\Psi(1)+y)|=0.$
(43)

The function $h_{BS}:[0,1]\times{\mathbb{R}}\to{\mathbb{R}}$, defined via
$h_{BS}(u,y):=u^{-2}(f(y-\log(1-u))-f(y)-\tfrac{u}{1-u}1_{[0,1/2]}(u)f^{\prime}(y)),$
$u\in[0,1],y\in{\mathbb{R}},$ is bounded. Let the random variables $Z_{k},$
$k\in{\mathbb{N}},$ have distribution given by

$\displaystyle{\mathbb{P}}(Z_{k}=j)=\binom{k+j-2}{j-1}\int_{[0,1]}u^{j-1}(1-u)^{k}\lambda({\rm
d}u),\qquad j,k\in{\mathbb{N}},$

i.e., $Z_{k}-1$ has a mixed negative binomial distribution. Observe that
$h_{BS}(1-(1+\tfrac{j}{k})^{-1},y)=(\tfrac{j}{k+j})^{-2}(f(y+\log(1+\tfrac{j}{k}))-f(y)-\tfrac{j}{k}1_{[0,1]}(\tfrac{j}{k})f^{\prime}(y)),$
$y\in{\mathbb{R}},$ and
$\gamma_{k,k+j}^{\lambda}=(\tfrac{j}{k+j})^{-2}(1+(k+j)^{-1})(1-(j+1)^{-1}){\mathbb{P}}(Z_{k}=j)$
for $j,k\in{\mathbb{N}}$. Hence,

$\displaystyle
S_{BS}(k,y)={\mathbb{E}}(h_{BS}(1-(1+Z_{k}/k)^{-1},y)(1+(k+Z_{k})^{-1})(1-(Z_{k}+1)^{-1})).$

Let $Z$ have uniform distribution on $(0,1)$ such that
$I_{BS}(y)={\mathbb{E}}(h_{BS}(Z,y))$ for $y\in{\mathbb{R}}$. Here it is used
that $\int_{[0,1]}u^{-2}(u-\tfrac{u}{1-u}1_{[0,1/2]}(u))\lambda({\rm
d}u)=\int_{0}^{1/2}-(1-u)^{-1}\lambda({\rm
d}u)+\int_{1/2}^{1}u^{-1}\lambda({\rm d}u)=0.$ The function
$g:(0,\infty)\to(0,1)$, defined via $g(u):=1-(1+u)^{-1},$ $u\in(0,\infty),$ is
bounded and continuous. Since $Z_{k}/k\to Z/(1-Z)$ in distribution as
$k\to\infty$, $1-(1+Z_{k}/k)^{-1}=g(Z_{k}/k)\to g(Z/(1-Z))=Z$ in distribution
as $k\to\infty$. In particular, the random variables have values in $[0,1]$.
When considering the limit $k\to\infty$, the factor
$(1+(k+Z_{k})^{-1})(1-(Z_{k}+1)^{-1})$ has no influence on $S_{BS}(k,y)$. From
Lemma 9 it follows that

$\displaystyle\lim_{k\to\infty}\sup_{y\in{\mathbb{R}}}|S_{BS}(k,y)-I_{BS}(y)|=0.$
(44)

The measure $\Lambda_{D}$ is real-valued. Eq. (45) below can be proved when
$\Lambda_{D}$ is replaced by $\Lambda_{D}^{+}$ and $\Lambda_{D}^{-}$ in this
paragraph, and then holds true for $\Lambda_{D}$ by linearity. The function
$h_{D}:[0,1]\times{\mathbb{R}}\to{\mathbb{R}},$ defined via
$h_{D}(u,y):=u^{-1}(f(y-\log(1-u))-f(y)),$ $u\in[0,1],y\in{\mathbb{R}},$ is
bounded. By assumption $c:=\int_{[0,1]}u^{-1}\Lambda_{D}({\rm d}u)<\infty.$
Define the probability measure $Q$ on $([0,1],\mathcal{B}\cap[0,1])$ via
$Q(A):=c^{-1}\int_{A}u^{-1}\Lambda_{D}({\rm d}u),$ $A\in\mathcal{B}\cap[0,1].$
Let the random variables $Z_{k},$ $k\in{\mathbb{N}},$ have distribution given
by

$\displaystyle{\mathbb{P}}(Z_{k}=j)=\binom{k+j-1}{j}\int_{[0,1]}u^{j}(1-u)^{k}Q({\rm
d}u),\qquad j\in{\mathbb{N}}_{0},k\in{\mathbb{N}},$

i.e., $Z_{k}$ has a mixed negative binomial distribution. Observe that
$h_{D}(1-(1+\tfrac{j}{k})^{-1},y)=(f(y+\log(1+\tfrac{j}{k}))-f(y))\tfrac{k+j}{j},$
$y\in{\mathbb{R}},$ and
$\gamma_{k,k+j}^{D}=c\tfrac{k+j}{j}(1-(1+j)^{-1}){\mathbb{P}}(Z_{k}=j)$ for
$j,k\in{\mathbb{N}}$. Hence,

$\displaystyle S_{D}(k,y)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}(f(y+\log(1+\tfrac{j}{k})-f(y))\gamma_{k,k+j}^{D}$
$\displaystyle=$ $\displaystyle
c{\mathbb{E}}(h_{D}(1-(1+Z_{k}/k)^{-1},y)(1-(1+Z_{k})^{-1})),\qquad
k\in{\mathbb{N}},y\in{\mathbb{R}}.$

Let the random variable $Z$ have distribution $Q$. In particular,
$I_{D}(y)=c{\mathbb{E}}(h_{D}(Z,y)),$ $y\in{\mathbb{R}}$. By Lemma 9 and since
$1-(1+Z_{k}/k)^{-1}$ converges in distribution to $Z$ as $k\to\infty$,
$\lim_{k\to\infty}\sup_{y\in{\mathbb{R}}}|{\mathbb{E}}(h_{D}(1-(1+Z_{k}/k)^{-1},y))-{\mathbb{E}}(h_{D}(Z,y))|=0.$
Thus,

$\displaystyle\lim_{k\to\infty}\sup_{y\in{\mathbb{R}}}|S_{D}(k,y)-I_{D}(y)|=0.$
(45)

Taking into account that $k=e^{y}n^{e^{bs}}\geq n$ for
$(s,y)\in\widetilde{E}_{n}$ and $n\in{\mathbb{N}}$, Eqs. (41)-(45) imply

$\displaystyle\lim_{n\to\infty}\sup_{(s,y)\in\widetilde{E}_{n}}|A_{s}^{(n)}f(y)-Af(y)|=0.$

The time-space variant of [8, IV, Corollary 8.7] as implemented in the proof
of Theorem 1 yields the desired convergence of $Y^{(n)}\to Y$ in
$D_{{\mathbb{R}}}[0,\infty)$ as $n\to\infty$. $\Box$

###  8 Appendix

###### Lemma 8

The $\Lambda$-coalescent does not come down from infinity under Assumption A.

  * Proof. Suppose that $\Lambda$ satisfies Assumption A. Define $\Lambda_{D}:=\Lambda-b\lambda$, let $\Lambda_{D}^{+}$ and $\Lambda_{D}^{-}$ denote the nonnegative measures constituting the Jordan decomposition $\Lambda_{D}=\Lambda_{D}^{+}-\Lambda_{D}^{-}$ of $\Lambda_{D}$ and let $|\Lambda_{D}|:=\Lambda_{D}^{+}+\Lambda_{D}^{-}$ denote the (total) variation of $\Lambda_{D}$. Define $\eta_{k}^{\Lambda}:=k\sum_{j=0}^{k-2}\int_{[0,1]}(1-u)^{j}\Lambda({\rm d}u)$ and $\eta_{k}^{b\lambda}$ and $\eta_{k}^{|\Lambda_{D}|}$ similarly with $b\lambda$ and $|\Lambda_{D}|$ in place of $\Lambda$ for $k\geq 2$. By assumption,

$\displaystyle\lim_{k\to\infty}k^{-1}\eta_{k}^{|\Lambda_{D}|}=\int_{[0,1]}u^{-1}|\Lambda_{D}|({\rm
d}u)<\infty.$

Since

$\displaystyle(k\log k)^{-1}\eta_{k}^{b\lambda}=b(\log
k)^{-1}\sum_{j=0}^{k-2}\int_{0}^{1}(1-u)^{j}{\rm d}u=b(\log
k)^{-1}\sum_{j=0}^{k-2}(j+1)^{-1}\rightarrow b,\qquad k\to\infty,$

it follows that $\eta_{k}^{b\lambda}+\eta_{k}^{|\Lambda_{D}|}\sim bk\log k$ as
$k\to\infty$. Due to $\Lambda\leq b\lambda+|\Lambda_{D}|$, it holds that
$\eta_{k}^{\Lambda}\leq\eta_{k}^{b\lambda}+\eta_{k}^{|\Lambda_{D}|}$ for
$k\geq 2$. Hence,

$\displaystyle\sum_{k=2}^{\infty}\big{(}\eta_{k}^{\Lambda}\big{)}^{-1}\geq\sum_{k=2}^{\infty}\big{(}\eta_{k}^{b\lambda}+\eta_{k}^{|\Lambda_{D}|}\big{)}^{-1}=\infty.$

The claim then follows from Schweinsberg’s criterion [21, Corollary 2]. $\Box$

The following lemma is a generalization of the integral criterion of
convergence in distribution and is used in Sections 4-7 to prove the uniform
convergence of generators.

###### Lemma 9

Let $X,X_{1},X_{2},\ldots$ be random variables on a probability space
$(\Omega,\mathcal{F},{\mathbb{P}})$ with values in $[0,1]$ such that
${\mathbb{P}}(X=0)={\mathbb{P}}(X=1)=0$ and $X_{n}\to X$ in distribution as
$n\to\infty$. Suppose that the family $F$ of functions
$f:[0,1]\to{\mathbb{R}}$ is uniformly bounded on $[0,1]$ and equicontinuous on
$[\delta,1-\delta]$ for every $0<\delta<1/2$. In particular, $f\in F$ is
bounded and continuous on $(0,1)$. Then

$\displaystyle\lim_{n\to\infty}\sup_{f\in
F}|{\mathbb{E}}(f(X_{n}))-{\mathbb{E}}(f(X))|=0.$

  * Proof. Define $M:=\sup_{f\in F}\|f\|<\infty$ and let $\varepsilon>0$ be arbitrary. The assumption ${\mathbb{P}}(X=0)={\mathbb{P}}(X=1)=0$ and the convergence of $X_{n}$ to $X$ in distribution as $n\to\infty$ provide the existence of $0<\delta<1/2$ and $n_{0}\in{\mathbb{N}}$ such that ${\mathbb{P}}(X_{n}\not\in[\delta,1-\delta])<\varepsilon/(4M)$ for $n\geq n_{0}$ and ${\mathbb{P}}(X\not\in[\delta,1-\delta])<\varepsilon/(4M)$. For $f\in F$ define $\tilde{f}:[0,1]\to{\mathbb{R}}$ via $\tilde{f}(u):=f(\delta),$ $0\leq u\leq\delta,$ $\tilde{f}(u):=f(u),$ $\delta\leq u\leq 1-\delta,$ and $\tilde{f}(u):=f(1-\delta),$ $1-\delta\leq u\leq 1$. Then $\\{\tilde{f}|f\in F\\}$ is bounded (by $M$) and equicontinuous on $[0,1]$. [17, Theorem 3.1] yields

$\displaystyle\lim_{n\to\infty}\sup_{f\in
F}|{\mathbb{E}}(\tilde{f}(X_{n}))-{\mathbb{E}}(\tilde{f}(X))|=0.$

From

$\displaystyle|{\mathbb{E}}(f(X_{n}))-{\mathbb{E}}(f(X))|$ $\displaystyle\leq$
$\displaystyle{\mathbb{E}}(|f(X_{n})-\tilde{f}(X_{n})|)$
$\displaystyle+|{\mathbb{E}}(\tilde{f}(X_{n}))-{\mathbb{E}}(\tilde{f}(X))|+{\mathbb{E}}(|\tilde{f}(X)-f(X)|)$
$\displaystyle\leq$ $\displaystyle
2M{\mathbb{P}}(X_{n}\not\in[\delta,1-\delta])$
$\displaystyle+2M{\mathbb{P}}(X\not\in[\delta,1-\delta])+|{\mathbb{E}}(\tilde{f}(X_{n}))-{\mathbb{E}}(\tilde{f}(X))|,\qquad
n\in{\mathbb{N}},f\in F,$

it follows that $\lim_{n\to\infty}\sup_{f\in
F}|{\mathbb{E}}(f(X_{n}))-{\mathbb{E}}(f(X))|\leq\varepsilon$. Since
$\varepsilon>0$ is arbitrary the proof is complete. $\Box$

  * Remark. In [17, Theorem 3.1] the state space is more generally a separable metric space, but equicontinuity of $F$ is required to hold on the whole state space.

Let $E$ be a complete separable metric space and equip
$\widetilde{E}:=[0,\infty)\times E$ with the product metric. The following
proposition treats the generator of time-space processes of time-homogeneous
Feller processes.

###### Proposition 10

Suppose that $(T_{t})_{t\geq 0}$ is a Feller semigroup on $\widehat{C}(E)$
with generator $A$ and that $D$ is a core for $A$. For
$f\in\widehat{C}(\widetilde{E})$ and $s\in[0,\infty)$ let $\pi f(s,x)$ denote
the function $x\mapsto f(s,x),$ $x\in E$. The semigroup
$(\widetilde{T}_{t})_{t\geq 0}$, defined via

$\displaystyle\widetilde{T}_{t}f(s,x):=T_{t}\pi
f(s+t,x),\qquad(s,x)\in\widetilde{E},f\in B(\widetilde{E}),t\geq 0,$

is a Feller semigroup on $\widehat{C}(\widetilde{E})$. Let $\widetilde{D}$
denote the space of functions $f\in\widehat{C}(\widetilde{E})$ of the form
$f(s,x)=\sum_{i=1}^{l}g_{i}(s)h_{i}(x),$ $(s,x)\in\widetilde{E}$, where
$l\in{\mathbb{N}},h_{i}\in D$ and $g_{i}\in C_{1}([0,\infty))$ such that
$g_{i},g_{i}^{\prime}\in\widehat{C}([0,\infty))$ for $i=1,\ldots,l$. Then
$\widetilde{D}$ is a core for the generator $\widetilde{A}$ of
$(\widetilde{T}_{t})_{t\geq 0}$ and

$\displaystyle\widetilde{A}f(s,x)=\frac{\partial}{\partial s}f(s,x)+A\pi
f(s,x),\qquad(s,x)\in\widetilde{E},f\in\widetilde{D}.$ (46)

  * Proof. Observe that all functions involved in the proof are bounded and uniformly continuous. Clearly, the right-hand side of (46) lies in $\widehat{C}(\widetilde{E})$. The core $D$ is a dense subset of $\widehat{C}(E)$. Hence $\widetilde{D}$ is a dense subset of the space $D_{0}$ of functions $f\in\widehat{C}(\widetilde{E})$ of the form $f(s,x)=\sum_{i=1}^{l}g_{i}(s)h_{i}(x),$ $(s,x)\in\widetilde{E},$ where $l\in{\mathbb{N}},h_{i}\in\widehat{C}(E)$ and $g_{i}\in\widehat{C}([0,\infty))$ for $i=1,\ldots,l$. The algebra $D_{0}$ separates points and vanishes nowhere. The Stone–Weierstrass theorem for locally compact spaces (e.g. [7]) ensures that $D_{0}$ is a dense subset of $\widehat{C}(\widetilde{E})$. In [7] the theorem is stated for complex-valued functions, but it remains true for real-valued functions. To see this, let $f\in\widehat{C}(E)\subseteq\widehat{C}(E,{\mathbb{C}})$ be arbitrary. By the theorem there exist a sequence $(k_{n})_{n\in{\mathbb{N}}}\subseteq\widehat{C}(E,{\mathbb{C}})$ such that $\lim_{n\to\infty}||k_{n}-f\|=0$. Then $f_{n}:={\rm Re}(k_{n})\in\widehat{C}(E),$ $n\in{\mathbb{N}},$ and $\|f_{n}-f\|\leq\|k_{n}-f\|\to 0$ as $n\to\infty$. Thus $\widetilde{D}$ is a dense subset of $\widehat{C}(\widetilde{E})$ as well. If $h\in D$ and $g\in C_{1}([0,\infty))$ such that $g,g^{\prime}\in\widehat{C}([0,\infty))$, then

$\displaystyle
t^{-1}(\widetilde{T}_{t}g(s)h(x)-g(s)h(x))=t^{-1}(g(s+t)-g(s))h(x)+g(s+t)t^{-1}(T_{t}h(x)-h(x))$

converges uniformly in $(s,x)\in\widetilde{E}$ to
$g^{\prime}(s)h(x)+g(s)Ah(x)$ as $t\searrow 0$, thus $\widetilde{D}$ lies in
the domain of $\widetilde{A}$ and (46) holds true. By the same argument as
above, the space $D_{1}$ of functions $f\in\widehat{C}(\widetilde{E})$ of the
form $f(s,x)=\sum_{i=1}^{l}g_{i}(s)h_{i}(x),$ $(s,x)\in\widetilde{E}$, where
$g_{i}(s)=c_{i}\exp(-a_{i}s),$ $s\in[0,\infty)$ with $c_{i}\in{\mathbb{R}}$
and $a_{i}>0$ and $h_{i}\in D$ for $i=1,\ldots,l$, is a dense subset of
$\widehat{C}(\widetilde{E})$. By Hille–Yosida theory (see e.g. [8, I,
Proposition 3.1]) it now suffices to show that the image of $\lambda
I-\widetilde{A}|_{\widetilde{D}}$ is a dense subspace of
$\widehat{C}(\widetilde{E})$ for some $\lambda>0$ in order to prove that
$\widetilde{D}$ is a core for $\widetilde{A}$. Here $I$ denotes the identity
map on $\widehat{C}(E)$ or $\widehat{C}(\widetilde{E})$. Let $\varepsilon>0$
and $f\in\widehat{C}(\widetilde{E})$ be arbitrary. By density of $D_{1}$ in
$\widehat{C}(\widetilde{E})$, there exists $f_{1}\in D_{1}$ of the form
$f_{1}(s,x)=\sum_{i=1}^{l}g_{i}(s)h_{i}(x),$ $(s,x)\in\widetilde{E}$, such
that $\|f_{1}-f\|<\varepsilon/2$. Since $D$ is a core for $A$, the image of
$\lambda I-A|_{D}$ is a dense subset of $\widehat{C}(E)$ for every
$\lambda>0$, in particular for $\lambda+a_{i}$ in place of $\lambda$. Hence
there exists $r_{i}\in D$ such that
$\|(\lambda+a_{i})r_{i}-Ar_{i}-h_{i}\|<\varepsilon/(2l\|g_{i}\|)$ for
$i=1,\ldots,l$. Clearly, the function
$(s,x)\mapsto\sum_{i=1}^{l}g_{i}(s)r_{i}(x),$ $(s,x)\in\widetilde{E}$, belongs
to $\widetilde{D}$ and, by (46),

$\displaystyle\|(\lambda
I-\widetilde{A})\sum_{i=1}^{l}g_{i}(s)r_{i}(x)-f(s,x)\|$ $\displaystyle\leq$
$\displaystyle\|(\lambda
I-\widetilde{A})\sum_{i=1}^{l}g_{i}(s)r_{i}(x)-\sum_{i=1}^{l}g_{i}(s)h_{i}(x)\|+\|f_{1}-f\|$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{l}\|g_{i}((\lambda+a_{i})r_{i}-Ar_{i}-h_{i})\|+\varepsilon/2\leq\varepsilon.$

In the second last step it is used that $g_{i}^{\prime}(s)=-a_{i}g_{i}(s),$
$s\in[0,\infty)$ for $i=1,\ldots,l$. Since $\varepsilon>0$ has been arbitrary
the proof is complete. $\Box$

  * Remark. The last part of the proof of Proposition 10 can be simplified under the additional assumption that $T_{t}D\subseteq D$ for every $t>0$. Then $\widetilde{T}_{t}\widetilde{D}\subseteq\widetilde{D}$ for every $t\geq 0$ and the claim follows by applying the core theorem [8, I, Proposition 3.3].

The computations in the proof of the following proposition are based on Gauß’
representation [24, p. 247] for the digamma function

$\displaystyle\Psi(z)=\int_{0}^{\infty}\bigg{(}\frac{e^{-u}}{u}-\frac{e^{-zu}}{1-e^{-u}}\bigg{)}{\rm
d}u,\qquad{\rm Re}(z)>0.$

###### Proposition 11

Suppose that $\Lambda=\beta(1,b)$ with $b>0$. Then the measure $\varrho$,
defined by (4), has density $f$ with respect to Lebesgue measure on
${\mathbb{R}}\setminus\\{0\\}$ given by $f(u):=be^{bu}(1-e^{u})^{-2}$ for
$u<0$ and $f(u):=0$ for $u>0$. Let $a$ and $\psi$ be given by (2) and (3).
Then

$\displaystyle a=b(1+\Psi(b))$ (47)

and

$\displaystyle\psi(x)=b((1-b)\Psi(b)-(1-b-ix)\Psi(b+ix)),\qquad
x\in{\mathbb{R}}.$

  * Proof. It is easily verified that $\varrho$ has density as stated in the proposition. Eq. (47) follows from

$\displaystyle\int_{[0,1]}u^{-1}(\Lambda-b\lambda){\rm d}u$ $\displaystyle=$
$\displaystyle b\int_{0}^{1}u^{-1}((1-u)^{b-1}-1){\rm d}u$ $\displaystyle=$
$\displaystyle
b\int_{0}^{\infty}\Big{(}\frac{e^{-bu}}{1-e^{-u}}-\frac{e^{-u}}{1-e^{-u}}\Big{)}{\rm
d}u=b(\Psi(1)-\Psi(b)).$

Next, note that

$\displaystyle\Psi(b)-\Psi(b+ix)=\int_{0}^{\infty}(e^{-ixu}-1)\frac{e^{-bu}}{1-e^{-u}}{\rm
d}u,\qquad x\in{\mathbb{R}},$

is the characteristic exponent of the negative of a drift-free subordinator,
whose Lévy measure has density $u\mapsto e^{bu}(1-e^{u})^{-1},$ $u<0,$ with
respect to Lebesgue measure on $(-\infty,0)$. If $b<1$ and $Z$ has
characteristic function $\exp((1-b)(\Psi(b)-\Psi(b+ix)),$ $x\in{\mathbb{R}},$
then ${\mathbb{E}}(\log(1+|Z|))<\infty$. This fact is required in order to use
[23, V, Theorem 6.7] in Example 2. Integration by parts yields

$\displaystyle ix(\Psi(b+ix)-\Psi(b))$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}(ix-ixe^{-ixu})\frac{e^{-bu}}{1-e^{-u}}{\rm
d}u$ $\displaystyle=$
$\displaystyle(ixu+e^{-ixu}-1)\frac{e^{-bu}}{1-e^{-u}}\bigg{|}_{u=0}^{u=\infty}$
$\displaystyle~{}~{}-\int_{0}^{\infty}(ixu+e^{-ixu}-1)\bigg{(}\frac{-be^{-bu}}{1-e^{-u}}-\frac{e^{-bu}}{(1-e^{-u})^{2}}e^{-u}\bigg{)}{\rm
d}u$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}(e^{-ixu}-1+ixu)\frac{e^{-bu}}{(1-e^{-u})^{2}}(1-(1-b)(1-e^{-u})){\rm
d}u,\qquad x\in{\mathbb{R}}.$

Hence,

$\displaystyle(1-b)\Psi(b)-(1-b-ix)\Psi(b+ix)$
$\displaystyle=ix\Psi(b)+(1-b)(\Psi(b)-\Psi(b+ix))+ix(\Psi(b+ix)-\Psi(b))$
$\displaystyle=ix\Psi(b)+(1-b)\int_{0}^{\infty}(e^{-ixu}-1)\frac{e^{-bu}}{1-e^{-u}}{\rm
d}u$
$\displaystyle~{}~{}~{}~{}~{}+\int_{0}^{\infty}(e^{-ixu}-1+ixu)\frac{e^{-bu}}{(1-e^{-u})^{2}}(1-(1-b)(1-e^{-u})){\rm
d}u$
$\displaystyle=ix\Psi(b)+\int_{0}^{\infty}(e^{-ixu}-1+ixu)\frac{e^{-bu}}{(1-e^{-u})^{2}}{\rm
d}u-ix(1-b)\int_{0}^{\infty}u\frac{e^{-bu}}{(1-e^{-u})}{\rm d}u$
$\displaystyle=ix(\Psi(b)-(1-b)\Psi^{\prime}(b))+b^{-1}\int_{{\mathbb{R}}\setminus\\{0\\}}(e^{ixu}-1-ixu)\varrho({\rm
d}u)$
$\displaystyle=ix\Big{(}\Psi(b)-(1-b)\Psi^{\prime}(b)+b^{-1}\int_{{\mathbb{R}}\setminus\\{0\\}}(e^{u}-1-u)\varrho({\rm
d}u)\Big{)}$
$\displaystyle~{}~{}~{}~{}~{}+b^{-1}\int_{{\mathbb{R}}\setminus\\{0\\}}(e^{ixu}-1+ix(1-e^{u}))\varrho({\rm
d}u).$

The calculation

$\displaystyle-(1-b)\Psi^{\prime}(b)+b^{-1}\int_{{\mathbb{R}}\setminus\\{0\\}}(e^{u}-1-u)\varrho({\rm
d}u)$
$\displaystyle~{}~{}~{}=\int_{0}^{\infty}(-(1-b)u(1-e^{-u})+e^{-u}-1+u)\frac{e^{-bu}}{(1-e^{-u})^{2}}{\rm
d}u=-\frac{e^{-bu}}{1-e^{-u}}u\Big{|}_{u=0}^{u=\infty}=1$

and multiplication with $b$ complete the proof. $\Box$

## References

  * [1] Applebaum, D. (2010) _Lévy Processes and Stochastic Calculus._ Cambridge University Press, Second Edition. MR2512800
  * [2] Applebaum, D. (2007) On the infinitesimal generators of Ornstein–Uhlenbeck processes with jumps in Hilbert space. _Potential Anal._ 26, no. 1, 79–100. MR2276526
  * [3] Berestycki, N. (2009) Recent progress in coalescent theory. _Ensaios Matemáticos_ 16. MR2574323
  * [4] Bogachev, V. I., Röckner, M. and Schmuland, B. (1996) Generalized Mehler semigroups and applications. _Probab. Theory Related Fields_ 105, no. 2, 193–225. MR1392452
  * [5] Bolthausen, E. and Sznitman, A.-S. (1998) On Ruelle’s probability cascades and an abstract cavity method. _Comm. Math. Phys._ 197, no. 2, 247–276. MR1652734
  * [6] Böttcher, B. (2014) Feller evolution systems: generators and approximation. _Stoch. Dyn._ 14, no. 3, 1350025, 15 pp. MR3213184
  * [7] de Branges, L. (1959) The Stone–Weierstrass theorem. Proc. Amer. Math. Soc. 10, 822–824. MR0113131
  * [8] Ethier, S. N. and Kurtz, T. G. (1986) _Markov Processes, Characterization and Convergence_. Wiley. MR0838085
  * [9] Gaiser, F. and Möhle, M. (2016) On the block counting process and the fixation line of exchangeable coalescents. _ALEA Lat. Am. J. Probab. Math. Stat._ 13, no. 2, 809–833. MR3546382
  * [10] Grigelionis, B. (2003) On the self-decomposability of Euler’s gamma function. _Lithuanian Math. J._ 43, no. 3, 295–305. MR2019545
  * [11] Hénard, O. (2015) The fixation line in the $\Lambda$-coalescent. _Ann. Appl. Probab._ 25, no. 5, 3007–3032. MR3375893
  * [12] Jurek, Z. J. and Vervaat, W. (1983) An integral representation for self-decomposable Banach space valued random variables. _Z. Wahrsch. Verw. Gebiete_ 62, no. 2, 247–262. MR0688989
  * [13] Kukla, J. and Möhle, M. (2018) On the block counting process and the fixation line of the Bolthausen–Sznitman coalescent. _Stochastic Process. Appl._ 128, no. 3, 939–962. MR3758343
  * [14] Möhle, M. (2015) The Mittag–Leffler process and a scaling limit for the block counting process of the Bolthausen–Sznitman coalescent. _ALEA Lat. Am. J. Probab. Math. Stat._ 12, no. 1, 35–53. MR3333734
  * [15] Möhle, M. (2021) The rate of convergence of the block counting process of exchangeable coalescents with dust. _ALEA Lat. Am. J. Probab. Math. Stat._ 18 (to appear)
  * [16] Pitman, J. (1999) Coalescents with multiple collisions. _Ann. Probab._ 27, no. 4, 1870–1902. MR1742892
  * [17] Ranga Rao, R. (1962) Relations between weak and uniform convergence of measures with applications. _Ann. Math. Statist._ 33, no. 2, 659–680. MR0137809
  * [18] Sagitov, S. (1999) The general coalescent with asynchronous mergers of ancestral lines. _J. Appl. Probab._ 36, no. 4, 1116–1125. MR1742154
  * [19] Sato, K. and Yamazato, M. (1984) Operator-selfdecomposable distributions as limit distributions of processes of Ornstein–Uhlenbeck type. _Stochastic Process. Appl._ 17, no. 1, 73–100. MR0738769
  * [20] Sato, K. (1999) _Lévy Processes and Infinitely Divisible Distributions._ Cambridge University Press. MR1739520
  * [21] Schweinsberg, J. (2000) A necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity. _Electron. Comm. Probab._ 5, 1–11. MR1736720
  * [22] Siegmund, D. (1976) The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes. _The Annals of Probability_ 4, no. 6, 914–924.
  * [23] Steutel, F. W. and Van Harn, K. (2004) _Infinite Divisibility of Probability Distributions on the Real Line._ Dekker, New York. MR2011862
  * [24] Whittaker, E. T. and Watson, G. N. (1996) _A Course of Modern Analysis._ Reprint of the fourth (1927) edition. Cambridge University Press. MR1424469
  * [25] Wolfe, S. J. (1982) On a continuous analogue of the stochastic difference equation $X_{n}=\rho X_{n-1}+B_{n}$. _Stochastic Process. Appl._ 12, no. 3, 301–312. MR0656279