Choosing $\delta_{1}=1/(4+8\alpha)$ yields $|G(\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})|\leq 1-\alpha(\theta/\eta)^{2}/2$, as claimed and being very
generous in the bound for $\theta$. Note that for $\alpha>1$ we have
$1/(24\alpha^{2})\leq\sqrt{\delta_{1}}/\sqrt{(\alpha+2)(\alpha+3)}\leq\sqrt{\delta_{1}}/(\alpha+1)$
so that (4.25) follows.

We continue with the proof of (4.26). Here we will use a basic trick that was
used in similar forms already long ago, see [18], where
$|G(\mathrm{e}^{\mathrm{i}\mkern 1.0mu\theta})|$ is related to the sum of the
differences of consecutive terms. Here we use the following construction. Note
that

$(1-\mathrm{e}^{-\eta}\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})G(\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})=\frac{1}{C(\omega)}\sum_{k\geq 1}\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta k-\eta k}(\rho^{k}c_{k}-\rho^{k-1}c_{k-1}).$

Note that
$\rho^{-1}c_{k-1}/c_{k}=h(k-1)/h(k)\cdot(1-k^{-1})^{\alpha-1}\leq(1-k^{-1})^{\alpha-1+o(1)}$
due to (A.2). Accordingly, since $\alpha>1$, we have for sufficiently large
$k$ that $\rho^{-1}c_{k-1}\leq c_{k}$. Let $\delta>0$. Then, using (A.2),
whenever $k$ is sufficiently large,

$\displaystyle|\rho^{k}c_{k}-\rho^{k-1}c_{k-1}|$
$\displaystyle=\rho^{k}c_{k}\left(1-\frac{\rho^{-1}c_{k-1}}{c_{k}}\right)=\rho^{k}c_{k}\left(1-\frac{h(k-1)}{h(k)}\Big{(}1-\frac{1}{k}\Big{)}^{\alpha-1}\right)\leq\rho^{k}c_{k}\left(1-\Big{(}1-\frac{1}{k}\Big{)}^{\alpha-1+\delta}\right).$

Note that $(1-x)^{a}\geq 1-(1+\delta)ax$ for any $a,\delta>0$ and $x$
sufficiently small. We thus obtain for sufficiently large $k$ that

$|\rho^{k}c_{k}-\rho^{k-1}c_{k-1}|\leq\frac{\alpha-1+\delta(\alpha+\delta)}{k}\rho^{k}c_{k}.$

Using this and the triangle inequality we obtain that for any $\delta>0$ there
is some $K\in\mathbb{N}$ and $d>0$ such that

$|G(\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})|\leq\frac{1}{C(\omega)|1-\mathrm{e}^{-\eta}\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta}|}\left((\alpha-1+\delta(\alpha+\delta))\sum_{k\geq
K}\mathrm{e}^{-\eta k}\frac{\rho^{k}c_{k}}{k}+d\right).$ (4.28)

A simple calculation reveals that

$|1-\mathrm{e}^{-\eta}\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta}|^{2}=1+\mathrm{e}^{-2\eta}-2\mathrm{e}^{-\eta}\cos(\theta).$

Since $|\theta|\in[\eta/24\alpha^{2},\pi]$ the $\cos$ is maximized for
$|\theta|=\eta/24\alpha^{2}$; using
$\mathrm{e}^{-x}=1-x+{x^{2}}/2+\mathcal{O}\left(x^{3}\right)$ and
$\cos(x)=1-x^{2}/2+\mathcal{O}\left(x^{4}\right)$ for $x\to 0$ and
abbreviating $a=2(24\alpha^{2})^{2}$ we get for sufficiently small $\eta$

$\displaystyle|1-\mathrm{e}^{-\eta}\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta}|^{2}$ $\displaystyle\geq
1+\mathrm{e}^{-2\eta}-2\mathrm{e}^{-\eta}\cos(\eta/24\alpha^{2})$
$\displaystyle=1+(1-2\eta+2\eta^{2})-2(1-\eta+\eta^{2}/2)(1-\eta^{2}/a)+\mathcal{O}\left(\eta^{3}\right)$
$\displaystyle=(1+2/a)\eta^{2}+\mathcal{O}\left(\eta^{3}\right)$
$\displaystyle\geq(1+1/a)\eta^{2}.$ (4.29)

Moreover, since $\alpha>1$ and using (4.24) as $\eta\to 0$

$\sum_{k\geq K}\mathrm{e}^{-\eta k}\frac{\rho^{k}c_{k}}{k}\leq\sum_{k\geq
1}\mathrm{e}^{-\eta
k}\frac{\rho^{k}c_{k}}{k}\sim\Gamma(\alpha-1)h(\eta^{-1})\eta^{-\alpha+1}.$

All in all, if $\alpha>1$ and using the latter inequality as well as
$C(\omega)\sim\Gamma(\alpha)h(\eta^{-1})\eta^{-\alpha}$ we obtain by plugging
in (4.29) into (4.28) that for any $\delta>0$ and $\eta$ sufficiently small

$\displaystyle|G(\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})|\leq(1+\delta)\frac{\alpha-1+\delta(\alpha+\delta)}{\Gamma(\alpha)h(\eta^{-1})\eta^{-\alpha}}\cdot\frac{\Gamma(\alpha-1)h(\eta^{-1})\eta^{-\alpha+1}}{(1+1/a)^{1/2}\eta}\leq\frac{1+\delta(1+(\alpha+\delta)/(\alpha-1))}{(1+1/a)^{1/2}}.$
(4.30)

Since $\delta>0$ was arbitrary and $a>0$ the claim in (4.26) is established by
choosing $\delta$ sufficiently small.

We complete the proof by showing (4.27). Note that there is a constant $b>0$
such that for all $0\leq x\leq\pi$

$|\mathrm{e}^{-x}-(1-x)|\leq
bx^{2}\quad\text{and}\quad|\cos(x)-(1-x^{2}/2)|\leq bx^{4}.$

Applying this to any occurrence of $\mathrm{e}^{x}$ and $\cos(x)$ below we
obtain that there is a $b>0$ such that for all $|\theta\rvert\leq\pi$ and
$\eta$ sufficiently small

$\big{|}|1-\mathrm{e}^{-\eta}\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta}|^{2}-\theta^{2}\big{|}=\big{|}\big{(}1+\mathrm{e}^{-2\eta}-2\mathrm{e}^{-\eta}\cos(\theta)\big{)}-\theta^{2}\big{|}\leq
b(\eta^{2}+\eta\theta^{2}+\theta^{4}).$

In particular, if $|\theta|\geq\eta$ we obtain that

$\big{|}|1-\mathrm{e}^{-\eta}\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta}|^{2}-\theta^{2}\big{|}\leq 3b\theta^{4}.$

Especially, if $|\theta|\leq(6b)^{-1/2}$, then
$|1-\mathrm{e}^{-\eta}\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta}|^{2}\geq\theta^{2}/2$. Moreover, since
$|1-\mathrm{e}^{-\eta}\mathrm{e}^{\mathrm{i}\mkern 1.0mu\theta}|^{2}$ is
monotone increasing for $\theta\in[0,\pi]$ we obtain
$|1-\mathrm{e}^{-\eta}\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta}|^{2}\geq(12b)^{-1}$ for all $|\theta|\geq(6b)^{-1/2}$. Using
these two last statements instead of (4.29) in (4.28) we arrive similarly to
(4.30) at the desired estimate (4.27). ∎

The previous statement applies only when $\alpha>1$. To handle the case
$0<\alpha\leq 1$ we show the following property, which establishes that
summing up iid random variables with probability generating function $G(x)$
sufficiently often we obtain a random variable based on the modified sequence
$\tilde{c}_{n}=\tilde{h}(n)n^{\tilde{\alpha}-1}\rho^{n}$ for $n\in\mathbb{N}$
where $\tilde{h}$ is slowly varying and $\tilde{\alpha}>1$. With this trick we
will be able to apply Lemma 4.5 even in the case $0<\alpha\leq 1$.

######  Lemma 4.6.

Let $\alpha>0$, $0<\rho<1$. Let $s\in\mathbb{N}$ and $Y_{s}:=\sum_{1\leq i\leq
s}X_{i}$ where $X_{1},X_{2},\dots$ are iid with probability generating
function $G(x)=C(\omega x)/C(\omega)$, where $\omega=\rho e^{-\eta}$. Then
there exists a eventually positive, continuous and slowly varying function
$\tilde{h}$ such that

$\Pr[Y_{s}=n]=\frac{\tilde{h}(n)n^{s\alpha-1}\mathrm{e}^{-\eta
n}}{C(\omega)^{k}},\quad n\in\mathbb{N}.$

###### Proof.

Let $\alpha^{\prime}=\ell\alpha$ for some $\ell\in\mathbb{N}$ and set
$\beta=\alpha-1$ as well as $\beta^{\prime}=\alpha^{\prime}-1$. Let $f$ be an
eventually positive slowly varying function. We will show that

$S:=\sum_{k=1}^{n-1}f(k)k^{\beta^{\prime}}\cdot
h(n-k)(n-k)^{\beta}=\tilde{h}(n)n^{\beta^{\prime}+\beta+1}$ (4.31)

for some slowly varying $\tilde{h}$ and $n\in\mathbb{N}$ such that
$\tilde{h}(n)\sim I(\beta^{\prime})\cdot f(n)h(n)$ for
$I(\beta^{\prime})=\int_{0}^{1}x^{\beta^{\prime}}(1-x)^{\beta}dx$. Then the
claimed statement follows readily by induction. In order to see (4.31) let
$\varepsilon>0$ be arbitrary and consider the (_middle_) sum

$\displaystyle M$ $\displaystyle:=\sum_{k=\varepsilon
n}^{(1-\varepsilon)n}f(k)k^{\beta^{\prime}}\cdot
h(n-k)(n-k)^{\beta}=f(n)h(n)n^{\beta^{\prime}+\beta}\sum_{k=\varepsilon
n}^{(1-\varepsilon)n}\frac{f(k)}{f(n)}\frac{h(n-k)}{h(n)}\left(\frac{k}{n}\right)^{\beta^{\prime}}\left(1-\frac{k}{n}\right)^{\beta}.$

By applying the Uniform Convergence Theorem A.1 we obtain that $f(k)/f(n)$ and
$h(n-k)/h(n)$ both tend to $1$ for any $k\in\\{\varepsilon
n,\dots,(1-\varepsilon)n\\}$ as $n\to\infty$. Further note that

$\frac{1}{(1-2\varepsilon)n+2}\sum_{k=\varepsilon
n}^{(1-\varepsilon)n}\left(\frac{k}{n}\right)^{\beta^{\prime}}\left(1-\frac{k}{n}\right)^{\beta}$

is a Riemann sum of
$I_{\varepsilon}:=\int_{\varepsilon}^{1-\varepsilon}x^{\beta^{\prime}}(1-x)^{\beta}dx\in\mathbb{R}$.
As $\varepsilon>0$ was arbitrary and $I_{0}<\infty$

$M\sim I_{\varepsilon}\cdot f(n)h(n)n^{\beta^{\prime}+\beta+1},\quad
n\to\infty.$ (4.32)

Next consider the (_tail_) sum. With (A.3), Corollary A.3 and $h(\varepsilon
n)\sim h(n)$ as well as $f(\varepsilon n)\sim f(n)$ we obtain

$\displaystyle T:=\sum_{k=1}^{\varepsilon n}f(k)k^{\beta^{\prime}}\cdot
h(n-k)(n-k)^{\beta}$ $\displaystyle\leq\sup_{(1-\varepsilon)n\leq k\leq
n}h(k)k^{\beta}\cdot\sum_{k=1}^{\varepsilon
n}f(k)k^{\beta^{\prime}}=\varepsilon^{\beta^{\prime}+1}\cdot\mathcal{O}\left(f(n)h(n)n^{\beta^{\prime}+\beta+1}\right).$

Analogously, consider the remaining terms

$T^{\prime}:=\sum_{t=1}^{\varepsilon n}f(n-k)(n-k)^{\beta^{\prime}}\cdot
h(k)k^{\beta}=\varepsilon^{\beta+1}\cdot\mathcal{O}\left(f(n)h(n)n^{\beta^{\prime}+\beta+1}\right).$

Comparing the estimates for $T$ and $T^{\prime}$ with (4.32) and letting
$\varepsilon\to 0$ we see that $S\sim
I_{0}f(n)h(n)n^{\beta^{\prime}+\beta+1}$. Moreover, we readily see that
$\tilde{h}(n)=I_{0}\cdot f(n)h(n)+o(f(n)h(n))$, which is obviously slowly
varying as $f,h$ are.

∎

Combining all these statements we are finally able to prove Lemma 3.7.

###### Proof of Lemma 3.7.

For the ease of reading we repeat some definitions. For $p\in\mathbb{N}$ let
$K_{p}:=\sum_{1\leq i\leq p}C_{1,i}$ and define
$\nu:=\mathbb{E}[C_{1_{i}}]={x}_{n}C^{\prime}({x}_{n})/C({x}_{n})$. Further we
need
$\sigma_{p}^{2}:=\textrm{Var}[K_{p}]=p(x_{n}^{2}C^{\prime\prime}(x_{n})/C(x_{n})-\nu+\nu^{2})$.
Then (3.12),(3.13) assert that $\nu\sim\alpha\chi_{n}^{-1}$ and
$\sigma_{p}^{2}\sim\alpha p\chi_{n}^{-2}$. We will use these properties
several times without explicitly referencing them. Set $M:=p\nu+t\sigma_{p}$.
Then

$\mathrm{Pr}\left[L_{p}=\mu_{p}+t\sigma_{p}\right]=\mathrm{Pr}\left[K_{p}=M\right]=[z^{M}]H(z)^{p},\qquad
H(z)=\frac{C({x}_{n}z)}{C({x}_{n})}.$

With Cauchy’s integral formula, where we integrate over an arbitrary closed
curve encircling the origin,

$\displaystyle\mathrm{Pr}\left[L_{p}=\mu_{p}+t\sigma_{p}\right]=\frac{C({x}_{n})^{-p}}{2\pi\mathrm{i}\mkern
1.0mu}\oint\mathrm{e}^{f(z)}\frac{dz}{z},\qquad f(z)=p\ln C({x}_{n}z)-M\ln z.$
(4.33)

Since we will need that several times, let us note that

$\begin{split}f^{\prime}(z)=p\frac{x_{n}C^{\prime}(x_{n}z)}{C(x_{n}z)}-\frac{M}{z},\quad
f^{\prime\prime}(z)=p\left(\frac{{x}_{n}^{2}C^{\prime\prime}({x}_{n}z)}{C({x}_{n}z)}-\left(\frac{{x}_{n}C^{\prime}({x}_{n}z)}{C({x}_{n}z)}\right)^{2}\right)+\frac{M}{z^{2}}\quad\text{and}\\\
f^{\prime\prime\prime}(z)=p\left(\frac{{x}_{n}^{3}C^{\prime\prime\prime}({x}_{n}z)}{C({x}_{n}z)}-3\frac{{x}_{n}^{3}C^{\prime\prime}({x}_{n}z)C^{\prime}({x}_{n}z)}{C({x}_{n}z)^{2}}+\left(\frac{{x}_{n}C^{\prime}({x}_{n}z)}{C({x}_{n}z)}\right)^{3}\right)-2\frac{M}{z^{3}}.\end{split}$
(4.34)

The subsequent proof follows a very clear route that is strewn with several
technical statements (as it is typical in this area). These statements, which
have self-contained proofs are clearly marked, and the proofs are presented at
the end of the section. We have in mind to apply the saddle-point method, that
is, we will split the integral in (4.33) up into one dominating part, where a
quadratic expansion of $f(z)$ is valid, and one negligible part. To this end,
consider the saddle-point equation

$\displaystyle f^{\prime}(z)=0\leavevmode\nobreak\
\Leftrightarrow\leavevmode\nobreak\
p\frac{{x}_{n}zC^{\prime}({x}_{n}z)}{C({x}_{n}z)}=M.$ (4.35)

This equation has obviously a unique solution that we call $w\equiv w(n,p,t)$,
the saddle-point. Note that $w=1$ if $t=0$. We claim that in general $w$
satisfies

$\displaystyle
w=\mathrm{e}^{\xi},\quad\text{where}\quad\xi=t\sigma_{p}^{-1}+\xi_{1},\quad\xi_{1}=\mathcal{O}\left(t^{2}p^{-1}\chi_{n}\right)$
(SaddleAsym)

uniformly for all $t=o(p^{1/6})$; the self-contained proof is at the end of
the section. We proceed by specifying in (4.33) the curve over which
integrate. We choose the simplest curve that passes the saddle point, i.e.,
the circle with radius $w$. By switching to polar coordinates

$\displaystyle\frac{C({x}_{n})^{-p}}{2\pi\mathrm{i}\mkern
1.0mu}\oint\mathrm{e}^{f(z)}\frac{dz}{z}$
$\displaystyle=\frac{C({x}_{n})^{-p}}{2\pi}\int_{-\pi}^{\pi}\mathrm{e}^{f(w\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})}d\theta.$ (4.36)

In the next step we study $f(w\mathrm{e}^{\mathrm{i}\mkern 1.0mu\theta})$ by
considering the Taylor series around $\theta=0$. Then $f^{\prime}(w)=0$
guarantees that $f(w\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})=f(w)-w^{2}f^{\prime\prime}(w)\theta^{2}/2+R$ for some remainder
$R$ that should be negligible in an appropriate interval around $0$. Let us
substantiate this. Set for the remainder of the proof

$\eta:=\chi_{n}-\xi\sim\chi_{n},\qquad\theta_{0}:=p^{-1/2+\varepsilon}\eta\textrm{
for some }0<\varepsilon<1/6.$

By applying (4.24) to (4.34) we obtain

$\displaystyle f^{\prime\prime}(w)\sim\alpha
p\eta^{-2}=\Theta(p\eta^{-2})\qquad\text{and}\qquad
f^{\prime\prime\prime}(w)=\mathcal{O}\left(p\eta^{-3}\right),$ (4.37)

so that the saddle-point heuristic
$f^{\prime\prime}(w)\theta_{0}^{2}=\omega(1)$ and
$f^{\prime\prime\prime}(w)\theta_{0}^{3}=o(1)$ is fulfilled. We claim that a
Gaussian expansion holds for $f(w\mathrm{e}^{\mathrm{i}\mkern 1.0mu\theta})$
as $\eta\to 0$ uniformly in $\lvert\theta\rvert<\theta_{0}$, that is, for
$t=o(p^{1/6})$,

$\displaystyle f(w\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})=f(w)-w^{2}f^{\prime\prime}(w){\theta^{2}}/{2}+o(1),$ (GaussExp)

the proof of which is at the end of this section. With this at hand, we split
up (4.36) into the integral over the – as we will establish, dominant – arc
$\\{w\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta}:-\theta_{0}\leq\theta\leq\theta_{0}\\}$ and the remainder, i.e.,

$\displaystyle\mathrm{Pr}\left[L_{p}=\mu_{p}+t\sigma_{p}\right]=\frac{C({x}_{n})^{-p}}{2\pi\mathrm{i}\mkern
1.0mu}\oint\mathrm{e}^{f(z)}\frac{dz}{z}=\frac{C({x}_{n})^{-p}}{2\pi}\left(\int_{-\theta_{0}}^{\theta_{0}}+\int_{\theta_{0}\leq\lvert\theta\rvert\leq\pi}\right)\mathrm{e}^{f(w\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})}d\theta=:I_{1}+I_{2}.$ (4.38)

We start with $I_{1}$. Since $\theta_{0}$ is such that
$\theta_{0}^{2}f^{\prime\prime}(w)\to\infty$ and
$\int_{\mathbb{R}}\mathrm{e}^{-y^{2}/2}dy=\sqrt{2\pi}$ we obtain due to
(GaussExp)

$\displaystyle
I_{1}\sim\frac{C({x}_{n})^{-p}}{2\pi}\int_{-\theta_{0}}^{\theta_{0}}\mathrm{e}^{f(w)-w^{2}f^{\prime\prime}(w)\frac{\theta^{2}}{2}}d\theta\sim\frac{1}{\sqrt{2\pi}}\left(\frac{C({x}_{n}w)}{C({x}_{n})}\right)^{p}w^{-M}\frac{1}{\sqrt{w^{2}f^{\prime\prime}(w)}}.$
(4.39)

We claim that for $t=o(p^{1/6})$ the first order of $I_{1}$ – the alleged
dominating integral – satisfies

$\displaystyle I_{1}\sim e^{-{t^{2}}/{2}}(2\pi
w^{2}f^{\prime\prime}(w))^{-1/2}.$ (DomInt)

Due to (4.37) and using that $w\sim 1$ as well as $\eta\sim\chi_{n}$ we obtain
that $w^{2}f^{\prime\prime}(w)\sim\alpha p\chi_{n}^{-2}\sim\sigma_{p}^{2}$.
Plugging this into (DomInt) we conclude that

$I_{1}\sim\mathrm{e}^{-{t^{2}}/{2}}\frac{1}{\sqrt{2\pi}\sigma_{p}}\sim\mathrm{e}^{-{t^{2}}/{2}}\frac{1}{\sqrt{2\pi}}\frac{\chi_{n}}{\sqrt{p\alpha}}.$

This establishes the claimed first order asymptotic of
$\mathrm{Pr}\left[K_{p}=M\right]=\mathrm{Pr}\left[L_{p}=\mu_{p}+t\sigma_{p}\right]$.
Hence, to finish the proof, we need to show that $I_{2}$ in (4.38) is
negligible compared to $I_{1}$. We first reformulate

$\displaystyle
I_{2}=\left(\frac{C({x}_{n}w)}{C({x}_{n})}\right)^{p}w^{-M}\frac{1}{2\pi}\int_{\theta_{0}\leq\lvert\theta\rvert\leq\pi}G(\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})^{p}\mathrm{e}^{-\mathrm{i}\mkern 1.0mu\theta M}d\theta,\qquad
G(y)=\frac{C({x}_{n}wy)}{C({x}_{n}w)}.$ (4.40)

In light of (4.39) and (4.37), in order to show that $I_{2}=o(I_{1})$ we just
need to show that

$\displaystyle\left\lvert\int_{\theta_{0}}^{\pi}G(\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})^{p}\mathrm{e}^{-\mathrm{i}\mkern 1.0mu\theta
M}d\theta\right\rvert=o\left(\eta p^{-1/2}\right).$ (4.41)

We first consider the case $\alpha>1$. With Lemma 4.5 we obtain that there are
$c<1,A,b>0$ such that

$\displaystyle\lvert G(\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})\rvert\leq\begin{cases}1-\alpha\left({\theta}/{\eta}\right)^{2}/2,&\theta\leq
b\eta\\\ c,&\theta\geq b\eta\\\
A\max\\{\eta/|\theta|,\eta\\},&\theta=\omega(\eta)\end{cases}.$ (4.42)

Applying the triangle inequality yields

$\displaystyle\left\lvert\int_{\theta_{0}}^{\pi}G(\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})^{p}\mathrm{e}^{-\mathrm{i}\mkern 1.0mu\theta
M}d\theta\right\rvert\leq\left(\int_{\theta_{0}}^{b\eta}+\int_{b\eta}^{\min\\{\pi,\eta\ln
p\\}}+\int_{\min\\{\pi,\eta\ln p\\}}^{\pi}\right)\lvert
G(\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})\rvert^{p}d\theta=:R_{1}+R_{2}+R_{3}.$

Due to (4.41) all that is left to show in order to obtain $I_{2}=o(I_{1})$ is

$\displaystyle R_{i}=o\left(\eta p^{-1/2}\right),\qquad i=1,2,3.$ (4.43)

Applying the first inequality of (4.42) and $1-x\leq\mathrm{e}^{-x}$ for
sufficiently small $x>0$ entails that

$\displaystyle
R_{1}\leq\int_{\theta_{0}}^{b\eta}\exp\left\\{-\frac{p\alpha}{2}\left(\frac{\theta}{\eta}\right)^{2}\right\\}d\theta=\frac{\eta}{\sqrt{p\alpha}}\int_{\sqrt{\alpha}p^{\varepsilon}}^{b\sqrt{\alpha}p^{1/2}}\mathrm{e}^{-t^{2}/2}dx=o\left(\eta
p^{-1/2}\right)$

showing (4.43) for $i=1$. We proceed to $R_{2}$, where we apply the second
inequality in (4.42) to obtain

$R_{2}\leq\eta\ln p\cdot c^{p}=({\eta}p^{-1/2})\cdot(p^{1/2}\ln p\cdot
c^{p}).$

Since $0<c<1$ this expression is $o(\eta p^{-1/2})$. For the remaining case we
apply the third inequality in (4.42) to obtain that

$\displaystyle R_{3}\leq\left(\int_{\min\\{1,\eta\ln
p\\}}^{1}+\int_{1}^{\pi}\right)\lvert G(\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})\rvert^{p}d\theta\leq(A\eta)^{p}\int_{\min\\{1,\eta\ln
p\\}}^{1}\theta^{-p}d\theta+\pi\cdot(C\eta)^{p}.$ (4.44)

Then, as $p\to\infty$,

$\displaystyle 0\leq(A\eta)^{p}\int_{\eta\ln
p}^{1}\theta^{-p}d\theta=(A\eta)^{p}\left[-\frac{\theta^{-p+1}}{p}\right]_{\eta\ln
p}^{1}\leq\frac{A^{p}\eta(\ln
p)^{-p+1}}{p}\leq\frac{\eta}{\sqrt{p}}\cdot\left(\frac{A}{\ln
p}\right)^{p}=o\left({\eta}p^{-1/2}\right).$

Hence all the terms on the right hand side of (4.44) are in $o(\eta p^{-1/2})$
validating (4.43) for $i=3$. We have just demonstrated the validity of (4.43)
for $i=1,2,3$ and thus the assertion of Lemma 3.7 is fully proven for
$\alpha>1$.

In the case $0<\alpha\leq 1$ we need to apply a trick in order to be able to
make use of Lemma 4.5. Instead of (4.40) we write for some $s\in\mathbb{N}$
such that $s\alpha>1$

$I_{2}=\left(\frac{C({x}_{n}w)}{C({x}_{n})}\right)^{p}w^{-M}\frac{1}{2\pi}\int_{\theta_{0}}^{\pi}G(\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})^{p/s}\mathrm{e}^{-\mathrm{i}\mkern 1.0mu\theta M}d\theta,\qquad
G(y)=\left(\frac{C({x}_{n}wy)}{C({x}_{n}w)}\right)^{s}.$

Now $G(y)$ is the generating function of a sum $Y_{s}$ of $s$ iid random
variables with probability generating function $C({x}_{n}wy)/C({x}_{n}w)$.
Hence we know from Lemma 4.6 that $Y_{s}$ has probability generating function
$\tilde{C}({x}_{n}wy)/\tilde{C}({x}_{n}w)$, where the coefficients of
$\tilde{C}$ are given by

$\tilde{c}_{n}=\tilde{h}(n)n^{s\alpha-1}\rho^{-n},\qquad\tilde{h}\text{
eventually positive, continuous and slowly varying}.$

As $s\alpha>1$ this is exactly the setting considered in Lemma 4.5 so that
adapting the proof after (4.41) is straightforward.

Proof of (SaddleAsym). Define for some (large) constant $A>0$ the quantities
$\xi_{\pm}=t\sigma_{p}^{-1}\pm A\cdot t^{2}p^{-1}\chi_{n}$ and note that
$x_{n}e^{\xi_{\pm}}=\rho e^{-\chi_{n}+{\cal O}(t\chi_{n}/\sqrt{p})}<\rho$. We
will show that $f^{\prime}(e^{\xi_{-}})<0<f^{\prime}(e^{\xi_{+}})$, from which
(SaddleAsym) follows immediately by the continuity of $f^{\prime}$. To this
end we compute the Taylor series of $x_{n}zC^{\prime}(x_{n}z)/C(x_{n}z)$
around $z=1$. Let $\beta>0$ such that $x_{n}\mathrm{e}^{\beta}<\rho$. Then we
obtain that there is some $\lvert\delta\rvert\in[0,\beta]$ such that

$\frac{{x}_{n}\mathrm{e}^{\beta}C^{\prime}({x}_{n}\mathrm{e}^{\beta})}{C({x}_{n}\mathrm{e}^{\beta})}=\nu+\frac{\sigma_{p}^{2}}{p}(\mathrm{e}^{\beta}-1)+R(\delta,\mathrm{e}^{\beta}),\qquad
R(\delta,\mathrm{e}^{\beta}):=\frac{d^{2}}{dz^{2}}\frac{{x}_{n}zC^{\prime}({x}_{n}z)}{C({x}_{n}z)}\bigg{|}_{z=\mathrm{e}^{\delta}}\frac{(\mathrm{e}^{\beta}-1)^{2}}{2}.$

Hence, there are $\lvert\delta_{\pm}\rvert\in[0,\xi_{\pm}]$ such that plugging
$\beta=\xi_{\pm}$ into the previous equation yiels

$\displaystyle
p\frac{{x}_{n}\mathrm{e}^{\xi_{\pm}}C^{\prime}({x}_{n}\mathrm{e}^{\xi_{\pm}})}{C({x}_{n}\mathrm{e}^{\xi_{\pm}})}=M\pm
At^{2}\sigma_{p}^{2}p^{-1}\chi_{n}+\mathcal{O}\left(t^{2}\right)+pR(\delta_{\pm},\mathrm{e}^{\xi_{\pm}}).$
(4.45)

Moreover,

$R(\delta_{\pm},\mathrm{e}^{\xi_{\pm}})=\Theta\left(\frac{C^{\prime\prime\prime}({x}_{n}\mathrm{e}^{\delta_{\pm}})}{C({x}_{n}\mathrm{e}^{\delta_{\pm}})}+\frac{C^{\prime\prime}({x}_{n}\mathrm{e}^{\delta_{\pm}})C^{\prime}({x}_{n}\mathrm{e}^{\delta_{\pm}})}{C({x}_{n}\mathrm{e}^{\delta_{\pm}})^{2}}+\frac{C^{\prime}({x}_{n}\mathrm{e}^{\delta_{\pm}})^{3}}{C({x}_{n}\mathrm{e}^{\delta_{\pm}})^{3}}\right)\cdot{\xi_{\pm}^{2}},$

where, crucially, the constants implicit in $\Theta$ do not depend on $A$.
Using (4.24) and since
$\lvert\delta_{\pm}\rvert\leq\lvert\xi_{\pm}\rvert=o(\chi_{n})$ we get that

$pR(\delta_{\pm},\mathrm{e}^{\xi_{\pm}})=\mathcal{O}\left(p\chi_{n}^{-3}\cdot\xi_{\pm}^{2}\right)=\mathcal{O}\left(p\chi_{n}^{-3}\cdot
t^{2}\sigma_{p}^{-2}\right)=\mathcal{O}\left(\chi_{n}^{-1}t^{2}\right).$

Moreover, since $1=o(\sigma_{p}^{2}p^{-1}\chi_{n})$ we obtain from (4.45)

$p\frac{{x}_{n}\mathrm{e}^{\xi_{\pm}}C^{\prime}({x}_{n}\mathrm{e}^{\xi_{\pm}})}{C({x}_{n}\mathrm{e}^{\xi_{\pm}})}=M\pm
At^{2}\chi_{n}^{-1}+\mathcal{O}\left(\chi_{n}^{-1}t^{2}\right),$

where, again, the constant in $\cal O$ does not depend on $A$. So, we may
choose $A$ large enough such that for $\xi_{+}$ this expression is $>M$ and
for $\xi_{-}$ it is $<M$. Together with
$f^{\prime}(z)=p{x_{n}C^{\prime}(x_{n}z)}/{C(x_{n}z)}-{M}/{z}$ we have thus
established $f^{\prime}(e^{\xi_{-}})<0<f^{\prime}(e^{\xi_{+}})$, as claimed.

Proof of (GaussExp). By applying Taylor’s theorem we obtain that for every
$|\theta|\leq\theta_{0}$ there is a $|\zeta|\leq|\theta|$ such that

$\displaystyle f(w\mathrm{e}^{\mathrm{i}\mkern
1.0mu\theta})=f(w)-w^{2}f^{\prime\prime}(w)\frac{\theta^{2}}{2}+(w^{3}f^{\prime\prime\prime}(w\mathrm{e}^{\mathrm{i}\mkern
1.0mu\zeta})+2w^{2}f^{\prime\prime}(w\mathrm{e}^{\mathrm{i}\mkern
1.0mu\zeta})+wf^{\prime}(w(\mathrm{e}^{\mathrm{i}\mkern
1.0mu\zeta}))\frac{(\mathrm{i}\mkern 1.0mu\theta)^{3}}{6}.$ (4.46)

To show that the terms involving $\zeta$ are $o(1)$, we claim that

$\displaystyle\lvert C(x_{n}w\mathrm{e}^{\mathrm{i}\mkern
1.0mu\zeta})\rvert\sim C(x_{n}w).$ (4.47)

With this at hand, and as $C$ and all its derivatives have only non-negative
coefficients,

$|\theta|^{3}\lvert w^{3}f^{\prime\prime\prime}(w\mathrm{e}^{\mathrm{i}\mkern
1.0mu\zeta})+2w^{2}f^{\prime\prime}(w\mathrm{e}^{\mathrm{i}\mkern
1.0mu\zeta})+wf^{\prime}(w(\mathrm{e}^{\mathrm{i}\mkern
1.0mu\zeta})\rvert=|\theta|^{3}p\cdot\mathcal{O}\left(\frac{C^{\prime\prime\prime}(x_{n}w)}{C(x_{n}w)}+\frac{C^{\prime\prime}(x_{n}w)C^{\prime}(x_{n}w)}{C(x_{n}w)^{2}}+\frac{C^{\prime}(x_{n}w)^{3}}{C(x_{n}w)^{3}}\right).$

Applying (4.24) and using that
$\lvert\zeta\rvert\leq\lvert\theta\rvert\leq\theta_{0}=p^{-1/2+\varepsilon}\eta$
we obtain that the last term is
$\mathcal{O}\left(\theta_{0}^{3}p\cdot\eta^{-3}\right)={\cal
O}(p^{-1/2+3\varepsilon})$ and, since $\varepsilon<1/6$, we are done with the
proof of (GaussExp). All is left to show is (4.47). Clearly, $\lvert
C(x_{n}w\mathrm{e}^{\mathrm{i}\mkern 1.0mu\zeta})\rvert\leq C(x_{n}w)$ since
$C$ has only non-negative coefficients. Let $A>0$ be a (large) constant. For
$1\leq k\leq A\eta^{-1}$ we obtain that $k\zeta=o(1)$ for
$\lvert\zeta\rvert\leq\theta_{0}=o(\eta)$ so that in this regime $\cos(\zeta
k)\sim 1$. Thus

$\lvert C(x_{n}w\mathrm{e}^{\mathrm{i}\mkern
1.0mu\zeta})\rvert\geq\bigg{\lvert}\sum_{k\geq 1}c_{k}(x_{n}w)^{k}\cos(\zeta
k)\bigg{\rvert}\sim\lvert C(x_{n}w)-R_{0}+R_{1}\rvert,\quad
R_{i}=\sum_{k>A\eta^{-1}}c_{k}(x_{n}w)^{k}\cos(\zeta k)^{i},i=0,1.$

We estimate for $i=0,1$ and assisted by (A.3)

$\displaystyle\lvert R_{i}\rvert$ $\displaystyle\leq\sum_{k\geq
A\eta^{-1}}h(k)k^{\alpha-1}\mathrm{e}^{-\eta k}\leq\sum_{k\geq
A\eta^{-1}}\sup_{\ell\geq A\eta^{-1}}h(\ell)\ell^{-1}\cdot
k^{\alpha}\mathrm{e}^{-\eta k}$
$\displaystyle\sim\frac{h(A\eta^{-1})}{A\eta^{-1}}\cdot\eta^{-\alpha}\cdot\sum_{k\geq
A\eta^{-1}}(k\eta)^{\alpha}\mathrm{e}^{-\eta k}\sim
A^{-1}h(\eta^{-1})\eta^{-\alpha}\int_{A}^{\infty}x^{\alpha}\mathrm{e}^{-x}dx.$

The integral in the previous display is finite and so, letting $A\to\infty$,
we obtain that $\lvert
R_{i}\rvert=o(h(\eta^{-1})\eta^{-\alpha})=o(C(x_{n}w))$. It follows that
$\lvert C(x_{n}w\mathrm{e}^{\mathrm{i}\mkern 1.0mu\zeta})\rvert\sim
C(x_{n}w)$, that is, (4.47) is valid which finishes the proof.

Proof of (DomInt). We need to show that

$\left(\frac{C({x}_{n}w)}{C({x}_{n})}\right)^{p}w^{-M}\sim\mathrm{e}^{-t^{2}/2}.$

Keeping in mind that $w=\mathrm{e}^{\xi}$ we have due to a standard Taylor
expansion of $C(x_{n}z)/C(x_{n})$ around $z=1$ that there is
$\delta_{1}\in[0,\xi]$ such that

$\displaystyle\frac{C({x}_{n}w)}{C({x}_{n})}=1+\frac{{x}_{n}C^{\prime}({x}_{n})}{C({x}_{n})}(\mathrm{e}^{\xi}-1)+\frac{{x}_{n}^{2}C^{\prime\prime}({x}_{n})}{C({x}_{n})}\frac{(\mathrm{e}^{\xi}-1)^{2}}{2}+R^{\prime},\qquad
R^{\prime}:=\frac{{x}_{n}^{3}C^{\prime\prime\prime}({x}_{n}\mathrm{e}^{\delta_{1}})}{C({x}_{n})}\frac{(\mathrm{e}^{\xi}-1)^{3}}{6}.$
(4.48)

Since $\lvert\delta_{1}\rvert\leq\lvert\xi\rvert=o(\chi_{n})$, we obtain by
applying Lemma 2.1 that
$R^{\prime}=\mathcal{O}\left(\xi^{3}\chi_{n}^{-3}\right)=o(p^{-1})$ for
$t=o(p^{1/6})$. Further, we obtain with Lemma 2.1 that for $t=o(p^{1/6})$

$\frac{{x}_{n}C^{\prime}({x}_{n})}{C({x}_{n})}\xi^{3}=o(p^{-1})\quad\text{and}\quad\frac{{x}_{n}^{2}C^{\prime\prime}({x}_{n})}{C({x}_{n})}\xi^{3}=o(p^{-1}).$

Hence we may rewrite (4.48) to

$\frac{C({x}_{n}w)}{C({x}_{n})}=1+\xi\frac{{x}_{n}C^{\prime}({x}_{n})}{C({x}_{n})}+\frac{\xi^{2}}{2}\frac{{x}_{n}C^{\prime}({x}_{n})}{C({x}_{n})}+\frac{\xi^{2}}{2}\frac{{x}_{n}^{2}C^{\prime\prime}({x}_{n})}{C({x}_{n})}+o(p^{-1}).$

All the terms involving $\xi$ in the latter equation are $o(1)$ for
$t=o(p^{1/6})$. With this at hand, we obtain for such $t$

$\displaystyle\left(\frac{C({x}_{n}w)}{C({x}_{n})}\right)^{p}$
$\displaystyle=\exp\left\\{p\ln\left(1+\xi\frac{{x}_{n}C^{\prime}({x}_{n})}{C({x}_{n})}+\frac{\xi^{2}}{2}\frac{{x}_{n}C^{\prime}({x}_{n})}{C({x}_{n})}+\frac{\xi^{2}}{2}\frac{{x}_{n}^{2}C^{\prime\prime}({x}_{n})}{C({x}_{n})}+o(p^{-1})\right)\right\\}$
$\displaystyle\sim\exp\left\\{p\xi\frac{{x}_{n}C^{\prime}({x}_{n})}{C({x}_{n})}+p\frac{\xi^{2}}{2}\left(\frac{{x}_{n}^{2}C^{\prime\prime}({x}_{n})}{C({x}_{n})}-\left(\frac{{x}_{n}C^{\prime}({x}_{n})}{C({x}_{n})}\right)^{2}\right)\right\\}=\exp\left\\{\xi
p\nu+{\xi^{2}}\sigma_{p}^{2}/2\right\\}.$

Hence, recalling that $M=p\nu+t\sigma_{p}$,

$\displaystyle\left(\frac{C({x}_{n}w)}{C({x}_{n})}\right)^{p}w^{-M}$
$\displaystyle\sim\exp\left\\{\xi
p\nu+{\xi^{2}}\sigma_{p}^{2}/2-\xi(p\nu+t\sigma_{p})\right\\}=\exp\left\\{{\xi^{2}}\sigma_{p}^{2}/2-\xi
t\sigma_{p}\right\\}.$

Next we replace $\xi$ by the expressions given in (SaddleAsym) to obtain

${\xi^{2}}\sigma_{p}^{2}/2={t^{2}}/{2}+{\cal
O}(t^{4}p^{-1})\quad\text{and}\quad\xi t\sigma_{p}=t^{2}+{\cal
O}(t^{3}p^{-1/2}).$

For $t=o(p^{1/6})$ both $\mathcal{O}$-terms vanish and (DomInt) follows. ∎

###  4.7 Proof of Corollaries 3.8 and 3.9

###### Proof of Corollary 3.8.

We have that $x_{n}y_{n}C^{\prime}(x_{n})=n+\mathcal{O}\left(1\right)$ and
$y_{n}C(x_{n})=n+\mathcal{O}\left(1\right)$ due to (3.4) and Lemma 3.3(I).
Hence $\mu_{N_{n}}=\mathbb{E}[\sum_{1\leq i\leq
N_{n}}C_{1,i}]=N_{n}{x}_{n}C^{\prime}({x}_{n})/C({x}_{n})=n+\mathcal{O}\left(n/N_{n}\right)$
and further $\sigma_{N_{n}}^{2}=\textrm{Var}(\sum_{1\leq i\leq
N_{n}}C_{1,i})\sim n^{2}/(\alpha N_{n})$ due to (3.13) and Lemma 4.1. We
conclude that

$\mathrm{Pr}\left[\sum_{1\leq i\leq
N_{n}}C_{1,i}=n\right]=\mathrm{Pr}\left[\sum_{1\leq i\leq
N_{n}}C_{1,i}=\mu_{N_{n}}+\sigma_{N_{n}}\cdot\mathcal{O}\left(N_{n}^{-1/2}\right)\right].$

Then apply Lemma 3.7 for some properly chosen $t={\cal O}(N_{n}^{-1/2})$.
Since $\sigma_{N_{n}}^{2}\sim
N_{n}(x_{n}^{2}C^{\prime\prime}(x_{n})/C(x_{n})-(x_{n}C^{\prime}(x_{n})/C(x_{n}))^{2})\sim
y_{n}(x_{n}^{2}C^{\prime\prime}(x_{n})-(x_{n}C^{\prime}(x_{n}))^{2}/C(x_{n}))\sim
y_{n}x_{n}^{2}C^{\prime\prime}(x_{n})/(\alpha+1)$ due to Lemmas 2.1 and 4.1
the claim follows. ∎

###### Proof of Corollary 3.9.

Set $\tau:=\mathbb{E}\left[P_{1}\right]=y_{n}C({x}_{n})$ and
$\tilde{n}=n-mN_{n}$. Let

$B_{\leq}:=\\{p\in\mathbb{N}_{0}:\lvert\tau-p\rvert\leq\sqrt{\tau}\ln\tau\\}\quad\text{and}\quad
B_{>}:=\\{p\in\mathbb{N}_{0}:\lvert\tau-p\rvert>\sqrt{\tau}\ln\tau\\}.$

Then we split up

$\mathrm{Pr}\left[L=\tilde{n}\right]=\left(\sum_{p\in B_{\leq}}+\sum_{p\in
B_{>}}\right)\mathrm{Pr}\left[L_{p}=\tilde{n}\right]\mathrm{Pr}\left[P_{1}=p\right]=:I_{1}+I_{2}.$

Recall that according to (3.13) the mean and variance of $L_{p}$ are such that
$\mu_{p}=p({x}_{n}C^{\prime}({x}_{n})/C({x}_{n})-m)\sim\alpha p\chi_{n}^{-1}$
and $\sigma_{p}^{2}\sim\alpha p\chi_{n}^{-2}$, respectively. Due to (3.4) we
see

$\displaystyle\mathbb{E}\left[L\right]=\tau(\mathbb{E}\left[C_{1,1}-m\right])=\tau\left(\frac{{x}_{n}C^{\prime}({x}_{n})}{C({x}_{n})}-m\right)=\tilde{n}.$
(4.49)

With this at hand we reformulate for $p\in\mathbb{N}$

$\displaystyle\mathrm{Pr}\left[L_{p}=\tilde{n}\right]=\mathrm{Pr}\left[L_{p}=\mu_{p}+t\sigma_{p}\right],\qquad
t=t(p)=\frac{\tilde{n}-\mu_{p}}{\sigma_{p}}=(\tau-p)\frac{{x}_{n}C^{\prime}({x}_{n})/C({x}_{n})-m}{\sigma_{p}}.$
(4.50)

Let us first study $I_{1}$. For $p\in B_{\leq}$ we obtain from Lemma 2.1, the
asymptotics for $\sigma_{p}$, and $p\sim\tau$ that
$t=\mathcal{O}\left(\ln\tau\right)$. Since $\ln\tau=o(p^{1/6})$ we can apply
Lemma 3.7 to (4.50). We obtain for all $p\in B_{\leq}$

$\displaystyle\mathrm{Pr}\left[L_{p}=\tilde{n}\right]\sim\mathrm{e}^{-{t^{2}}/{2}}(2\pi\sigma_{p}^{2})^{-1/2}\sim\mathrm{e}^{-{t^{2}}/{2}}(2\pi\sigma_{\tau}^{2})^{-1/2}$

Further (2.10) yields for such $p$

$\mathrm{Pr}\left[P_{1}=p\right]=\mathrm{Pr}\left[P_{1}=\tau+y\sqrt{\tau}\right]\sim\mathrm{e}^{-{y^{2}}/{2}}(2\pi\tau)^{-1/2},\qquad
y=y(p)=\frac{p-\tau}{\sqrt{\tau}}.$

With this at hand we obtain

$\displaystyle I_{1}\frac{1}{2\pi\sqrt{\tau\sigma_{\tau}^{2}}}\sum_{p\in
B_{\leq}}\mathrm{e}^{-{(t^{2}+y^{2})}/{2}}.$

Set

$\displaystyle\Delta:=\frac{({x}_{n}C^{\prime}({x}_{n})/C({x}_{n})-m)^{2}}{\sigma_{\tau}^{2}}+\frac{1}{\tau}\sim\frac{\alpha+1}{\tau}.$
(4.51)

Since for all $p\in B_{\leq}$ we have due to (3.12) and (3.13)

$\lvert
t^{2}+y^{2}-(\tau-p)^{2}\Delta\rvert=(\tau-p)^{2}\left(\frac{{x}_{n}C^{\prime}({x}_{n})}{C({x}_{n})}-m\right)^{2}\cdot\left\lvert\frac{1}{\sigma_{p}^{2}}-\frac{1}{\sigma_{\tau}^{2}}\right\rvert=\mathcal{O}\left((\tau-p)^{3}\tau^{-2}\right)=o(1).$

Accordingly, we obtain

$\displaystyle
I_{1}\sim\frac{1}{2\pi\sqrt{\tau\sigma_{\tau}^{2}}}\sum_{|p|\leq\sqrt{\tau}\ln\tau}\mathrm{e}^{-p^{2}\Delta/2}.$
(4.52)

Applying (2.18) there exists $Q$ with $\lvert Q\rvert\leq 3$ such that

$\sum_{|p|\leq\sqrt{\tau}\ln\tau}\mathrm{e}^{-p^{2}\Delta/2}=\int_{-\sqrt{\tau}\ln\tau}^{\sqrt{\tau}\ln\tau}\mathrm{e}^{-x^{2}\Delta/2}dx+Q.$

Since (4.51) guarantees that
$\sqrt{\tau}\ln\tau\sqrt{\Delta}=\Theta(\ln\tau)=\omega(1)$ we compute

$\displaystyle\int_{-\sqrt{\tau}\ln\tau}^{\sqrt{\tau}\ln\tau}\mathrm{e}^{-x^{2}\Delta/2}dx={\Delta}^{-1/2}\int_{-\sqrt{\tau}\ln\tau\sqrt{\Delta}}^{\sqrt{\tau}\ln\tau\sqrt{\Delta}}\mathrm{e}^{-x^{2}/2}dx\sim\sqrt{\frac{2\pi\tau}{\alpha+1}}=\omega(1).$

Hence (4.52) yields together with the expressions for the asymptotic behaviour
of $\chi_{n}$ and $\tau$ in Lemma 4.2 as well as $\sigma_{\tau}^{2}\sim\alpha
p\chi_{n}^{-2}$

$I_{1}\sim\frac{1}{\sqrt{2\pi(\alpha+1)}\sigma_{\tau}}\sim\sqrt{\frac{\alpha
C_{0}}{2\pi(\alpha+1)}\cdot
g(n-mN_{n})\cdot(n-mN_{n})^{-(\alpha+2)/(\alpha+1)}}.$

By plugging in the asymptotics from Lemma 2.1 we also obtain
$(\alpha+1)\sigma_{\tau}^{2}\sim\rho^{-m}(\alpha+1)(x_{n}C^{\prime\prime}(x_{n})-(x_{n}C^{\prime}(x_{n}))^{2}/C(x_{n}))\sim\rho^{-m}x_{n}^{2}C^{\prime\prime}(x_{n})$.
To show that $I_{2}$ is negligible compared to $I_{1}$ we apply (2.9) to
obtain the existence of $d>0$ such that
$I_{2}\leq\mathrm{e}^{-d(\ln\tau)^{2}}$. Moreover, by applying Lemma 4.2 we
obtain that for some $\delta>0$ eventually $\tau\geq(n-mN_{n})^{\delta}$ and
in addition $I_{1}\geq(n-mN_{n})^{-\delta}$. From this we infer that
$I_{2}\leq\mathrm{e}^{-d(\ln\tau)^{2}}=o(1)$ and the proof is finished. ∎

###  4.8 Proof of Lemma 3.9

####  4.8.1 Proof of Lemma 3.10(I)

###### Proof of Lemma 3.10(I).

We write $\tilde{n}:=n-mN_{n}$. Assisted by (3.9) and using the independece of
$L,R$ we obtain

$\displaystyle\mathrm{Pr}\left[{\mathcal{E}}_{n}\mid{\mathcal{P}}_{N_{n}}\right]$
$\displaystyle=\sum_{p,r\geq
0}\mathrm{Pr}\left[L_{p}=\tilde{n}-r\right]\mathrm{Pr}\left[P_{1}=p,R=r\mid{\mathcal{P}}_{N_{n}}\right].$

We partition the summation into three parts. For some $b>0$ (that we will
choose appropriately) set

$\displaystyle B_{\leq}$ $\displaystyle:=\\{(p,r)\in\mathbb{N}_{0}^{2}:\lvert
N_{n}-p\rvert\leq\sqrt{N_{n}}\ln N_{n},r\leq b\ln n\\},$ $\displaystyle
B_{\cdot,>}$ $\displaystyle:=\\{(p,r)\in\mathbb{N}_{0}^{2}:r>b\ln
n\\}\qquad\text{and}$ $\displaystyle B_{>,\leq}$
$\displaystyle:=\\{(p,r)\in\mathbb{N}_{0}^{2}:\lvert
N_{n}-p\rvert>\sqrt{N_{n}}\ln N_{n},r\leq b\ln n\\}.$

Then we obtain the three partial sums

$\displaystyle\mathrm{Pr}\left[{\mathcal{E}}_{n}\mid{\mathcal{P}}_{N_{n}}\right]$
$\displaystyle=\Big{(}\sum_{(p,r)\in B_{\leq}}+\sum_{(p,r)\in
B_{\cdot,>}}+\sum_{(p,r)\in
B_{>,\leq}}\Big{)}\mathrm{Pr}\left[L_{p}=\tilde{n}-r\right]\mathrm{Pr}\left[P_{1}=p,R=r\mid{\mathcal{P}}_{N_{n}}\right]=:I_{1}+I_{2}+I_{3}.$

It will turn out that the sum over $B_{\leq}$ is essentially the whole sum. We
will argue that

$\displaystyle I_{1}\sim\mathrm{Pr}\left[L_{N_{n}}=n\right],\quad
I_{2}=o(I_{1})\quad\text{and}\quad I_{3}=o(I_{1}).$ (4.53)

Let us start by showing that $I_{1}\sim\mathrm{Pr}\left[L_{N_{n}}=n\right]$.
By (3.4) and (3.11)

$\displaystyle\tilde{n}=y_{n}C({x}_{n})\left(\frac{{x}_{n}C^{\prime}({x}_{n})}{C({x}_{n})}-m\right)\quad\text{and}\quad\mu_{p}=\mathbb{E}\left[L_{p}\right]=p\left(\frac{{x}_{n}C^{\prime}({x}_{n})}{C({x}_{n})}-m\right).$
(4.54)

Also recall (3.13) which says

$\displaystyle\mu_{p}\sim\alpha
p\chi_{n}^{-1}\quad\text{and}\quad\sigma_{p}^{2}:=\mathrm{Var}\left(L_{p}\right)\sim\alpha
p\chi_{n}^{-2}.$ (4.55)

Then

$\mathrm{Pr}\left[L_{p}=\tilde{n}-r\right]=\mathrm{Pr}\left[L_{p}=\mu_{p}+(t-\tilde{r})\sigma_{p}\right],\qquad
t=t(p):=\frac{\tilde{n}-\mu_{p}}{\sigma_{p}},\leavevmode\nobreak\
\leavevmode\nobreak\ \tilde{r}=\tilde{r}(p):=\frac{r}{\sigma_{p}}.$

According to Lemma 4.1 $y_{n}C({x}_{n})=N_{n}+\mathcal{O}\left(1\right)$.
Hence with (4.54), (4.55) we obtain for $(p,r)\in B_{\leq}$ that

$t=(y_{n}C({x}_{n})-p)({x}_{n}C^{\prime}({x}_{n})/C({x}_{n})-m){\sigma_{p}}^{-1}\sim(N_{n}-p)\sqrt{{\alpha}/{p}}=\mathcal{O}\left(\ln
N_{n}\right).$

Further, as $0\leq r\leq b\ln n$ and $\chi_{n}\sim\alpha n/N_{n}$ according to
Lemma 4.1,

$\tilde{r}=\mathcal{O}\left({r}/{\sqrt{p}\chi_{n}^{-1}}\right)=o(1)\quad\text{and}\quad
t\tilde{r}=o(1).$

Since $t-\tilde{r}=o(p^{1/6})$ we may apply Lemma 3.7 and get for all
$(p,r)\in B_{\leq}$

$\mathrm{Pr}\left[L_{p}=\tilde{n}-r\right]\sim\mathrm{e}^{-{(t-\tilde{r})^{2}}/{2}}\mathrm{Pr}\left[L_{p}=\mu_{p}\right]\sim\mathrm{e}^{-t^{2}/2}\mathrm{Pr}\left[L_{p}=\mu_{p}\right].$

We also deduce from Lemma 3.7 and by plugging in $\chi_{n}\sim\alpha N_{n}/n$
from Lemma 4.1 that $\mathrm{Pr}\left[L_{p}=\mu_{p}\right]\sim\sqrt{\alpha
N_{n}/(2\pi))}/n$ whenever $p\sim N_{n}$. Then Corollary 3.8 reveals that
$\mathrm{Pr}\left[L_{p}=\mu_{p}\right]\sim\mathrm{Pr}\left[L_{N_{n}}=\tilde{n}\right]$.
Hence for all $(p,r)\in B_{\leq}$

$\displaystyle\mathrm{Pr}\left[L_{p}=\tilde{n}-r\right]\sim\mathrm{e}^{-{t^{2}}/{2}}\mathrm{Pr}\left[L_{N_{n}}=\tilde{n}\right].$
(4.56)

With this at hand, we simplify

$\mathrm{Pr}\left[L_{N_{n}}=\tilde{n}\right]^{-1}\cdot I_{1}=\sum_{\lvert
N_{n}-p\rvert\leq\sqrt{N_{n}}\ln
N_{n}}\mathrm{e}^{-{t^{2}}/{2}}\mathrm{Pr}\left[P_{1}=p\mid{\mathcal{P}}_{N_{n}}\right]+\mathcal{O}\left(\mathrm{Pr}\left[R\geq
b\ln n\mid{\mathcal{P}}_{N_{n}}\right]\right).$

Due to Lemma 3.6 there is some $0<a<1$ such that
$\mathrm{Pr}\left[P_{1}=p,R=r\mid{\mathcal{P}}_{N_{n}}\right]\leq a^{b\ln n}$
for all $r>b\ln n$, so that the $\cal O$-term is
$\mathcal{O}\left(n^{-1}\right)$ for $b$ sufficiently large. Hence

$\displaystyle
I_{1}\sim\mathrm{Pr}\left[L_{N_{n}}=\tilde{n}\right]\sum_{\lvert
N_{n}-p\rvert\leq\sqrt{N_{n}}\ln
N_{n}}\mathrm{e}^{-t^{2}/2}\mathrm{Pr}\left[P_{1}=p\mid{\mathcal{P}}_{N_{n}}\right]+o\left(\frac{\sqrt{N_{n}}}{n}\right).$
(4.57)

Next we show that the main contribution to the sum in (4.57) is given by a
very small range, namely $\lvert N_{n}-p\lvert\leq\ln N_{n}$. For that recall
(4.54) and set

$\Delta_{p}:=\frac{t^{2}}{(\tau-p)^{2}}=\frac{\big{(}{x}_{n}C^{\prime}({x}_{n})/C({x}_{n})-m\big{)}^{2}}{\sigma_{p}^{2}}.$

It is true that $\mathrm{Pr}\left[P_{1}=p\mid{\mathcal{P}}_{N_{n}}\right]=0$
for $p>N_{n}$. Hence we can rewrite the sum in (4.57) as

$\displaystyle\sum_{\lvert N_{n}-p\rvert\leq\sqrt{N_{n}}\log
N_{n}}\mathrm{e}^{-t^{2}/2}\mathrm{Pr}\left[P_{1}=p\mid{\mathcal{P}}_{N_{n}}\right]$
$\displaystyle=\sum_{0\leq q\leq\sqrt{N_{n}}\ln N_{n}}\mathrm{e}^{-(\tau-
N_{n}+q)^{2}\Delta_{N_{n}-q}/2}\mathrm{Pr}\left[P_{1}=N_{n}-q\mid{\mathcal{P}}_{N_{n}}\right].$
(4.58)

Disassemble

$\mathrm{Pr}\left[P_{1}=N_{n}-q\mid{\mathcal{P}}_{N_{n}}\right]=\frac{1}{\mathrm{Pr}\left[{\mathcal{P}}_{N_{n}}\right]}\mathrm{Pr}\left[P_{1}=N_{n}-q\right]\mathrm{Pr}\left[\sum_{j\geq
2}jP_{j}=q\right].$

We known that the density of a Poisson random variable $P_{1}$ is maximised at
its mean, and so
$\mathrm{Pr}\left[P_{1}=N_{n}-q\right]\leq\mathrm{Pr}\left[P_{1}=\tau\right]$
for any $q\in\mathbb{N}$ and further
$\mathrm{Pr}\left[P_{1}=\tau\right]\sim\mathrm{Pr}\left[{\mathcal{P}}_{N_{n}}\right]$
with Lemma 3.5(I). With Lemma 3.5(I) it also follows that
$\mathrm{Pr}\left[\sum_{j\geq 2}jP_{j}=q\right]=\mathcal{O}\left(a^{q}\right)$
for some $0<a<1$ as $q\to\infty$. Thus, we obtain

$\displaystyle\sum_{\ln N_{n}<q<\sqrt{N_{n}}\ln N_{n}}\mathrm{e}^{-(\tau-
N_{n}+q)^{2}\Delta_{N_{n}-q}/2}\mathrm{Pr}\left[P_{1}=N_{n}-q\mid{\mathcal{P}}_{N_{n}}\right]\leq\sum_{q>\ln
N_{n}}\mathrm{Pr}\left[\sum_{j\geq 2}jP_{j}=q\right]=o(1).$ (4.59)

Next we consider the range $0\leq q\leq\ln N_{n}$. Here we have that
$\mathrm{Pr}\left[P_{1}=N_{n}-q\right]\sim\mathrm{Pr}\left[P_{1}=\tau\right]$
since $\tau=N_{n}+\mathcal{O}\left(1\right)$ and by applying (2.10). Further
note that $\Delta_{N_{n}-q}\sim\alpha/N_{n}$ for $\lvert N_{n}-p\rvert\leq\ln
N_{n}$ according to (4.55) for that range of $q$ such that $(\tau-
N_{n}+q)^{2}\cdot\Delta_{N_{n}-q}\sim\alpha(\log N_{n})^{2}/N_{n}=o(1)$.
Consequently, with the same $0<a<1$ from (4.59),

$\displaystyle\sum_{0\leq q\leq\ln N_{n}}\mathrm{e}^{-(\tau-
N_{n}+q)^{2}\Delta_{N-q}/2}\mathrm{Pr}\left[P_{1}=N_{n}-q\mid{\mathcal{P}}_{N_{n}}\right]\sim\sum_{0\leq
q\leq\ln N_{n}}\mathrm{Pr}\left[\sum_{j\geq
2}jP_{j}=q\right]=1-\mathcal{O}\left(a^{\ln N_{n}}\right)\sim 1.$ (4.60)

Plugging (4.59) and (4.60) into (4.58) and then into (4.57) yields
$I_{1}\sim\mathrm{Pr}\left[L_{N_{n}}=\tilde{n}\right]+o(\sqrt{N_{n}}/n)$.
Since $\mathrm{Pr}\left[L_{N_{n}}=n\right]=\Theta(\sqrt{N_{n}}/n)$ due to
Corollary 3.8 the first part in (4.53) follows.

Next we dedicate ourselves to showing the remaining claims in (4.53). Since
$I_{1}=\Theta(\sqrt{N_{n}}/n)$ we need to prove that $I_{2}$ and $I_{3}$ are
in $o(\sqrt{N_{n}}/n)$. Start with $I_{2}$. According to Lemma 3.6 we obtain
that there exists $0<a<1$ yielding for $b$ sufficiently large

$\displaystyle I_{2}\leq\sum_{(p,r)\in
B_{\cdot,>}}\mathrm{Pr}\left[P_{1}=p,R=r\mid{\mathcal{P}}_{N_{n}}\right]\leq\mathrm{Pr}\left[R\geq
r\mid{\mathcal{P}}_{N_{n}}\right]\leq a^{b\ln
n}=o\left(\frac{\sqrt{N_{n}}}{n}\right).$ (4.61)

Next we treat $I_{3}$. We observe that $\mathrm{Pr}\left[\lvert
N_{n}-P_{1}\rvert>\sqrt{N_{n}}\ln N_{n}\right]/{\mathcal{P}}_{N_{n}}=o(1)$
according to Lemma 3.5(I) and (2.9). Then Lemma 3.5(I) entails that there is
some $0<a<1$ such that

$\displaystyle I_{3}\leq\sum_{\lvert N_{n}-p\rvert>\sqrt{N_{n}}\ln
N_{n}}\mathrm{Pr}\left[P_{1}=p\mid{\mathcal{P}}_{n}\right]=o\big{(}\min\\{a,y_{n}\\}^{\sqrt{N_{n}}\ln
N_{n}}\big{)}.$ (4.62)

If $N_{n}\geq(\ln n)^{3}$, then this is certainly in $o(\sqrt{N_{n}}/n)$. If
$N_{n}\leq(\ln n)^{3}$ and $N_{n}\to\infty$, then $\lambda_{n}\leq(\ln
n)^{3}/N_{n}^{*}$ and from Lemma 4.1 we obtain that $y_{n}\leq n^{-c}$ for
some $c>0$ and all sufficiently large $n$. Plugging this into (4.62) yields
$I_{3}=n^{-\omega(1)}$ and the proof is finished. ∎

####  4.8.2 Proof of Lemma 3.10(II)

###### Proof of Lemma 3.10(II).

Let $\tilde{n}=n-mN_{n}$. With (3.9) we get

$\displaystyle\mathrm{Pr}\left[{\mathcal{E}}_{n}\mid{\mathcal{P}}_{N_{n}}\right]=\sum_{p,r\geq
0}\mathrm{Pr}\left[L_{p}=\tilde{n}-r\right]\mathrm{Pr}\left[P_{1}=p,R=r\mid{\mathcal{P}}_{N_{n}}\right].$

Set $\tau:=\mathbb{E}\left[P_{1}\right]=y_{n}C({x}_{n})$. We partition the
summation regime into three parts, namely, for some constant $b>0$ (which we
need to choose sufficiently large later) we define

$\displaystyle B_{\leq}$ $\displaystyle:=\\{(p,r)\in\mathbb{N}_{0}^{2}:\lvert
p-\tau\rvert\leq\sqrt{\tau}\ln\tau,r\leq b(\ln\tilde{n})^{2}\\},$
$\displaystyle B_{\cdot,>}$
$\displaystyle:=\\{(p,r)\in\mathbb{N}_{0}^{2}:r>b(\ln\tilde{n})^{2}\\}\quad\text{and}$
$\displaystyle B_{>,\leq}$
$\displaystyle:=\\{(p,r)\in\mathbb{N}_{0}^{2}:\lvert
p-\tau\rvert>\sqrt{\tau}\ln\tau,r\leq b(\ln\tilde{n})^{2}\\}.$

Then

$\displaystyle\mathrm{Pr}\left[{\mathcal{E}}_{n}\mid{\mathcal{P}}_{N_{n}}\right]$
$\displaystyle=\left(\sum_{(p,r)\in B_{\leq}}+\sum_{(p,r)\in
B_{\cdot,>}}+\sum_{(p,r)\in
B_{>,\leq}}\right)\mathrm{Pr}\left[L_{p}=\tilde{n}-r\right]\mathrm{Pr}\left[P_{1}=p,R=r\mid{\mathcal{P}}_{N_{n}}\right]=:I_{1}+I_{2}+I_{3}.$

We will show that the sum over $B_{\leq}$ is dominant compared to the
negligible sums over $B_{\cdot,>}$ and $B_{>,\leq}$ as $n$ tends to infinity,
that is,

$\displaystyle I_{1}\sim\mathrm{Pr}\left[L=\tilde{n}\right],\quad
I_{2}=o(I_{1})\quad\text{and}\quad I_{3}=o(I_{1}).$ (4.63)

Let us first determine $I_{1}$. Set
$\mu_{p}:=\mathbb{E}\left[L_{p}\right]=p(x_{n}C^{\prime}(x_{n})/C(x_{n})-m)$
and $\sigma_{p}^{2}:=\mathrm{Var}\left(L_{p}\right)$. Due to (3.13) the
asymptotics of these expressions are $\mu_{p}\sim\alpha p\chi_{n}^{-1}$ and
$\sigma_{p}^{2}\sim\alpha p\chi_{n}^{-2}$. We observe that for any $p,r$,
compare also to (4.49),

$\mathrm{Pr}\left[L_{p}=\tilde{n}-r\right]=\mathrm{Pr}\left[L_{p}=\mu_{p}+(t-\tilde{r})\sigma_{p}\right],\qquad
t:=t(p)=(\tau-p)\frac{{x}_{n}C^{\prime}({x}_{n})/C({x}_{n})-m}{\sigma_{p}},\leavevmode\nobreak\
\tilde{r}:=\tilde{r}(p)=\frac{r}{\sigma_{p}}.$

We want to apply Lemma 3.7. Using (2.6), for $(p,r)\in B_{\leq}$ we obtain
that $t=\mathcal{O}\left(\ln\tau\right)=o(p^{1/6})$. Moreover, plugging in the
asymptotics of $\tau$ and $\chi_{n}$ from Lemma 4.2 and noting that $g$ grows
slower than any polynomial we obtain

$\displaystyle\tilde{r}$
$\displaystyle=\mathcal{O}\left((\ln\tilde{n})^{2}\tau^{-1/2}\chi_{n}\right)=\mathcal{O}\left((\ln\tilde{n})^{2}\sqrt{g(\tilde{n})}\tilde{n}^{-(\alpha+1)/(2(\alpha+1))}\right)=o(1)$
(4.64)

implying $t-\tilde{r}=o(p^{1/6})$. So, by Lemma 3.7 we obtain for all
$(p,r)\in B_{\leq}$

$\displaystyle\frac{\mathrm{Pr}\left[L_{p}=\tilde{n}-r\right]}{\mathrm{Pr}\left[L_{p}=\tilde{n}\right]}=\frac{\mathrm{Pr}\left[L_{p}=\mu_{p}+(t-\tilde{r})\sigma_{p}\right]}{\mathrm{Pr}\left[L_{p}=\mu_{p}+t\sigma_{p}\right]}\sim\mathrm{e}^{-{((t-\tilde{r})^{2}+t^{2})}/{2}}.$
(4.65)

Since $t=\mathcal{O}\left(\ln\tau\right)=\mathcal{O}\left(\ln\tilde{n}\right)$
due to Lemma 4.2, Equation (4.64) also implies that $t\tilde{r}=o(1)$. Thus we
get due to (4.65) for all $(p,r)\in B_{\leq}$

$\mathrm{Pr}\left[L_{p}=\tilde{n}-r\right]\sim\mathrm{Pr}\left[L_{p}=\tilde{n}\right].$

Accordingly,

$\displaystyle I_{1}\sim\sum_{\lvert
p-\tau\rvert\leq\sqrt{\tau}\ln\tau}\mathrm{Pr}\left[L_{p}=\tilde{n}\right]\sum_{0\leq
r\leq
b(\ln\tilde{n})^{2}}\mathrm{Pr}\left[P_{1}=p,R=r\mid{\mathcal{P}}_{N_{n}}\right].$
(4.66)

Next we claim that the sum over $r$ equals asymptotically
$\mathrm{Pr}\left[P_{1}=p\right]$. For any $p$ we have

$\displaystyle\sum_{0\leq r\leq
b(\ln\tilde{n})^{2}}\mathrm{Pr}\left[P_{1}=p,R=r\mid{\mathcal{P}}_{N_{n}}\right]=\mathrm{Pr}\left[P_{1}=p\mid{\mathcal{P}}_{N_{n}}\right]-\sum_{r>b(\ln\tilde{n})^{2}}\mathrm{Pr}\left[P_{1}=p,R=r\mid{\mathcal{P}}_{N_{n}}\right].$
(4.67)

According to Lemma 3.6 there is some $0<a<1$ such that

$\sum_{r>b(\ln\tilde{n})^{2}}\mathrm{Pr}\left[P_{1}=p,R=r\mid{\mathcal{P}}_{N_{n}}\right]=\mathcal{O}\left(a^{b(\ln\tilde{n})^{2}}\right).$

We further obtain with Lemma 3.5(II) for any $(p,r)\in B_{\leq}$

$\displaystyle\mathrm{Pr}\left[P_{1}=p\mid{\mathcal{P}}_{N_{n}}\right]$
$\displaystyle=\frac{\mathrm{Pr}\left[\sum_{j\geq
2}jP_{j}=N_{n}-p\right]}{\mathrm{Pr}\left[{\mathcal{P}}_{N_{n}}\right]}\mathrm{Pr}\left[P_{1}=p\right]$
$\displaystyle\sim\frac{\exp\left\\{-c_{m}\frac{a_{n}}{1-a_{n}}\right\\}}{(1-a_{n})^{c_{m}-1}}\cdot({x}_{n}^{m}y_{n})^{-p}\cdot\left(\frac{N_{n}-p}{N_{n}}\right)^{c_{m}-1}\cdot\mathrm{Pr}\left[P_{1}=p\right].$
(4.68)

Since according to Lemma 4.2 we have $p\sim\tau=y_{n}C({x}_{n})\sim a_{n}\cdot
N_{n}$ we get that

$\left(\frac{N_{n}-p}{(1-a_{n})N_{n}}\right)^{c_{m}-1}\sim 1.$

Further, Lemma 4.2 reveals that $\limsup a_{n}\leq\limsup\lambda_{n}^{-1}<1$
and $x_{n}^{m}y_{n}/(1-x_{n}^{m}y_{n})\sim(1-a_{n})N_{n}/c_{m}$ implying
${x}_{n}^{m}y_{n}=1-c_{m}(1-a_{n})^{-1}N_{n}^{-1}+o((1-a_{n})^{-1}N_{n}^{-1})$,
so that

$({x}_{n}^{m}y_{n})^{-p}=\exp\left\\{-p\ln({x}_{n}^{m}y_{n})\right\\}=\exp\left\\{c_{m}\frac{a_{n}}{1-a_{n}}+\mathcal{O}\left(\frac{a_{n}}{(1-a_{n})^{2}}N_{n}^{-1}\right)\right\\}\sim\exp\left\\{c_{m}\frac{a_{n}}{1-a_{n}}\right\\}.$

Plugging the asymptotic identities in the previous two displays into (4.68) we
deduce for $(p,r)\in B_{\leq}$ that
$\mathrm{Pr}\left[P_{1}=p\mid{\mathcal{P}}_{N_{n}}\right]\sim\mathrm{Pr}\left[P_{1}=p\right]$.
Since $p$ differs at most by $\sqrt{\tau}\ln\tau$ from the mean $\tau$ of
$P_{1}$ we further obtain by (2.10) that $\mathrm{Pr}\left[P_{1}=p\right]\sim
1/\sqrt{2\pi\tau}\exp\left\\{-\mathcal{O}\left((\ln\tau)^{2}\right)\right\\}$.
From (A.2) and Lemma 4.2 we deduce that for any $\delta>0$ eventually
$\tau\leq\tilde{n}^{\alpha/(\alpha+1)+\delta}$. Hence
$\mathrm{Pr}\left[P_{1}=p\right]=\exp\left\\{-\mathcal{O}\left((\log\tilde{n})^{2}\right)\right\\}=\omega(a^{b(\ln\tilde{n})^{2}})$
for $b$ sufficiently large so that the expression in (4.67) is asymptotically
given by $\mathrm{Pr}\left[P_{1}=p\right]$ for
$\lvert\tau-p\rvert\leq\sqrt{\tau}\ln\tau$. So far we have shown that (4.66)
can be asymptotically simplified to

$I_{1}\sim\sum_{\lvert
p-\tau\rvert\leq\sqrt{\tau}\ln\tau}\mathrm{Pr}\left[L_{p}=\tilde{n}\right]\mathrm{Pr}\left[P_{1}=p\right],$

where this sum in turn equals

$\displaystyle I_{1}\sim\mathrm{Pr}\left[L=\tilde{n}\right]-\sum_{\lvert
p-\tau\rvert>\sqrt{\tau}\ln\tau}\mathrm{Pr}\left[L_{p}=\tilde{n}\right]\mathrm{Pr}\left[P_{1}=p\right].$
(4.69)

With (2.9) and again using that $\ln\tau=\Omega(\ln\tilde{n})$ we obtain for
some $d>0$

$\displaystyle\sum_{\lvert
p-\tau\rvert>\sqrt{\tau}\ln\tau}\mathrm{Pr}\left[L_{p}=\tilde{n}\right]\mathrm{Pr}\left[P_{1}=p\right]\leq\mathrm{Pr}\left[\lvert
P_{1}-\tau\rvert>\sqrt{\tau}\ln\tau\right]\leq\mathrm{e}^{-d(\ln\tau)^{2}}=\mathrm{e}^{-\Omega((\ln\tilde{n})^{2})}.$

Applying Corollary 3.9 and once again (A.2) we get for some $c>0$ that

$\mathrm{Pr}\left[L=\tilde{n}\right]=\omega(\tilde{n}^{-c})=\omega(\mathrm{e}^{-\Omega((\ln\tilde{n})^{2})}),$

delivering the first part of (4.63) in light of (4.69). We continue by
applying Lemma 3.6 to $I_{2}$, i.e. with $0<a<1$ from before we have

$I_{2}\leq\sum_{r>b(\ln\tilde{n})^{2}}\mathrm{Pr}\left[R=r\mid{\mathcal{P}}_{N_{n}}\right]=\mathcal{O}\left(a^{b(\ln\tilde{n})^{2}}\right)=o(\tilde{n}^{-c})=o\left(\mathrm{Pr}\left[L=\tilde{n}\right]\right)$

showing the second part of (4.63). Next we show that $I_{3}=o(I_{1})$. Let
$\varepsilon>0$ be such that $(1+\varepsilon)\limsup\tau/N_{n}<1$. We can find
such an $\varepsilon$ because $\limsup\tau/N_{n}\leq\limsup\lambda_{n}^{-1}<1$
due to Lemma 4.2. Then

$\displaystyle I_{3}\leq\left(\sum_{\lvert
p-\tau\rvert>\sqrt{\tau}\ln\tau,p<N_{n}/(1+\varepsilon)}+\sum_{N_{n}/(1+\varepsilon)\leq
p\leq
N_{n}}\right)\mathrm{Pr}\left[P_{1}=p\mid{\mathcal{P}}_{N_{n}}\right]=:R_{1}+R_{2}.$

Since $\liminf(1-a_{n})>0$, see Lemma 4.2, we get analogous to (4.68) that

$R_{1}=\mathcal{O}\left(\sum_{\lvert
p-\tau\rvert>\sqrt{\tau}\ln\tau,p<N_{n}/(1+\varepsilon)}\mathrm{Pr}\left[P_{1}=p\right]({x}_{n}^{m}y_{n})^{-p}\left(1-\frac{p}{N_{n}}\right)^{c_{m}-1}\right).$

Further ${x}_{n}^{m}y_{n}=1-\Theta(N_{n}^{-1})$ due to Lemma 4.2 by which we
conclude for $p<N_{n}/(1+\varepsilon)$ that
$({x}_{n}^{m}y_{n})^{-p}=\mathcal{O}\left(1\right)$. For that range of $p$ we
also have that $1-p/N_{n}=\Theta(1)$ so that with (2.9) there is some $d>0$
yielding

$R_{1}=\mathcal{O}\left(\mathrm{Pr}\left[\lvert
P_{1}-\tau\rvert>\sqrt{\tau}\ln\tau\right]\right)=\mathcal{O}\left(\mathrm{e}^{-d(\ln\tau)^{2}}\right)=o\left(\mathrm{Pr}\left[L=\tilde{n}\right]\right).$

We proceed by treating $R_{2}$. Here we estimate
$\mathrm{Pr}\left[P_{1}=p,R=r\mid{\mathcal{P}}_{N_{n}}\right]\leq\mathrm{Pr}\left[P_{1}=p\right]/\mathrm{Pr}\left[{\mathcal{P}}_{N_{n}}\right]$.
With Lemma 3.5(II) we further obtain
$\mathrm{Pr}\left[{\mathcal{P}}_{N_{n}}\right]=\Theta(N_{n}^{-1})$. Setting
$s=s(n):=(N_{n}/(1+\varepsilon)-\tau)/\sqrt{\tau}$ an application of (2.9)
gives for some $d>0$

$R_{2}\leq\frac{\mathrm{Pr}\left[P_{1}>N_{n}/(1+\varepsilon)\right]}{\mathrm{Pr}\left[{\mathcal{P}}_{N_{n}}\right]}=\mathcal{O}\left(N_{n}\mathrm{Pr}\left[P_{1}>\tau+s\sqrt{\tau}\right]\right)=\mathcal{O}\left(N_{n}\mathrm{e}^{-ds\min\\{s,\sqrt{\tau}\\}}\right).$

By the choice of $\varepsilon$ we have that
$s=N_{n}/\sqrt{\tau}((1+\varepsilon)^{-1}-\tau/N_{n})=\Theta(N_{n}/\sqrt{\tau})$.
This lets us conclude that
$s\min\\{s,\sqrt{\tau}\\}=\Theta(\min\\{N_{n}^{2}/\tau,N_{n}\\})=\Omega(N_{n})=\Omega(N_{n}^{*})$.
Due to (1.6) and (A.2) we know that $N_{n}^{*}=n^{\alpha/(\alpha+1)+o(1)}$
leading to $R_{2}=n^{-\omega(1)}$ and the proof is finished. ∎

###  4.9 Proof of Lemma 3.10(II)

###### Proof of Lemma 3.11(I).

Together with the basic fact that $[x^{n}]A(ax)=a^{n}[x^{n}]A(x)$ for any
power series $A$ we obtain with Lemma 3.10(I)

$\displaystyle\mathrm{Pr}\left[\sum_{1\leq i\leq
N_{n}}C_{1,i}=n\right]=[x^{n}]\frac{C({x}_{n}x)^{N_{n}}}{C({x}_{n})^{N_{n}}}=\frac{{x}_{n}^{n}}{C({x}_{n})^{N_{n}}}[x^{n}]C(x)^{N_{n}}.$
(4.70)

Further as $S_{n}=\mathcal{O}\left(1\right)$ according to Lemma 4.1

$\displaystyle(y_{n}C({x}_{n}))^{N_{n}}=(N_{n}-c_{m}S)^{N_{n}}=N_{n}^{N_{n}}\left(1-c_{m}\frac{S_{n}}{N_{n}}\right)^{N_{n}}\sim
N_{n}^{N_{n}}\exp\left\\{-c_{m}S_{n}\right\\}.$ (4.71)

Lemma 3.11(I) follows since Lemma 4.1 implies that
$\mathrm{e}^{c_{m}S}\sim\mathrm{e}^{c_{m}\rho^{m}y_{n}/(1-\rho^{m}y_{n})}$ and
by applying Stirling’s formula to $N_{n}^{N_{n}}$. ∎

###### Proof of Lemma 3.11(II).

First we argue that $\tau=y_{n}C({x}_{n})$ and
$\tilde{\tau}:=C({x}_{n})/{x}_{n}^{m}$ differ by
$-c_{m}a_{n}/(1-a_{n})+o(1)=\mathcal{O}\left(1\right)$. This is true since
$\tau-\tilde{\tau}=y_{n}C(x_{n})(1-(x_{n}^{m}y_{n})^{-1})$ and Lemma 4.2 gives
$y_{n}C(x_{n})\sim a_{n}N_{n}$ as well as
$1-(x_{n}^{m}y_{n})^{-1}=-S_{n}^{-1}\sim-c_{m}/((1-a_{n})N_{n})$. Thus by
replacing $\tau$ by $\tilde{\tau}$, $P_{1}$ by
$\tilde{P}_{1}\sim\mathrm{Po}\left(\tilde{\tau}\right)$ and $L$ by
$\tilde{L}:=\sum_{1\leq i\leq\tilde{\tau}}(C_{1,i}-m)$ in the proof of
Corollary 3.9 we obtain

$\mathrm{Pr}\left[\tilde{L}=n-mN_{n}\right]\sim\frac{1}{\sqrt{2\pi\rho^{-m}x_{n}^{2}C^{\prime\prime}(x_{n})}}\sim\mathrm{Pr}\left[L=n-mN_{n}\right].$

As the probability generating function of $C_{1,1}-m$ is given by
$x^{-m}C({x}_{n}x)/C({x}_{n})$ we obtain

$\mathrm{Pr}\left[L=n-mN_{n}\right]\sim\mathrm{Pr}\left[\tilde{L}=n-mN_{n}\right]=[x^{n-mN_{n}}]\exp\left\\{\frac{C({x}_{n})}{{x}_{n}^{m}}\left(x^{-m}\frac{C({x}_{n}x)}{C({x}_{n})}-1\right)\right\\}.$

Next we use the basic fact that $[x^{n}]bF(ax)=a^{n}b[x^{n}]F(x)$ for any
series $F$ and $a,b\in\mathbb{R}$. Hence

$\mathrm{Pr}\left[L=n-mN_{n}\right]\sim{x}_{n}^{n-mN_{n}}\exp\left\\{-\frac{C({x}_{n})}{{x}_{n}^{m}}\right\\}[x^{n-mN_{n}}]\exp\left\\{\frac{C(x)}{x^{m}}\right\\}.$

∎

###  4.10 Proof of Theorem 1.1

Let $z_{n}$ be the solution to $z_{n}C^{\prime}(z_{n})=n$. Clearly
$z_{n}=\rho\mathrm{e}^{-\eta_{n}}$ with $\eta_{n}\to 0$ as $n\to\infty$, so
that by (4.24)

$\eta_{n}\sim(\Gamma(\alpha+1)h(\eta_{n}^{-1}))^{1/(\alpha+1)}n^{-1/(\alpha+1)}.$

Since (1.2) implies that $c_{n}/c_{n-1}\sim\rho^{-1}$ we have due to [4, Cor.
4.3] that

$\frac{[z^{n}]\exp\left\\{C(z)\right\\}}{[z^{n-1}]\exp\left\\{C(z)\right\\}}\sim\rho^{-1}.$

Further $0<\rho<1$ implies that the radius of convergence of $\sum_{j\geq
2}{C(z^{j})}/{j}$ is greater than $\rho$. Since the radius of convergence of
$\exp\left\\{C(z)\right\\}$ is $\rho$ we deduce from Lemma 2.3

$\displaystyle g_{n}=[z^{n}]\exp\left\\{C(z)+\sum_{j\geq
2}\frac{C(z^{j})}{j}\right\\}\sim\exp\left\\{\sum_{j\geq
2}\frac{C(\rho^{j})}{j}\right\\}[z^{n}]\exp\left\\{C(z)\right\\},\qquad\text{as
}n\to\infty.$ (4.72)

By virtue of this the task of determining $g_{n}$ reduces to computing the
coefficient of $\exp\left\\{C(z)\right\\}$. In what follows we consider the
one-parametric Boltzmann model with the parameter $z_{n}$ as explained at the
beginning of Section 3.1. We need the following notation: Let $P$ be a
$\mathrm{Po}\left(C(z_{n})\right)$ distributed random variable and
$C_{1},C_{2},\dots$ iid copies of $\Gamma C(z_{n})$, that is,

$\mathrm{Pr}\left[C_{1}=k\right]=\frac{c_{k}z_{n}^{k}}{C(z_{n})},\qquad
k\in\mathbb{N}.$

Further we need the sum of these random variables $K_{p}:=\sum_{1\leq i\leq
p}C_{i}$ for $p\in\mathbb{N}_{0}$ and its randomly stopped version $K:=K_{P}$.
Noting that the probability generating functions of $P$ and $C_{1}$ are given
by $\exp\left\\{C(z_{n})(z-1)\right\\}$ and $C(z_{n}z)/C(z_{n})$,
respectively, we reformulate

$\displaystyle[z^{n}]\mathrm{e}^{C(z)}=z_{n}^{-n}\mathrm{e}^{C(z_{n})}[z^{n}]\exp\left\\{C(z_{n})\left(\frac{C(z_{n}z)}{C(z_{n})}-1\right)\right\\}=z_{n}^{-n}\mathrm{e}^{C(z_{n})}\mathrm{Pr}\left[K=n\right].$
(4.73)

Set $\tau:=C(z_{n})$. Define

$B_{\leq}:=\\{p\in\mathbb{N}_{0}:\lvert p-\tau\rvert\leq\sqrt{\tau}\ln
n\\}\quad\text{and}\quad B_{>}:=\\{p\in\mathbb{N}_{0}:\lvert
p-\tau\rvert>\sqrt{\tau}\ln n\\}.$

With these definitions at hand we split up

$\displaystyle\mathrm{Pr}\left[K=n\right]=\left(\sum_{p\in
B_{\leq}}+\sum_{p\in
B_{>}}\right)\mathrm{Pr}\left[K_{p}=n\right]\mathrm{Pr}\left[P=p\right]=:I_{1}+I_{2}.$
(4.74)

We start with $I_{1}$. With the help of (4.24) we establish the identities

$\displaystyle\nu_{p}$
$\displaystyle:=\mathbb{E}\left[K_{p}\right]=p\frac{z_{n}C^{\prime}(z_{n})}{C(z_{n})}\sim\alpha
p\eta_{n}^{-1}\quad\text{and}$ $\displaystyle\sigma_{p}^{2}$
$\displaystyle:=\mathrm{Var}\left(K_{p}\right)=p\left(\frac{z_{n}^{2}C^{\prime\prime}(z_{n})+z_{n}C^{\prime}(z_{n})}{C(z_{n})}-\left(\frac{z_{n}C^{\prime}(z_{n})}{C(z_{n})}\right)^{2}\right)\sim\alpha
p\eta_{n}^{-2}.$

Define $L_{p}=K_{p}-mp$ and $\mu_{p}=\mathbb{E}\left[L_{p}\right]=\nu_{p}-mp$
implying $\\{L_{p}=\mu_{p}+d\\}=\\{K_{p}=\nu_{p}+d\\}$ for any
$d\in\mathbb{R}$. Let

$t=t(p):=\frac{(\tau-p)z_{n}C^{\prime}(z_{n})/C(z_{n})}{\sigma_{p}}.$

With this definition of $t$ we have $n=\tau
z_{n}C^{\prime}(z_{n})/C(z_{n})=\nu_{p}+t\sigma_{p}$. Further
$t=\mathcal{O}\left(\ln n\right)$ for $p\in B_{\leq}$ so that Lemma 3.7 is
applicable and we obtain

$\displaystyle\mathrm{Pr}\left[K_{p}=n\right]=\mathrm{Pr}\left[L_{p}=\mu_{p}+t\sigma_{p}\right]\sim\mathrm{e}^{-{t^{2}}/{2}}(2\pi\sigma_{\tau}^{2})^{-1/2}.$
(4.75)

Note that we used $\sigma_{p}\sim\sigma_{\tau}$ for $p\sim\tau$ in the latter
display. Next we observe due to (2.10) that for $p\in B_{\leq}$ and
$s=s(p):=(p-\tau)/\sqrt{\tau}=\mathcal{O}\left(\ln n\right)$

$\displaystyle\mathrm{Pr}\left[P=p\right]=\mathrm{Pr}\left[P=\tau+s\sqrt{\tau}\right]\sim({2\pi\tau})^{-1/2}\mathrm{e}^{-{s^{2}}/{2}}.$

Plugging this and (4.75) into $I_{1}$ defined in (4.74) yields

$I_{1}\sim\frac{1}{2\pi}(\tau\sigma_{\tau}^{2})^{-1/2}\sum_{p\in
B_{\leq}}\mathrm{e}^{-(s^{2}+t^{2})/2}.$ (4.76)

Set

$\Delta:=\frac{(z_{n}C^{\prime}(z_{n})/C(z_{n}))^{2}}{\sigma_{\tau}^{2}}+\frac{1}{\tau}.$

By

$\lvert
s^{2}+t^{2}-(\tau-p)^{2}\Delta\rvert=(\tau-p)^{2}\left(\frac{z_{n}C^{\prime}(z_{n})}{C(z_{n})}\right)^{2}\left\lvert\frac{1}{\sigma_{p}^{2}}-\frac{1}{\sigma_{\tau}^{2}}\right\rvert=\mathcal{O}\left((\ln
n)^{3}\frac{\tau}{\tau^{3/2}}\right)=o(1)$

and (4.76) we obtain the asymptotic identity

$\displaystyle I_{1}\sim\frac{1}{2\pi}(\tau\sigma_{\tau}^{2})^{-1/2}\sum_{p\in
B_{\leq}}\mathrm{e}^{-(\tau-p)^{2}\Delta/2}=\frac{1}{2\pi}(\tau\sigma_{\tau}^{2})^{-1/2}\sum_{|p|\leq\sqrt{\tau}\ln
n}\mathrm{e}^{-p^{2}\Delta/2}.$ (4.77)

By (2.18) we obtain that there exists $Q$ with $\lvert Q\rvert\leq 3$ such
that

$\displaystyle\sum_{|p|\leq\sqrt{\tau}\ln
n}\mathrm{e}^{-p^{2}\Delta/2}=\int_{-\sqrt{\tau}\ln n}^{\sqrt{\tau}\ln
n}\mathrm{e}^{-x^{2}\Delta/2}dx+Q.$ (4.78)

By a change of variables and since $\sqrt{\Delta\tau}\ln n=\Theta(\ln
n)=\omega(1)$

$\int_{-\sqrt{\tau}\ln n}^{\sqrt{\tau}\ln
n}\mathrm{e}^{-x^{2}\Delta/2}dx=\Delta^{-1/2}\int_{-\sqrt{\Delta\tau}\ln
n}^{\sqrt{\Delta\tau}\ln
n}\mathrm{e}^{-{x^{2}}/{2}}dx\sim\Delta^{-1/2}\sqrt{2\pi}.$

Combined with Equations (4.77) and (4.78) we readily obtain that
$I_{1}\sim(2\pi\tau\sigma_{\tau}^{2}\Delta)^{-1/2}$. Moreover, since
$\tau\sigma_{\tau}^{2}\Delta=\tau\cdot(\sigma_{\tau}^{2}\Delta)=z_{n}C^{\prime\prime}(z_{n})+z_{n}C^{\prime}(z_{n})\sim
z_{n}C^{\prime\prime}(z_{n})$ the proof is finished.

## References

  * Arratia and Tavaré [1994] R. Arratia and S. Tavaré.  Independent Process Approximations for Random Combinatorial Structures.  _Adv. Math._ , 104(1):90–154, 1994.  ISSN 0001-8708.  doi: 10.1006/aima.1994.1022.  URL https://doi-org.org/10.1006/aima.1994.1022.
  * Arratia et al. [2003] R. Arratia, A. Barbour, and S. Tavaré.  _Logarithmic Combinatorial Structures: A Probabilistic Approach_.  EMS monographs in mathematics. European Mathematical Society, 2003.  ISBN 9783037190005.  URL https://books.google.de/books?id=oBPvAAAAMAAJ.
  * Barbour and Granovsky [2005] A. Barbour and B. L. Granovsky.  Random Combinatorial Structures: the Convergent Case.  _Journal of Combinatorial Theory, Series A_ , 109(2):203–220, 2005.  ISSN 0097-3165.  doi: 10.1016/j.jcta.2004.09.001.  URL http://www.sciencedirect.com/science/article/pii/S0097316504001347.
  * Bell and Burris [2003] J. P. Bell and S. N. Burris.  Asymptotics for Logical Limit Laws: when the Growth of the Components is in an RT Class.  _Trans. Amer. Math. Soc._ , 355(9):3777–3794, 2003.  ISSN 0002-9947.  doi: 10.1090/S0002-9947-03-03299-9.  URL https://doi.org/10.1090/S0002-9947-03-03299-9.
  * Bell et al. [2000] J. P. Bell, E. A. Bender, P. J. Cameron, and L. B. Richmond.  Asymptotics for the Probability of Connectedness and the Distribution of Number of Components.  _Electron. J. Combin._ , 7:Research Paper 33, 22, 2000.  URL https://doi.org/10.37236/1511.
  * Bingham et al. [1987] N. H. Bingham, C. M. Goldie, and J. L. Teugels.  _Regular Variation_ , volume 27 of _Encyclopedia of Mathematics and its Applications_.  Cambridge University Press, Cambridge, 1987.  ISBN 0-521-30787-2.  doi: 10.1017/CBO9780511721434.  URL https://doi.org/10.1017/CBO9780511721434.
  * Bodirsky et al. [2007] M. Bodirsky, É. Fusy, M. Kang, and S. Vigerske.  Enumeration and Asymptotic Properties of Unlabeled Outerplanar Graphs.  _Electron. J. Combin._ , 14(1):Research Paper 66, 24, 2007.  URL http://www.combinatorics.org/Volume_14/Abstracts/v14i1r66.html.
  * Bodirsky et al. [2011] M. Bodirsky, É. Fusy, M. Kang, and S. Vigerske.  Boltzmann Samplers, Pólya Theory, and Cycle Pointing.  _SIAM Journal on Computing_ , 40(3):721–769, 2011\.  ISSN 0097-5397.  doi: 10.1137/100790082.  URL http://dx.doi.org/10.1137/100790082.
  * Burris [2001] S. N. Burris.  _Number Theoretic Density and Logical Limit Laws_ , volume 86 of _Mathematical Surveys and Monographs_.  American Mathematical Society, Providence, RI, 2001.  ISBN 0-8218-2666-2.  doi: 10.1090/surv/086.  URL https://doi.org/10.1090/surv/086.
  * Davis and McDonald [1995] B. Davis and D. McDonald.  An Elementary Proof of the Local Central Limit Theorem.  _J. Theoret. Probab._ , 8(3):693–701, 1995.  ISSN 0894-9840.  doi: 10.1007/BF02218051.  URL https://doi.org/10.1007/BF02218051.
  * de Bruijn [1981] N. G. de Bruijn.  _Asymptotic Methods in Analysis_.  Dover Publications, Inc., New York, third edition, 1981.  ISBN 0-486-64221-6.
  * Drmota [1994] M. Drmota.  A Bivariate Asymptotic Expansion of Coefficients of Powers of Generating Functions.  _European J. Combin._ , 15(2):139–152, 1994.  ISSN 0195-6698.  doi: 10.1006/eujc.1994.1016.  URL https://doi.org/10.1006/eujc.1994.1016.
  * Drmota and Soria [1995] M. Drmota and M. Soria.  Marking in Combinatorial Constructions: Generating Functions and Limiting Distributions.  _Theoretical Computer Science_ , 144(1):67–99, 1995.  ISSN 0304-3975.  doi: 10.1016/0304-3975(94)00294-S.  URL https://doi.org/10.1016/0304-3975(94)00294-S.
  * Duchon et al. [2004] P. Duchon, P. Flajolet, G. Louchard, and G. Schaeffer.  Boltzmann Samplers for the Random Generation of Combinatorial Structures.  _Combin. Probab. Comput._ , 13(4-5):577–625, 2004\.  ISSN 0963-5483.  doi: 10.1017/S0963548304006315.  URL https://doi.org/10.1017/S0963548304006315.
  * Flajolet and Sedgewick [2009] P. Flajolet and R. Sedgewick.  _Analytic Combinatorics_.  Cambridge University Press, Cambridge, 2009.  ISBN 978-0-521-89806-5.  doi: 10.1017/CBO9780511801655.  URL https://doi.org/10.1017/CBO9780511801655.
  * Freiman and Granovsky [2005] G. A. Freiman and B. L. Granovsky.  Clustering in Coagulation-fragmentation Processes, Random Combinatorial Structures and Additive Number Systems: Asymptotic Formulae and Limiting Laws.  _Trans. Amer. Math. Soc._ , 357(6):2483–2507, 2005.  ISSN 0002-9947.  doi: 10.1090/S0002-9947-04-03617-7.  URL https://doi.org/10.1090/S0002-9947-04-03617-7.
  * Fristedt [1993] B. Fristedt.  The Structure of Random Partitions of Large Integers.  _Trans. Amer. Math. Soc._ , 337(2):703–735, 1993\.  ISSN 0002-9947.  doi: 10.2307/2154239.  URL https://doi.org/10.2307/2154239.
  * Gamkrelidze [1988] N. G. Gamkrelidze.  Application of a Smoothness Function in the Proof of a Local Limit Theorem.  _Teor. Veroyatnost. i Primenen._ , 33(2):373–376, 1988.  ISSN 0040-361X.  doi: 10.1137/1133053.  URL https://doi.org/10.1137/1133053.
  * Graham et al. [1994] R. L. Graham, D. E. Knuth, and O. Patashnik.  _Concrete Mathematics_.  Addison-Wesley Publishing Company, Reading, MA, second edition, 1994.  ISBN 0-201-55802-5.  A Foundation for Computer Science.
  * Granovsky and Stark [2006] B. L. Granovsky and D. Stark.  Asymptotic Enumeration and Logical Limit Laws for Expansive Multisets and Selections.  _J. London Math. Soc. (2)_ , 73(1):252–272, 2006\.  ISSN 0024-6107.  doi: 10.1112/S0024610705022477.  URL https://doi.org/10.1112/S0024610705022477.
  * Granovsky and Stark [2015] B. L. Granovsky and D. Stark.  Developments in the Khintchine-Meinardus Probabilistic Method for Asymptotic Enumeration.  _Electron. J. Combin._ , 22(4):Paper 4.32, 26, 2015.  URL https://doi.org/10.37236/4581.
  * Granovsky et al. [2008] B. L. Granovsky, D. Stark, and M. Erlihson.  Meinardus’ Theorem on Weighted Partitions: Extensions and a Probabilistic Proof.  _Adv. in Appl. Math._ , 41(3):307–328, 2008.  ISSN 0196-8858.  doi: 10.1016/j.aam.2007.11.001.  URL https://doi.org/10.1016/j.aam.2007.11.001.
  * Hardy and Ramanujan [1918] G. H. Hardy and S. Ramanujan.  Asymptotic Formulae in Combinatory Analysis.  _Proc. London Math. Soc. (2)_ , 17:75–115, 1918.  ISSN 0024-6115.  doi: 10.1112/plms/s2-17.1.75.  URL https://doi.org/10.1112/plms/s2-17.1.75.
  * Hwang [1997] H.-K. Hwang.  Distribution of Integer Partitions with Large Number of Summands.  _Acta Arith._ , 78(4):351–365, 1997.  ISSN 0065-1036.  doi: 10.4064/aa-78-4-351-365.  URL https://doi.org/10.4064/aa-78-4-351-365.
  * Karamata [1930] J. Karamata.  Sur un Mode de Croissance Régulière des fonctions.  _Mathematica (Cluj)_ , 4:38–53, 1930.
  * Karamata [1931] J. Karamata.  Neuer Beweis und Verallgemeinerung der Tauberschen Sätze, welche die Laplacesche und Stieltjessche Transformation betreffen.  _J. Reine Angew. Math._ , 164:27–39, 1931.  ISSN 0075-4102.  doi: 10.1515/crll.1931.164.27.  URL https://doi.org/10.1515/crll.1931.164.27.
  * Karamata [1933] J. Karamata.  Sur un Mode de Croissance Régulière. Théorèmes Fondamentaux.  _Bull. Soc. Math. France_ , 61:55–62, 1933.  ISSN 0037-9484.  URL http://www.numdam.org/item?id=BSMF_1933__61__55_0.
  * Khinchin [1960] A. Y. Khinchin.  _Mathematical Foundations of Quantum Statistics_.  Graylock Press, Albany, N.Y., 1960.  Translation from the first (1951) Russian ed., edited by Irwin Shapiro.
  * Knessl and Keller [1990] C. Knessl and J. B. Keller.  Partition Asymptotics from Recursion Equations.  _SIAM J. Appl. Math._ , 50(2):323–338, 1990.  ISSN 0036-1399.  doi: 10.1137/0150020.  URL https://doi.org/10.1137/0150020.
  * Korevaar [2004] J. Korevaar.  _Tauberian Theory, A Century of Developments_ , volume 329 of _Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]_.  Springer-Verlag, Berlin, 2004.  ISBN 3-540-21058-X.  doi: 10.1007/978-3-662-10225-1.  URL https://doi.org/10.1007/978-3-662-10225-1.
  * Korevaar et al. [1949] J. Korevaar, T. van Aardenne-Ehrenfest, and N. G. de Bruijn.  A Note on Slowly Oscillating Functions.  _Nieuw Arch. Wiskunde (2)_ , 23:77–86, 1949.
  * Leroux et al. [1997] P. Leroux, F. Bergeron, and G. Labelle.  _Combinatorial Species and Tree-Like Structures_ , volume 67 of _Encyclopedia of Mathematics and its Applications_.  Cambridge University Press, Cambridge, 1997.  URL http://www.bibsonomy.org/bibtex/216d18979a45d1a17ca0db4e5f58d3f47/brouder.
  * Meinardus [1954] G. Meinardus.  Asymptotische Aussagen über Partitionen.  _Math. Z._ , 59:388–398, 1954.  ISSN 0025-5874.  doi: 10.1007/BF01180268.  URL https://doi.org/10.1007/BF01180268.
  * Miller [2006] P. D. Miller.  _Applied asymptotic analysis_ , volume 75.  American Mathematical Soc., 2006.
  * Mutafchiev [2011] L. Mutafchiev.  Limit Theorems for the Number of Parts in a Random Weighted Partition.  _Electron. J. Combin._ , 18(1):Paper 206, 27, 2011\.  ISSN 1077-8926.  URL https://doi.org/10.37236/693.
  * Mutafchiev [2013] L. Mutafchiev.  The Size of the Largest Part of Random Weighted Partitions of Large Integers.  _Combin. Probab. Comput._ , 22(3):433–454, 2013\.  ISSN 0963-5483.  doi: 10.1017/S0963548313000047.  URL https://doi.org/10.1017/S0963548313000047.
  * Panagiotou and Ramzews [2021] K. Panagiotou and L. Ramzews.  Asymptotic Enumeration and Limit Laws for Multisets: the Subexponential Case.  _arXiv preprint_ , 2021.  arxiv: 2007.08274.
  * Panagiotou and Ramzews [2022+] K. Panagiotou and L. Ramzews.  Expansive Multisets: Limit Laws.  _Manuscript_ , 2022+.
  * Panagiotou and Sinha [2012] K. Panagiotou and M. Sinha.  Vertices of Degree $k$ in Random Unlabeled Trees.  _J. Graph Theory_ , 69:114–130, 2012.  ISSN 0364-9024.  doi: 10.1002/jgt.20567.  URL https://doi.org/10.1002/jgt.20567.
  * Panagiotou and Stufler [2018] K. Panagiotou and B. Stufler.  Scaling Limits of Random Pólya Trees.  _Probability Theory and Related Fields_ , 170(3):801–820, 2018.  URL https://doi.org/10.1007/s00440-017-0770-4.
  * Pemantle and Wilson [2013] R. Pemantle and M. C. Wilson.  _Analytic Combinatorics in Several Variables_ , volume 140 of _Cambridge Studies in Advanced Mathematics_.  Cambridge University Press, Cambridge, 2013.  ISBN 978-1-107-03157-9.  doi: 10.1017/CBO9781139381864.  URL https://doi.org/10.1017/CBO9781139381864.
  * Stark [2021] D. Stark.  The Asymptotic Number of Weighted Partitions with a Given Number of Parts.  _The Ramanujan Journal_ , 2021.  doi: 10.1007/s11139-020-00350-2.  URL https://doi.org/10.1007/s11139-020-00350-2.
  * Stufler [2020] B. Stufler.  Unlabelled Gibbs Partitions.  _Combin. Probab. Comput._ , 29(2):293–309, 2020\.  ISSN 0963-5483.  doi: 10.1017/s0963548319000336.  URL https://doi.org/10.1017/s0963548319000336.

##  Appendix A Appendix: Slowly Varying Functions

Let $h:[1,\infty)\to(0,\infty)$ be a slowly varying function, that is, $h$ is
measurable and for any $\lambda>0$

$\displaystyle\lim_{x\to\infty}\frac{h(\lambda x)}{h(x)}=1.$ (A.1)

“Slowly varying” means essentially smaller than any polynomial, see also below
in (A.2) for formal variants of this statement. All the results in this
section for slowly varying strictly positive $h$ do straightforwardly hold for
$h[1,\infty)\to[0,\infty)$ such that $h$ is eventually positive and (A.1) is
valid.

##### Results on Slowly Varying Functions

The theory presented in this chapter goes back to Jovan Karamata, who proved
all the basic results in his works [25, 26, 27]. A thorough overview can be
found in the comprehensive textbooks [6] or [30, Chapter IV]. Let us begin
with the _Uniform Convergence Theorem_ that will be useful later.

######  Theorem A.1 ([6, Thm. 1.2.1]).

The convergence in (A.1) is uniform for $\lambda$ in any compact subset of
$(0,\infty)$.

Let us continue with the famous _Representation Theorem_ , first obtained by
[27] in the continuous setting and again for arbitrary measurable functions by
[31], c.f. [6, Theorem 1.3.1]. It states that there exist bounded measurable
functions $c(x)$ and $\varepsilon(x)$ such that

$h(x)=c(x)\exp\left(\int_{1}^{x}\frac{\varepsilon(t)}{t}dt\right),$

where, for some $c>0$,

$c(x)\to c\qquad\text{and}\qquad\varepsilon(x)\to 0\quad\text{ as
}x\to\infty.$

From this we immediately obtain for any $\delta>0$ that there is an ${x}_{n}$
such that

$x^{-\delta}\leq h(x)\leq
x^{\delta}\quad\text{and}\quad\Big{(}\frac{x}{x^{\prime}}\Big{)}^{\delta}\leq\frac{h(x^{\prime})}{h(x)}\leq\Big{(}\frac{x^{\prime}}{x}\Big{)}^{\delta}\quad\text{
for all }x^{\prime}\geq x\geq{x}_{n}.$ (A.2)

Moreover, see [6, Theorem 1.5.3], we obtain that for any $\mu>0$

$\sup_{1\leq y\leq x}h(y)y^{\mu}\sim
h(x)x^{\mu}\quad\text{and}\quad\sup_{y\geq x}h(y)y^{-\mu}\sim
h(x)x^{-\mu}\qquad\text{ as }x\to\infty.$ (A.3)

Let $\alpha>0$. From here we consider the function

$c(s)=h(s)s^{\alpha-1},\quad s\geq 1,$

where $h$ is continuous. Note that this is no restriction in our setting as
$c_{n}$ given in (1.2) is only defined for natural numbers, such that we can
simply interpolate linearly to obtain continuity. We proceed with the
following important result, known as _Karamata’s Theorem_.

######  Theorem A.2 ([6, Prop. 1.5.8]).

Let $h$ be slowly varying and $\alpha>0$. Then for any $a\geq 1$

$\int_{a}^{x}c(t)\textrm{d}t\sim\alpha^{-1}\,h(x)\,x^{\alpha},\quad\text{as
}x\to\infty.$

We will be interested in sums rather than integrals. A simple trick and the
“sub-polynomiality” of slowly varying functions will help us here. We doubt
that the following statement was not known before, but we know of no
reference.

######  Corollary A.3.

The previous theorem holds with $\int_{a}^{x}$ replaced by $\sum_{a}^{x-1}$
for $a\in\mathbb{N}$.

###### Proof.

The bounds in (A.2) guarantee that $c(s)^{-1}\sup_{0\leq x\leq
1}c(s+x)\leq\sup_{0\leq x\leq 1}(1-x/s)^{\alpha}=1$ and also
$c(x)^{-1}\inf_{0\leq x\leq 1}c(s+x)\geq\inf_{0\leq x\leq 1}(1+x/s)^{-1}\sim
1-x/s$ for $s\to\infty$. Hence for any $\varepsilon>0$ there is
$s_{0}\in\mathbb{N}$ such that

$\displaystyle\bigg{\lvert}\sup_{0\leq x\leq 1}c(s+x)-\inf_{0\leq x\leq
1}c(s+x)\bigg{\rvert}\leq c(s)\varepsilon\leavevmode\nobreak\ \text{ for all
}\leavevmode\nobreak\ s\in\mathbb{N},s\geq s_{0}.$ (A.4)

This helps us in proving the claimed statement as follows. Note that

$\displaystyle\left|\sum_{s=s_{0}}^{x-1}c(s)-\int_{s_{0}}^{x}c(t)dt\right|\leq\sum_{s=s_{0}}^{x-1}\left|c(s)-\int_{s}^{s+1}c(t)dt\right|\leq\sum_{s=s_{0}}^{x-1}\bigg{|}\sup_{0\leq
x\leq 1}c(s+x)-\inf_{0\leq x\leq 1}c(s+x)\bigg{|}.$

By using (A.4) we infer that this sum is at most
$\varepsilon\sum_{s=s_{0}}^{x-1}c(s)$. Hence

$\left\lvert
1-\frac{\sum_{a}^{x-1}c(s)}{\int_{a}^{x}c(t)dt}\right\rvert\sim\left\lvert
1-\frac{\sum_{s_{0}}^{x-1}c(s)}{\int_{s_{0}}^{x}c(t)dt}\right\rvert\leq\varepsilon.$

As $\varepsilon>0$ was arbitrary, the proof is completed. ∎

Another ingredient in our proofs is the following property of sums, where the
terms depend on some slowly varying function. Let $U$ be a non-decreasing
right-continuous function on $\mathbb{R}$ such that $U(x)=0$ for $x<0$.
Consider the Laplace-Stieltjes transform

$\hat{U}(\chi_{n})=\int_{0}^{\infty}\mathrm{e}^{-\chi x}\textrm{d}U(x).$

If $U$ is a step function with jumps at the integers, that is, $U(s)=U(\lfloor
s\rfloor)$ for all $s\in\mathbb{R}$, then

$\hat{U}(\chi_{n})=\sum_{s\geq 0}U(s)\mathrm{e}^{-\chi_{n}s}.$ (A.5)

The next result is referenced to as _Karamata’s Tauberian Theorem_ and was
derived in [26].

######  Theorem A.4 ([6, Thm. 1.7.1]).

Let $\alpha\geq 0,h$ slowly varying and $c>0$. Then the following statements
are equivalent.

  1. 1.

$U(x)\sim\frac{c}{\Gamma(\alpha+1)}\,h(x)\,x^{\alpha}$ as $x\to\infty$.

  2. 2.

$\hat{U}(\chi)\sim c\,h(\chi^{-1})\,\chi^{-(\alpha+1)}$ as $\chi\to 0$.