# A compound Poisson perspective of Ewens-Pitman sampling model

Emanuele Dolera<EMAIL_ADDRESS>Department of Mathematics, University
of Pavia, Italy Stefano Favaro<EMAIL_ADDRESS>Department of Economics
and Statistics, University of Torino and Collegio Carlo Alberto, Italy

###### Abstract

The Ewens-Pitman sampling model (EP-SM) is a distribution for random
partitions of the set $\\{1,\ldots,n\\}$, with $n\in\mathbb{N}$, which is
index by real parameters $\alpha$ and $\theta$ such that either
$\alpha\in[0,1)$ and $\theta>-\alpha$, or $\alpha<0$ and $\theta=-m\alpha$ for
some $m\in\mathbb{N}$. For $\alpha=0$ the EP-SM reduces to the celebrated
Ewens sampling model (E-SM), which admits a well-known compound Poisson
perspective in terms of the log-series compound Poisson sampling model (LS-
CPSM). In this paper, we consider a generalization of the LS-CPSM, which is
referred to as the negative Binomial compound Poisson sampling model (NB-
CPSM), and we show that it leads to extend the compound Poisson perspective of
the E-SM to the more general EP-SM for either $\alpha\in(0,1)$, or $\alpha<0$.
The interplay between the NB-CPSM and the EP-SM is then applied to the study
of the large $n$ asymptotic behaviour of the number of blocks in the
corresponding random partitions, leading to a new proof of Pitman’s $\alpha$
diversity. We discuss the proposed results, and conjecture that analogous
compound Poisson representations may hold for the class of $\alpha$-stable
Poisson-Kingman sampling models, of which the EP-SM is a noteworthy special
case.

Keywords: Berry-Esseen type theorem; Ewens-Pitman sampling model; exchangeable
random partitions; Log-series compound Poisson sampling model; Mittag-Leffler
distribution function; negative Binomial compound Poisson sampling model;
Pitman’s $\alpha$-diversity; Wright distribution function

##  1 Introduction

The Pitman-Yor process is a discrete random probability measure indexed by
real parameters $\alpha$ and $\theta$ such that either $\alpha\in[0,1)$ and
$\theta>-\alpha$, or $\alpha<0$ and $\theta=-m\alpha$ for some
$m\in\mathbb{N}$. See, e.g., the works of Perman et al. (1992), Pitman (1995)
and Pitman and Yor (1997). Let $\\{V_{i}\\}_{i\geq 1}$ be independent random
variables such that $V_{i}$ is distributed as a Beta distribution with
parameter $(1-\alpha,\theta+i\alpha)$, for $i\geq 1$, with the convention for
$\alpha<0$ that $V_{m}=1$ and $V_{i}$ is undefined for $i>m$. If
$P_{1}:=V_{1}$ and $P_{i}:=V_{i}\prod_{1\leq j\leq i-1}(1-V_{j})$ for $i\geq
2$, such that $\sum_{i\geq 1}P_{i}=1$ almost surely, then the Pitman-Yor
process is the random probability measure
$\tilde{\mathfrak{p}}_{\alpha,\theta}$ on $(\mathbb{N},2^{\mathbb{N}})$ such
that $\tilde{\mathfrak{p}}_{\alpha,\theta}(\\{i\\})=P_{i}$ for $i\geq 1$. The
Dirichlet process (Ferguson, 1973) arises for $\alpha=0$. Because of the
discreteness of $\tilde{\mathfrak{p}}_{\alpha,\theta}$, a random sample
$(X_{1},\ldots,X_{n})$ induces a random partition $\Pi_{n}$ of
$\\{1,\ldots,n\\}$ by means of the equivalence $i\sim j\iff X_{i}=X_{j}$
(Pitman, 2006). Let $K_{n}(\alpha,\theta):=K_{n}(X_{1},\ldots,X_{n})\leq n$ be
the number of blocks of $\Pi_{n}$ and let
$M_{r,n}(\alpha,\theta):=M_{r,n}(X_{1},\ldots,X_{n})$, for $r=1,\ldots,n$, be
the number of blocks with frequency $r$ of $\Pi_{n}$ with $\sum_{1\leq r\leq
n}M_{r,n}=K_{n}$ and $\sum_{1\leq r\leq n}rM_{r,n}=n$. Pitman (1995) showed
that

$\displaystyle\text{Pr}[(M_{1,n}(\alpha,\theta),\ldots,M_{n,n}(\alpha,\theta))=(x_{1},\ldots,x_{n})]=n!\frac{\left(\frac{\theta}{\alpha}\right)_{(\sum_{i=1}^{n}x_{i})}}{(\theta)_{(n)}}\prod_{i=1}^{n}\frac{\left(\frac{\alpha(1-\alpha)_{(i-1)}}{i!}\right)^{x_{i}}}{x_{i}!},$
(1)

with $(x)_{(n)}$ being the ascending factorial of $x$ of order $n$, i.e.
$(x)_{(n)}:=\prod_{0\leq i\leq n-1}(x+i)$. The distribution (1) is referred to
as Ewens-Pitman sampling model (EP-SM), and for $\alpha=0$ it reduces to the
celebrated Ewens sampling model (E-SM) introduced by Ewens (1972). The Pitman-
Yor process plays a critical role in a variety of research areas, such as
mathematical population genetics, Bayesian nonparametric statistics,
statistical machine learning, excursion theory, combinatorics and statistical
physics. See Pitman (2006) and Crane (2016) for a comprehensive treatment of
this subject.

The E-SM admits a well-known compound Poisson perspective in terms of the log-
series compound Poisson sampling model (LS-CPSM). See Charalambides (2007),
and references therein, for an overview on compound Poisson models and
generalizations thereof. In particular, we consider a generic population of
individuals with a random number $K$ of distinct types, and let $K$ be
distributed according to a Poisson distribution with parameter
$\lambda=-z\log(1-q)$ for $q\in(0,1)$ and $z>0$. For $i\in\mathbb{N}$ let
$N_{i}$ denote the random number of individuals of type $i$ in the population,
and let the $N_{i}$’s to be independent of $K$ and independent each other,
with the same distribution of the form

$\text{Pr}[N_{1}=x]=-\frac{1}{x\log(1-q)}q^{x}$ (2)

for $x\in\mathbb{N}$. Let $S=\sum_{1\leq i\leq K}N_{i}$ and let
$M_{r}=\sum_{1\leq i\leq K}\mathbbm{1}_{\\{N_{i}=r\\}}$ for $r=1,\ldots,S$,
that is $M_{r}$ is the random number of $N_{i}$’s equal to $r$ such that
$\sum_{r\geq 1}M_{r}=K$ and $\sum_{r\geq 1}rM_{r}=S$. If
$(M_{1}(z,n),\ldots,M_{n}(z,n))$ denotes a random variable whose distribution
coincides with the conditional distribution of $(M_{1},\ldots,M_{S})$ given
$S=n$, then (Charalambides, 2007, Section 3) it holds that

$\displaystyle\text{Pr}[(M_{1}(z,n),\ldots,M_{n}(z,n))=(x_{1},\ldots,x_{n})]=\frac{n!}{(z)_{(n)}}\prod_{i=1}^{n}\frac{\left(\frac{z}{i}\right)^{x_{i}}}{x_{i}!}.$
(3)

The distribution (3) is referred to as the LS-CPSM, and it equivalent to the
E-SM. That is, the distribution displayed in (3) coincides with the
distribution (1) with $\alpha=0$. Therefore, the distributions of
$K(z,n)=\sum_{1\leq r\leq n}M_{r}(z,n)$ and $M_{r}(z,n)$ coincide with the
distributions of $K_{n}(0,z)$ and $M_{r,n}(0,z)$, respectively. Let
$\stackrel{{\scriptstyle w}}{{\longrightarrow}}$ denote the weak convergence
for random variables. From the work of Korwar and Hollander (1973),
$K(z,n)/\log n\stackrel{{\scriptstyle w}}{{\longrightarrow}}z$ as
$n\rightarrow+\infty$, whereas from Ewens (1972) it follows that
$M_{r}(z,n)\stackrel{{\scriptstyle w}}{{\longrightarrow}}P_{z/r}$ as
$n\rightarrow+\infty$, where $P_{z}$ is a Poisson random variable with
parameter $z$.

In this paper, we consider a generalization of the LS-CPSM, which is referred
to as the negative Binomial compound Poisson sampling model (NB-CPSM). In
particular, the NB-CPSM is indexed by a pair of real parameters $\alpha$ and
$z$ such that either $\alpha\in(0,1)$ and $z>0$, or $\alpha<0$ and $z<0$. The
LS-CPSM is recovered by letting $\alpha\rightarrow 0$ and $z>0$. We show that
the NB-CPSM leads to extend the compound Poisson perspective of the E-SM to
the more general EP-SM for either $\alpha\in(0,1)$, or $\alpha<0$. That is, we
show that: i) for $\alpha\in(0,1)$ the EP-SM (1) admits a representation as a
randomized NB-CPSM with $\alpha\in(0,1)$ and $z>0$, where the randomization
acts on $z$ with respect a scale mixture between a Gamma and a scaled Mittag-
Leffler distribution (Pitman, 2006); ii) for $\alpha<0$ the NB-CPSM admits a
representation in terms of a randomized EP-SM with $\alpha<0$ and
$\theta=-m\alpha$ for some $m\in\mathbb{N}$, where the randomization acts on
$m$ with respect to a tilted Poisson distribution arising from the Wright
function (Wright, 1935). The interplay between the NB-CPSM and the EP-SM is
then applied to the study of the large $n$ asymptotic behaviour of the number
of distinct blocks in the random partitions induced by the corresponding
sampling models. In particular, by combining the randomized representation in
i) with the large $n$ asymptotic behaviour or the number of distinct blocks
under the NB-CPSM, we present a new proof of Pitman’s $\alpha$-diversity
(Pitman, 2006), namely the large $n$ asymptotic behaviour of
$K_{n}(\alpha,\theta)$ under the EP-SM.

##  2 A compound Poisson perspective of EP-SM

We start by introducing the NB-CPSM and investigating the large $n$ asymptotic
behaviour of some statistics of its induced random partition. To introduce the
NB-CPSM, we consider a generic population of individuals with a random number
$K$ of types, and let $K$ be distributed as a Poisson distribution with
parameter $\lambda=z[1-(1-q)^{\alpha}]$ such that either $q\in(0,1)$,
$\alpha\in(0,1)$ and $z>0$, or $q\in(0,1)$, $\alpha<0$ and $z<0$. For
$i\in\mathbb{N}$ let $N_{i}$ be the random number of individuals of type $i$
in the population, and let the $N_{i}$’s to be independent of $K$ and
independent each other, with the same distribution

$\text{Pr}[N_{1}=x]=-\frac{1}{[1-(1-q)^{\alpha}]}{\alpha\choose x}(-q)^{x}$
(4)

for $x\in\mathbb{N}$. Let $S=\sum_{1\leq i\leq K}N_{i}$ and $M_{r}=\sum_{1\leq
i\leq K}\mathbbm{1}_{\\{N_{i}=r\\}}$ for $r=1,\ldots,S$, that is $M_{r}$ is
the random number of $N_{i}$’s equal to $r$ such that $\sum_{r\geq 1}M_{r}=K$
and $\sum_{r\geq 1}rM_{r}=S$. If
$(M_{1}(\alpha,z,n),\ldots,M_{n}(\alpha,z,n))$ is a random variable whose
distribution coincides with the conditional distribution of
$(M_{1},\ldots,M_{S})$ given $S=n$, then it holds (Charalambides, 2007,
Section 3) that

$\displaystyle\text{Pr}[(M_{1}(\alpha,z,n),\ldots,M_{n}(\alpha,z,n))=(x_{1},\ldots,x_{n})]=\frac{n!}{\sum_{j=0}^{n}\mathscr{C}(n,j;\alpha)z^{j}}\prod_{i=1}^{n}\frac{\left[z\frac{\alpha(1-\alpha)_{(i-1)}}{i!}\right]^{x_{i}}}{x_{i}!},$
(5)

where $\mathscr{C}(n,j;\alpha)=\frac{1}{j!}\sum_{0\leq i\leq j}{j\choose
i}(-1)^{i}(-i\alpha)_{(n)}$ is the generalized factorial coefficient
(Charalambides, 2005), with the proviso $\mathscr{C}(n,0,\alpha)=0$ for all
$n\in\mathbb{N}$, $\mathscr{C}(n,j,\alpha)=0$ for all $j>n$ and
$\mathscr{C}(0,0,\alpha)=1$. The distribution (5) is referred to as the NB-
CPSM. In particular, as $\alpha\rightarrow 0$, it is easy to show that the
distribution (4) reduces to the distribution (2). Accordingly, as
$\alpha\rightarrow 0$, the NB-CPSM (5) reduces to the LS-CPSM (3). The next
theorem states the large $n$ asymptotic behaviour of the counting statistics
$K(\alpha,z,n)=\sum_{1\leq r\leq n}M_{r}(\alpha,z,n)$ and $M_{r}(\alpha,z,n)$
arising from the NB-CPSM.

######  Theorem 1.

Let $P_{\lambda}$ denote a Poisson random variable with parameter $\lambda>0$.
As $n\rightarrow+\infty$ it holds

  * i)

for $\alpha\in(0,1)$ and $z>0$

$K(\alpha,z,n)\stackrel{{\scriptstyle w}}{{\longrightarrow}}1+P_{z}$ (6)

and

$M_{r}(\alpha,z,n)\stackrel{{\scriptstyle
w}}{{\longrightarrow}}P_{\frac{\alpha(1-\alpha)_{(r-1)}}{r!}z};$ (7)

  * ii)

for $\alpha<0$ and $z<0$

$\frac{K(\alpha,z,n)}{n^{\frac{-\alpha}{1-\alpha}}}\stackrel{{\scriptstyle
w}}{{\longrightarrow}}\frac{(\alpha z)^{\frac{1}{1-\alpha}}}{-\alpha}$ (8)

and

$M_{r}(\alpha,z,n)\stackrel{{\scriptstyle
w}}{{\longrightarrow}}P_{\frac{\alpha(1-\alpha)_{(r-1)}}{r!}z}.$ (9)

###### Proof.

As regard the proof of the asymptotic behaviour in (6), we start by recalling
that the probability generating function $G(\cdot;\lambda)$ of $P_{\lambda}$
is $G(s;\lambda)=\exp\\{-\lambda(s-1)\\}$ for any $s>0$. Now, let
$G(\cdot;\alpha,z,n)$ be the probability generating function of
$K(\alpha,z,n)$. The distribution of $K(\alpha,z,n)$ follows by combining the
NB-CPSM (5) with Theorem 2.15 of Charalambides (2005). In particular, it
follows that

$G(s;\alpha,z,n)=\frac{\sum_{j=1}^{n}\mathscr{C}(n,j;\alpha)(sz)^{j}}{\sum_{j=1}^{n}\mathscr{C}(n,j;\alpha)z^{j}}.$

Hereafter, we show that $G(s;\alpha,z,n)\rightarrow s\exp\\{z(s-1)\\}$ as
$n\rightarrow+\infty$, for any $s>0$, which implies (6). In particular, by a
direct application of the definition of $\mathscr{C}(n,k;\alpha)$ we write the
following identities

$\displaystyle\sum_{j=1}^{n}\mathscr{C}(n,j;\alpha)z^{j}$
$\displaystyle=\sum_{i=1}^{n}(-1)^{i}(-i\alpha)_{(n)}\sum_{k=i}^{n}\frac{1}{k!}{k\choose
i}z^{k}=\sum_{i=1}^{n}(-1)^{i}(-i\alpha)_{(n)}e^{z}z^{i}\frac{\Gamma(n-i+1,z)}{i!\Gamma(n-i+1)},$

where $\Gamma(a,x):=\int_{x}^{+\infty}t^{a-1}e^{-t}\text{d}t$ denotes the
incomplete gamma function for $a,x>0$ and
$\Gamma(a):=\int_{0}^{+\infty}t^{a-1}e^{-t}\text{d}t$ denotes the Gamma
function for $a>0$. Accordingly, we can write the following identity

$G(s;\alpha,z,n)=e^{z(s-1)}\frac{-zs\frac{\Gamma(n,zs)}{\Gamma(n)}+\sum_{i=2}^{n}(-1)^{i}\frac{(-i\alpha)_{(n)}}{(-\alpha)_{(n)}}(zs)^{i}\frac{\Gamma(n-i+1,zs)}{i!\Gamma(n-i+1)}}{-z\frac{\Gamma(n,z)}{\Gamma(n)}+\sum_{i=2}^{n}(-1)^{i}\frac{(-i\alpha)_{(n)}}{(-\alpha)_{(n)}}z^{i}\frac{\Gamma(n-i+1,z)}{i!\Gamma(n-i+1)}}\
.$

Now, since $\lim_{n\rightarrow+\infty}\frac{\Gamma(n,x)}{\Gamma(n)}=1$ for any
$x>0$, the proof (6) is completed by showing that, for any $t>0$,

$\lim_{n\rightarrow+\infty}\sum_{i=2}^{n}(-1)^{i}\frac{(-i\alpha)_{(n)}}{(-\alpha)_{(n)}}\frac{\Gamma(n-i+1,t)}{\Gamma(n-i+1)}\frac{t^{i}}{i!}=0.$
(10)

By the definition of ascending factorials and the reflection formula of the
Gamma function, it holds true

$\frac{(-i\alpha)_{(n)}}{(-\alpha)_{(n)}}=\frac{\Gamma(n-i\alpha)}{\Gamma(n-\alpha)}\frac{\sin
i\pi\alpha}{\pi}\Gamma(i\alpha+1)\Gamma(-\alpha).$

In particular, by means of the monotonicity of the function $[1,+\infty)\ni
z\mapsto\Gamma(z)$, we can write the identity

$\frac{1}{i!}\Big{|}\frac{(-i\alpha)_{(n)}}{(-\alpha)_{(n)}}\Big{|}\leq\frac{|\Gamma(-\alpha)|}{\pi}\frac{\Gamma(n-2\alpha)}{\Gamma(n-\alpha)}\frac{\Gamma(i\alpha+1)}{i!}$
(11)

for any $n\in\mathbb{N}$ such that $n>1/(1-\alpha)$, and
$i\in\\{2,\dots,n\\}$. Note that $\frac{\Gamma(n,x)}{\Gamma(n)}\leq 1$. Then
we apply (11) to get

$\displaystyle\Big{|}\sum_{i=2}^{n}(-1)^{i}\frac{(-i\alpha)_{(n)}}{(-\alpha)_{(n)}}\frac{\Gamma(n-i+1,t)}{\Gamma(n-i+1)}\frac{t^{i}}{i!}\Big{|}$
$\displaystyle\leq\sum_{i=2}^{n}\frac{t^{i}}{i!}\Big{|}\frac{(-i\alpha)_{(n)}}{(-\alpha)_{(n)}}\Big{|}$
$\displaystyle\leq\frac{|\Gamma(-\alpha)|}{\pi}\frac{\Gamma(n-2\alpha)}{\Gamma(n-\alpha)}\sum_{i\geq
0}t^{i}\frac{\Gamma(i\alpha+1)}{i!}.$

Now, by means of Stirling approximation it holds
$\frac{\Gamma(n-2\alpha)}{\Gamma(n-\alpha)}\sim\frac{1}{n^{\alpha}}$ as
$n\rightarrow+\infty$. Moreover, we have that

$\sum_{i\geq
0}t^{i}\frac{\Gamma(i\alpha+1)}{i!}=\int_{0}^{+\infty}e^{tz^{\alpha}-z}\text{d}z<+\infty$

where the finiteness of the integral follows, for any fixed $t>0$, from the
fact that $tz^{\alpha}<\frac{1}{2}z$ if $z>(2t)^{\frac{1}{1-\alpha}}$. This
completes the proof of (10), and hence the proof of (6). As regard the proof
of (7), we make use of the falling factorial moments of $M_{r}(\alpha,z,n)$,
which follows by combining the NB-CPSM (5) with Theorem 2.15 of Charalambides
(2005). Let $(a)_{[n]}$ be the falling factorial of $a$ of order $n$, i.e.
$(a)_{[n]}=\prod_{0\leq i\leq n-1}(a-i)$, for any $a\in\mathbb{R}^{+}$ and
$n\in\mathbb{N}_{0}$ with the proviso $(a)_{[0]}=1$. Then, we write

$\displaystyle\mathbb{E}[(M_{r}(\alpha,z,n))_{[s]}]$
$\displaystyle\quad=(-1)^{rs}(n)_{[rs]}{\alpha\choose
r}^{s}(-z)^{s}\frac{\sum_{j=0}^{n-rs}\mathscr{C}(n-rs,j;\alpha)z^{j}}{\sum_{j=0}^{n}\mathscr{C}(n,j;\alpha)z^{j}}$
$\displaystyle\quad=(-1)^{rs}(n)_{[rs]}{\alpha\choose
r}^{s}(-z)^{s}\frac{(-z)\frac{\Gamma(n-rs,z)}{\Gamma(n-rs)}+\sum_{i=2}^{n-rs}(-1)^{i}\frac{(-i\alpha)_{(n-rs)}}{(-\alpha)_{(n-rs)}}(z)^{i}\frac{\Gamma(n-rs-i+1,z)}{i!\Gamma(n-rs-i+1)}}{(-z)\frac{\Gamma(n,z)}{\Gamma(n)}+\sum_{i=2}^{n}(-1)^{i}\frac{(-i\alpha)_{(n)}}{(-\alpha)_{(n)}}(z)^{i}\frac{\Gamma(n-i+1,z)}{\Gamma(n-i+1)}}$
$\displaystyle\quad=(-1)^{rs}(n)_{[rs]}{\alpha\choose r}^{s}(-z)^{s}$
$\displaystyle\quad\quad\times\frac{(-\alpha)_{(n-rs)}}{(-\alpha)_{(n)}}\frac{(-z)\frac{\Gamma(n-rs,z)}{\Gamma(n-rs)}+\sum_{i=2}^{n-rs}(-1)^{i}\frac{(-i\alpha)_{(n-rs)}}{(-\alpha)_{(n-lr)}}(z)^{i}\frac{\Gamma(n-rs-i+1,z)}{i!\Gamma(n-rs-i+1)}}{(-z)\frac{\Gamma(n,z)}{\Gamma(n)}+\sum_{i=2}^{n}(-1)^{i}\frac{(-i\alpha)_{(n)}}{(-\alpha)_{(n)}}(z)^{i}\frac{\Gamma(n-i+1,z)}{\Gamma(n-i+1)}}.$

Now, by means of the same argument applied in the proof of the statement in
(6), it holds true that

$\lim_{n\rightarrow+\infty}\frac{(-z)\frac{\Gamma(n-rs,z)}{\Gamma(n-rs)}+\sum_{i=2}^{n-rs}(-1)^{i}\frac{(-i\alpha)_{(n-rs)}}{(-\alpha)_{(n-lr)}}(z)^{i}\frac{\Gamma(n-rs-i+1,z)}{i!\Gamma(n-rs-i+1)}}{(-z)\frac{\Gamma(n,z)}{\Gamma(n)}+\sum_{i=2}^{n}(-1)^{i}\frac{(-i\alpha)_{(n)}}{(-\alpha)_{(n)}}(z)^{i}\frac{\Gamma(n-i+1,z)}{\Gamma(n-i+1)}}=1.$

Then,

$\lim_{n\rightarrow+\infty}\mathbb{E}[(M_{r}(\alpha,z,n))_{[s]}]=(-1)^{rs}{\alpha\choose
r}^{s}(-z)^{s}=\left[\frac{\alpha(1-\alpha)_{(r-1)}}{r!}z\right]^{s}$

follows from the fact that
$(n)_{[rs]}\sim\frac{(-\alpha)_{(n-rs)}}{(-\alpha)_{(n)}}$ as
$n\rightarrow+\infty$. Finally, the proof of the large $n$ asymptotics (7) is
completed by recalling that falling factorial moment of order $s$ of
$P_{\lambda}$ is $\mathbb{E}[(P_{\lambda})_{[s]}]=\lambda^{s}$.

As regard the proof of the asymptotic behaviour in (8), let $\alpha=-\sigma$
for any $\sigma>0$ and let $z=-\zeta$ for any $\zeta>0$. Then, by a direct
application of Equation 2.27 of Charalambides (2005), we write the identity

$\sum_{j=0}^{n}\mathscr{C}(n,j;-\sigma)(-\zeta)^{j}=(-1)^{n}\sum_{v=0}^{n}s(n,v)(-\sigma)^{v}\sum_{j=0}^{v}\zeta^{j}S(v,j),$

where $S(v,j)$ is the Stirling number of that second type. Now, note that
$\sum_{0\leq j\leq v}^{v}\zeta^{j}S(v,j)$ is the moment of order $v$ of a
Poisson random variable with parameter $\zeta>0$. Then, we write the following
identities

$\displaystyle\sum_{j=0}^{n}\mathscr{C}(n,j;-\sigma)(-\zeta)^{j}$
$\displaystyle=\sum_{v=0}^{n}|s(n,v)|\sigma^{v}\sum_{j\geq
0}j^{v}\text{e}^{-\zeta}\frac{\zeta^{j}}{j!}=\sum_{j\geq
0}\text{e}^{-\zeta}\frac{\zeta^{j}}{j!}\int_{0}^{+\infty}x^{n}f_{G_{\sigma
j,1}}(x)\text{d}x.$ (12)

That is,

$B_{n}(w)=\mathbb{E}[(G_{\sigma P_{w},1})^{n}],$ (13)

where $G_{a,1}$ and $P_{w}$ denote two independent random variables such that
$G_{a,1}$ is a Gamma random variable with shape parameter $a>0$ and scale
parameter $1$, and $P_{w}$ is a Poisson random variable with parameter $w$.
Accordingly, the distribution of the random variable $G_{\sigma P_{w},1}$, say
$\mu_{\sigma,w}$ is the following

$\mu_{\sigma,w}(\text{d}t)=e^{-w}\delta_{0}(\text{d}t)+\left(\sum_{j\geq
1}\frac{e^{-w}w^{j}}{j!}\frac{1}{\Gamma(j\sigma)}e^{-t}t^{j\sigma-1}\right)\text{d}t$

for $t>0$. The discrete component of the distribution $\mu_{\sigma,w}$ does
not contribute in the expectation (13), so that we focus on the absolutely
continuous component, whose density can be written as follows

$\sum_{j\geq
1}\frac{e^{-w}w^{j}}{j!}\frac{1}{\Gamma(j\sigma)}e^{-t}t^{j\sigma-1}=\frac{e^{-(w+t)}}{t}W_{\sigma,0}(wt^{\sigma}),$

where $W_{\sigma,\tau}(y):=\sum_{j\geq 0}\frac{y^{j}}{j!\Gamma(j\sigma+\tau)}$
is the Wright function (Wright, 1935). In particular, for $\tau=0$ it holds

$B_{n}(w)=\int_{0}^{+\infty}t^{n}\frac{e^{-(w+t)}}{t}W_{\sigma,0}(wt^{\sigma})\text{d}t\
.$ (14)

If we split the integral as $\int_{0}^{M}+\int_{M}^{+\infty}$ for any $M>0$,
the contribution of the latter integral is overwhelming with respect to the
contribution of the former. Then $W_{\sigma,0}$ can be equivalently replaced
by the asymptotics $W_{\sigma,0}(y)\sim
c(\sigma)y^{\frac{1}{2(1+\sigma)}}\exp\\{\sigma^{-1}(\sigma+1)(\sigma
y)^{\frac{1}{1+\sigma}}\\}$, as $y\rightarrow+\infty$, for some constant
$c(\sigma)$ depending solely on $\sigma$. See Theorem 2 in Wright (1935).
Hence, we can write the identity

$\displaystyle B_{n}(w)$ $\displaystyle\sim
c(\sigma)\int_{0}^{+\infty}t^{n-1}e^{-(w+t)}(wt^{\sigma})^{\frac{1}{2(1+\sigma)}}\exp\left\\{\frac{\sigma+1}{\sigma}(\sigma
wt^{\sigma})^{\frac{1}{1+\sigma}}\right\\}\text{d}t$
$\displaystyle=c(\sigma)e^{-w}w^{\frac{1}{2(1+\sigma)}}\int_{0}^{+\infty}t^{n+\frac{\sigma}{2(1+\sigma)}-1}\exp\\{A(w,\sigma)t^{\frac{\sigma}{1+\sigma}}-t\\}\text{d}t,$

where $A(w,\sigma):=\frac{\sigma+1}{\sigma}(\sigma w)^{\frac{1}{1+\sigma}}$.
Then, the problem is reduced to an integral whose asymptotic behaviour is
described in Berg (1958). From Equation 31 of Berg (1958) and Stirling
approximation, we have

$B_{n}(w)\sim
c(\sigma)e^{-w}w^{\frac{1}{2(1+\sigma)}}\Gamma(n)\exp\left\\{A(w,\sigma)n^{\frac{\sigma}{1+\sigma}}\right\\}\
.$ (15)

In particular, observe that such a last asymptotic expansion leads directly to
(8). Indeed let $G(\cdot;-\sigma,-\zeta,n)$ be the probability generating
function of the random variable $K(-\sigma,-\zeta,n)$, which reads as
$G(s;-\sigma,-\zeta,n)=B_{n}(s\zeta)/B_{n}(\zeta)$ for $s>0$. Then, by means
of (15), for any fixed $s>0$ we write

$G(s;-\sigma,-\zeta,n)\sim
e^{-w(s-1)}s^{\frac{1}{2(1+\sigma)}}\exp\left\\{n^{\frac{\sigma}{1+\sigma}}\frac{\sigma+1}{\sigma}(\sigma\zeta)^{\frac{1}{1+\sigma}}[s^{\frac{1}{1+\sigma}}-1]\right\\}\
.$ (16)

Note that (15) holds uniformly in $w$ in a compact set. Accordingly, we
consider the function $G(s;-\sigma,-\zeta,n)$ evaluated at some point $s_{n}$
and extend the validity of (16) with $s_{n}$ in the place of $s$, as long as
$\\{s_{n}\\}_{n\geq 1}$ varies in a compact subset of $[0,+\infty)$. Thus, we
can choose $s_{n}=s^{\beta(n)}$ and
$\beta(n)=\frac{1}{n^{\frac{\sigma}{1+\sigma}}}$ and notice that
$\beta(n)\rightarrow 0$ as $n\rightarrow+\infty$. Thus, $s_{n}\simeq
1+\beta(n)\log s\rightarrow 1$ and we have that

$n^{\frac{\sigma}{1+\sigma}}\frac{\sigma+1}{\sigma}(\sigma
w)^{\frac{1}{1+\sigma}}[s_{n}^{\frac{1}{1+\sigma}}-1]\rightarrow\frac{(\sigma\zeta)^{\frac{1}{1+\sigma}}}{\sigma}\log
s,$

which implies that
$K(-\sigma,-\zeta,n)\rightarrow\frac{(\sigma\zeta)^{\frac{1}{1+\sigma}}}{\sigma}$
as $n\rightarrow+\infty$. This completes the proof of (8). As regard the proof
(9), let $\alpha=-\sigma$ for any $\sigma>0$ and let $z=-\zeta$ for any
$\zeta>0$. Similarly to the proof of (7), here we make use of the falling
factorial moments of $M_{r}(-\sigma,-\zeta,n)$. In particular, we can write

$\displaystyle\mathbb{E}[(M_{r}(-\sigma,\zeta,n))_{[s]}]=(-1)^{rs}(n)_{[rs]}{-\sigma\choose
r}^{s}\zeta^{s}\frac{\sum_{j=0}^{n-rs}\mathscr{C}(n-rs,j;-\sigma)(-\zeta)^{j}}{\sum_{j=0}^{n}\mathscr{C}(n,j;-\sigma)(-\zeta)^{j}}.$

At this point, we can make use of the same large $n$ asymptotic arguments
applied in the proof of statement (7). In particular, by means of the large
$n$ asymptotic (15), as $n\rightarrow+\infty$, it holds true that

$\frac{\sum_{j=0}^{n-rs}\mathscr{C}(n-rs,j;-\sigma)(-\zeta)^{j}}{\sum_{j=0}^{n}\mathscr{C}(n,j;-\sigma)(-\zeta)^{j}}\sim
n^{-rs}.$

Then,

$\lim_{n\rightarrow+\infty}\mathbb{E}[(M_{r}(-\sigma,-\zeta,n))_{[s]}]=(-1)^{rs}{-\sigma\choose
r}^{s}\zeta^{s}=\left[\frac{-\sigma(1+\sigma)_{(r-1)}}{r!}(-\zeta)\right]^{s}$

follows from the fact that $(n)_{[rs]}\sim n^{rs}$ as $n\rightarrow+\infty$.
Finally, the proof of the large $n$ asymptotic behaviour in (9) is completed
by recalling that falling factorial moment of order $s$ of $P_{\lambda}$ is
$\mathbb{E}[(P_{\lambda})_{[s]}]=\lambda^{s}$. ∎

In the rest of the present section, we make use of the NB-CPSM displayed in
(5) to introduce a compound Poisson perspective of the EP-SM. In particular,
our main result extends the well-known compound Poisson perspective of the
E-SM to the EP-SM for either $\alpha\in(0,1)$, or $\alpha<0$. For
$\alpha\in(0,1)$ let $f_{\alpha}$ denote the density function of a positive
$\alpha$-stable random variable $X_{\alpha}$, that is $X_{\alpha}$ is a random
variable for which the moment generating function is
$\mathbb{E}[\exp\\{-tX_{\alpha}\\}]=\exp\\{-t^{\alpha}\\}$ for any $t>0$. For
$\alpha\in(0,1)$ and $\theta>-\alpha$ let $S_{\alpha,\theta}$ be a positive
random variable with density function

$f_{S_{\alpha,\theta}}(s)=\frac{\Gamma(\theta+1)}{\alpha\Gamma(\theta/\alpha+1)}s^{\frac{\theta-1}{\alpha}-1}f_{\alpha}(s^{-\frac{1}{\alpha}}).$

That is, the random variable $S_{\alpha,\theta}$ is a scaled Mittag-Leffler
random variable (Pitman, 2006, Chapter 1). Now, let $G_{a,b}$ be a Gamma
random variable with scale parameter $b>0$ and shape parameter $a>0$, and let
assume that $G_{a,b}$ is independent of $S_{\alpha,\theta}$. Then, for
$\alpha\in(0,1)$, $\theta>-\alpha$ and $n\in\mathbb{N}$ we define

$\bar{X}_{\alpha,\theta,n}\stackrel{{\scriptstyle
d}}{{=}}G^{\alpha}_{\theta+n,1}S_{\alpha,\theta}.$ (17)

Finally, for $\alpha<0$, $z<0$ and $n\in\mathbb{N}$ let
$\tilde{X}_{\alpha,z,n}$ be a random variable on $\mathbb{N}$ whose
distribution is a tilted Poisson distribution arising from the identity (12).
Precisely, for any $x\in\mathbb{N}$ the distribution of
$\tilde{X}_{\alpha,z,n}$ is

$\text{Pr}[\tilde{X}_{\alpha,z,n}=x]=\frac{1}{\sum_{j=1}^{n}\mathscr{C}(n,j;\alpha)z^{j}}\frac{\text{e}^{z}(-z)^{x}\Gamma(-x\alpha+n)}{x!\Gamma(-x\alpha)}.$
(18)

In the next theorem, we make use of the random variables
$\bar{X}_{\alpha,\theta,n}$ and $\tilde{X}_{\alpha,z,n}$ to set an interplay
between the NB-CPSM (5) and the EP-SM (1). This extends the compound Poisson
perspective of the E-SM.

######  Theorem 2.

Let $(M_{1,n}(\alpha,\theta),\ldots,M_{n,n}(\alpha,\theta))$ be distributed as
the EP-SM (1) and let $\bar{X}_{\alpha,\theta,n}$ be the random variable
defined in (17), which is independent of
$(M_{1,n}(\alpha,\theta),\ldots,M_{n,n}(\alpha,\theta))$. Moreover, let
$(M_{1}(\alpha,z,n),\ldots,M_{n}(\alpha,z,n))$ be distributed as the NB-CPSM
(5), and let $\tilde{X}_{\alpha,z,n}$ be the random variable defined in (18),
which is independent of $(M_{1}(\alpha,z,n),\ldots,M_{n}(\alpha,z,n))$. Then,
it holds true that

  * i)

for $\alpha\in(0,1)$ and $\theta>-\alpha$

$(M_{1,n}(\alpha,\theta),\ldots,M_{n,n}(\alpha,\theta))\stackrel{{\scriptstyle
d}}{{=}}(M_{1}(\alpha,\bar{X}_{\alpha,\theta,n},n),\ldots,M_{n}(\alpha,\bar{X}_{\alpha,\theta,n},n));$

  * ii)

for $\alpha<0$ and $z<0$

$(M_{1}(\alpha,z,n),\ldots,M_{n}(\alpha,z,n))\stackrel{{\scriptstyle
d}}{{=}}(M_{1,n}(\alpha,-\tilde{X}_{\alpha,z,n}\alpha),\ldots,M_{n,n}(\alpha,-\tilde{X}_{\alpha,z,n}\alpha)).$

###### Proof.

As regard the proof of statement i), its proof relies on the classical
integral representation of the Gamma function. That is, by applying the
integral representation of the function $\Gamma(\theta/\alpha+k)$ to the EP-SM
(1), for $x_{1},\ldots,x_{n}\in\\{0,\dots,n\\}$ with $\sum_{i=1}^{n}x_{i}=k$
and $\sum_{i=1}^{n}ix_{i}=n$, we can write that

$\displaystyle\text{Pr}[(M_{1,n}(\alpha,\theta),\ldots,M_{n,n}(\alpha,\theta))=(x_{1},\ldots,x_{n})]$
$\displaystyle\quad=n!\frac{\alpha^{k}}{\Gamma(\theta+n)}\prod_{i=1}^{n}\frac{\left(\frac{(1-\alpha)_{(i-1)}}{i!}\right)^{x_{i}}}{x_{i}!}\frac{\Gamma(\theta+1)}{\alpha\Gamma(\theta/\alpha+1)}$
$\displaystyle\quad\quad\times\int_{0}^{+\infty}z^{\theta/\alpha-1}\text{e}^{-z}\frac{z^{k}}{\sum_{j=1}^{n}\mathscr{C}(n,j;\alpha)z^{j}}\left(\sum_{j=1}^{n}\mathscr{C}(n,j;\alpha)z^{j}\right)\text{d}z$
[By Equation 13 of Favaro et al. (2015)]
$\displaystyle\quad=n!\frac{\alpha^{k}}{\Gamma(\theta+n)}\prod_{i=1}^{n}\frac{\left(\frac{(1-\alpha)_{(i-1)}}{i!}\right)^{x_{i}}}{x_{i}!}\frac{\Gamma(\theta+1)}{\alpha\Gamma(\theta/\alpha+1)}$
$\displaystyle\quad\quad\times\int_{0}^{+\infty}z^{\theta/\alpha-1}\text{e}^{-z}\frac{z^{k}}{\sum_{j=1}^{n}\mathscr{C}(n,j;\alpha)z^{j}}\left(\text{e}^{z}z^{n/\alpha}\int_{0}^{+\infty}y^{n}\text{e}^{-yz^{1/\alpha}}f_{\alpha}(y)\text{d}y\right)\text{d}z$
$\displaystyle\quad=\int_{0}^{+\infty}\frac{n!}{\sum_{j=0}^{n}\mathscr{C}(n,j,\alpha)z^{j}}\prod_{i=1}^{n}\frac{\left(z\frac{\alpha(1-\alpha)_{(i-1)}}{i!}\right)^{x_{i}}}{x_{i}!}$
$\displaystyle\quad\quad\times\frac{\Gamma(\theta+1)}{\alpha\Gamma(\theta+n)\Gamma(\theta/\alpha+1)}z^{\theta/\alpha+n/\alpha-1}\int_{0}^{+\infty}y^{n}\text{e}^{-yz^{1/\alpha}}f_{\alpha}(y)\text{d}y\text{d}z$
$\displaystyle\quad=\int_{0}^{+\infty}\text{Pr}[(M_{1}(\alpha,x,n),\ldots,M_{n}(\alpha,x,n))=(x_{1},\ldots,x_{n})]$
$\displaystyle\quad\quad\times\frac{\Gamma(\theta+1)}{\alpha\Gamma(\theta+n)\Gamma(\theta/\alpha+1)}z^{\theta/\alpha+n/\alpha-1}\int_{0}^{+\infty}y^{n}\text{e}^{-yz^{1/\alpha}}f_{\alpha}(y)\text{d}y\text{d}z$
[By the distribution of $\bar{X}_{\alpha,\theta,n}$]
$\displaystyle\quad=\int_{0}^{+\infty}\text{Pr}[(M_{1}(\alpha,z,n),\ldots,M_{n}(\alpha,z,n))=(x_{1},\ldots,x_{n})]f_{\bar{X}_{\alpha,\theta,n}}(z)\text{d}z,$

where $f_{\bar{X}_{\alpha,\theta,n}}$ is the density function of the random
variable $\bar{X}_{\alpha,\theta,n}$. This completes the proof of i).

As regard the proof of statement ii), for any $\alpha<0$, $m\in\mathbb{N}$,
$k\leq m$ and $n\in\mathbb{N}$ we start by defining the function $m\mapsto
A(m;k,\alpha,n)=\frac{m!}{(m-k)!}\frac{\Gamma(-m\alpha)}{\Gamma(-m\alpha+n)}$,
and then we consider the following identity

$\frac{(-z)^{k}}{\sum_{j=1}^{n}\mathscr{C}(n,j;\alpha)z^{j}}=\sum_{m\geq
k}A(m;k,\alpha,n)\text{Pr}[\tilde{X}_{\alpha,z,n}=m].$ (19)

By applying (19) to the NB-CPSM (5), for
$x_{1},\ldots,x_{n}\in\\{0,\dots,n\\}$ with $\sum_{i=1}^{n}x_{i}=k$ and
$\sum_{i=1}^{n}ix_{i}=n$, we write

$\displaystyle\text{Pr}[(M_{1}(\alpha,z,n),\ldots,M_{n}(\alpha,z,n))=(x_{1},\ldots,x_{n})]$
$\displaystyle\quad=\sum_{m\geq
k}n!(-1)^{k}A(m;k,\alpha,n)\text{Pr}[\tilde{X}_{\alpha,z,n}=m]\prod_{i=1}^{n}\frac{\left(\frac{\alpha(1-\alpha)_{(i-1)}}{i!}\right)^{x_{i}}}{x_{i}!}$
$\displaystyle\quad=\sum_{m\geq
k}n!(-1)^{k}\frac{m!}{(m-k)!}\frac{\Gamma(-m\alpha)}{\Gamma(-m\alpha+n)}\text{Pr}[\tilde{X}_{\alpha,z,n}=m]\prod_{i=1}^{n}\frac{\left(\frac{\alpha(1-\alpha)_{(i-1)}}{i!}\right)^{x_{i}}}{x_{i}!}$
$\displaystyle\quad=\sum_{m\geq
k}n!\frac{\left(\frac{-m\alpha}{\alpha}\right)_{(k)}}{(-m\alpha)_{(n)}}\prod_{i=1}^{n}\frac{\left(\frac{\alpha(1-\alpha)_{(i-1)}}{i!}\right)^{x_{i}}}{x_{i}!}\text{Pr}[\tilde{X}_{\alpha,z,n}=m]$
$\displaystyle\quad=\sum_{m\geq
k}\text{Pr}[(M_{1}(\alpha,-m\alpha),\ldots,M_{n}(\alpha,-m\alpha))=(x_{1},\ldots,x_{n})]\text{Pr}[\tilde{X}_{\alpha,z,n}=m].$

This completes the proof of ii). ∎

Theorem 2 presents a compound Poisson perspective of the EP-SM in terms of the
NB-CPSM, thus extending the well-known compound Poisson perspective of the
E-SM in terms of the LS-CPSM. Statement i) of Theorem 2 shows that for
$\alpha\in(0,1)$ and $\theta>-\alpha$ the EP-SM admits a representation in
terms of the NB-CPSM with $\alpha\in(0,1)$ and $z>0$, where the randomization
acts on the parameter $z$ with respect to the distribution (17). Precisely,
this is a compound mixed Poisson sampling model. That is, a compound sampling
model in which the distribution of the random number $K$ of distinct types in
the population is a mixture of Poisson distributions with respect to the law
of $\bar{X}_{\alpha,\theta,n}$. Statement ii) of Theorem 2 shows that for
$\alpha<0$ and $z<0$ the NB-CPSM admits a representation in terms of a
randomized EP-SM with $\alpha<0$ and $\theta=-m\alpha$ for some
$m\in\mathbb{N}$, where the randomization acts on the parameter $m$ with
respect to the distribution (17).

######  Remark 3.

The randomization procedure introduced in Theorem 2 is somehow reminiscent of
the definition of the class of Gibbs-type sampling models introduced in Gnedin
and Pitman (2006). This class is defined from the EP-SM with $\alpha<0$ and
$\theta=-m\alpha$, for some $m\in\mathbb{N}$, and then it assume that the
parameter $m$ is distributed according to an arbitrary distribution on
$\mathbb{N}$. See Theorem 12 of Gnedin and Pitman (2006), and Gnedin (2010)
for an example. However, differently from the definition of Gnedin and Pitman
(2006), in our context the distribution on $m$ depends on the sample size $n$.

For $\alpha\in(0,1)$ and $\theta>-\alpha$, Pitman (2006) first investigated
the large $n$ asymptotic behaviour of $K_{n}(\alpha,\theta)$. See also Gnedin
and Pitman (2006), and references therein. Let $\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}$ denote the almost sure convergence for random
variables, and let $S_{\alpha,\theta}$ be the scaled Mittag-Leffler random
variable defined above. Theorem 3.8 of Pitman (2006) exploited a martingale
convergence argument to show that

$\frac{K_{n}(\alpha,\theta)}{n^{\alpha}}\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}S_{\alpha,\theta}$ (20)

as $n\rightarrow+\infty$. The random variable $S_{\alpha,\theta}$ is typically
referred to as Pitman’s $\alpha$-diversity. For $\alpha<0$ and
$\theta=-m\alpha$, for some $m\in\mathbb{N}$, the large $n$ asymptotic
behaviour of $K_{n}(\alpha,\theta)$ is trivial, that is it holds

$K_{n}(\alpha,\theta)\stackrel{{\scriptstyle w}}{{\longrightarrow}}m$ (21)

as $n\rightarrow+\infty$. See Dolera and Favaro (2020a, b) for Berry-Esseen
type refinements of the large $n$ asymptotic behaviour (20), and to Favaro et
al. (2009, 2012) and Favaro et al. (2015) for generalizations of (20) with
applications to Bayesian nonparametric inference for species sampling
problems. See also Pitman (2006, Chapter 4) for a general treatment of (20).
According to Theorem 2, it is natural to ask weather there exists an interplay
between Theorem 1 and the large $n$ asymptotic behaviours (20) and (21).
Hereafter, we show that: i) (20), with the almost sure convergence replaced by
the convergence in distribution, arises by combining (6) with i) of Theorem 2;
ii) (8) arises by combining (21) with ii) of Theorem 2. This provides with an
alternative proof of Pitman’s $\alpha$-diversity.

######  Theorem 4.

Let $K_{n}(\alpha,\theta)$ and $K(\alpha,z,n)$ under the EP-SM and the NB-
CPSM, respectively. As $n\rightarrow+\infty$

  * i)

for $\alpha\in(0,1)$ and $\theta>-\alpha$

$\frac{K_{n}(\alpha,\theta)}{n^{\alpha}}\stackrel{{\scriptstyle
w}}{{\longrightarrow}}S_{\alpha,\theta}.$ (22)

  * ii)

for $\alpha<0$ and $z<0$

$\frac{K(\alpha,z,n)}{n^{\frac{-\alpha}{1-\alpha}}}\stackrel{{\scriptstyle
w}}{{\longrightarrow}}\frac{(\alpha z)^{\frac{1}{1-\alpha}}}{-\alpha}.$ (23)

###### Proof.

We show that (22) arises by combining (6) with statement i) of Theorem 2. For
any pair of $\mathbb{N}$-valued random variables $U$ and $V$, let
$\mathrm{d}_{TV}(U;V)$ be the total variation distance between the
distribution of the random variale $U$ and the distribution of the random
variable $V$. Also, let $P_{c}$ denote a Poisson random variable with
parameter $c>0$. For any $\alpha\in(0,1)$ and $t>0$, we show that as
$n\rightarrow+\infty$

$\mathrm{d}_{TV}(K(\alpha,tn^{\alpha},n);1+P_{tn^{\alpha}})\rightarrow 0.$
(24)

This implies (22). The proof of (24) requires a careful analysis of the
probability generating function of $K(\alpha,tn^{\alpha},n)$. In particular,
let us define
$\omega(t;n,\alpha):=tn^{\alpha}+\frac{tM^{\prime}_{\alpha}(t)}{M_{\alpha}(t)}$,
where
$M_{\alpha}(t):=\frac{1}{\pi}\sum_{m=1}^{\infty}\frac{(-t)^{m-1}}{(m-1)!}\Gamma(\alpha
m)\sin(\pi\alpha m)$ is the Wright-Mainardi function (Mainardi et al., 2010).
Then, we apply Corollary 2 of Dolera and Favaro (2020a) in order to conclude
that
$\mathrm{d}_{TV}(K(\alpha,tn^{\alpha},n);1+P_{\omega(t;n,\alpha)})\rightarrow
0$ as $n\rightarrow+\infty$. Finally, we apply the inequality (2.2) in Adell
and Jodrá (2006) to obtain

$\displaystyle\mathrm{d}_{TV}(1+P_{tn^{\alpha}};1+P_{\omega(t;n,\alpha)})$
$\displaystyle=\mathrm{d}_{TV}(P_{tn^{\alpha}};P_{\omega(t;n,\alpha)})\leq\frac{tM^{\prime}_{\alpha}(t)}{M_{\alpha}(t)}\min\left\\{1,\frac{\sqrt{(2/e)}}{\sqrt{\omega(t;n,\alpha)}+\sqrt{tn^{\alpha}}}\right\\}$

so that
$\mathrm{d}_{TV}(1+P_{tn^{\alpha}};1+P_{\omega(t;n,\alpha)})\rightarrow 0$ as
$n\rightarrow+\infty$, and (24) follows. Now, keeping $\alpha$ and $t$ fixed
as above, we show that (24) entails (22). To this aim, here we introduce the
Kolmogorov distance $\mathrm{d}_{K}$ which, for any pair of
$\mathbb{R}_{+}$-valued random variables $U$ and $V$, is defined by
$\mathrm{d}_{K}(U;V):=\sup_{x\geq 0}|\text{Pr}[U\leq x]-\text{Pr}[V\leq x]|$.
In particular, the claim that has to be proved is equivalent to the following
claim

$\mathrm{d}_{K}(K_{n}(\alpha,\theta)/n^{\alpha};S_{\alpha,\theta})\rightarrow
0$

as $n\rightarrow+\infty$. In particular, we exploit statement i) of Theorem 2.
This leads to the distributional identity
$K_{n}(\alpha,\theta)\stackrel{{\scriptstyle
d}}{{=}}K(\alpha,\bar{X}_{\alpha,\theta,n},n)$. Thus, in view of the basic
properties of the Kolmogorov distance,

$\displaystyle\mathrm{d}_{K}(K_{n}(\alpha,\theta)/n^{\alpha};S_{\alpha,\theta})$
$\displaystyle\leq\mathrm{d}_{K}(K_{n}(\alpha,\theta);K(\alpha,n^{\alpha}S_{\alpha,\theta},n))$
(25)
$\displaystyle\quad\quad+\mathrm{d}_{K}(K(\alpha,n^{\alpha}S_{\alpha,\theta},n);1+P_{n^{\alpha}S_{\alpha,\theta}})$
$\displaystyle\quad\quad\quad+\mathrm{d}_{K}([1+P_{n^{\alpha}S_{\alpha,\theta}}]/n^{\alpha};S_{\alpha,\theta}),$

where the $\\{P_{\lambda}\\}_{\lambda\geq 0}$ is thought of here as a
homogeneous Poisson process with rate 1, independent of $S_{\alpha,\theta}$.
The desired conclusion will be reached as soon as we will prove that all the
three summands on the right-hand side of (25) go to zero as
$n\rightarrow+\infty$. Before proceeding, we recall that
$\mathrm{d}_{K}(U;V)\leq\mathrm{d}_{TV}(U;V)$. Therefore,

$\displaystyle\mathrm{d}_{K}(K_{n}(\alpha,\theta);K(\alpha,n^{\alpha}S_{\alpha,\theta},n))$
$\displaystyle\leq\frac{1}{2}\sum_{k=1}^{n}\Big{|}\mathscr{C}(n,k;\alpha)\frac{\Gamma(k+\theta/\alpha)}{\alpha\Gamma(\theta/\alpha+1)}\frac{\Gamma(\theta+1)}{\Gamma(n+\theta)}-\int_{0}^{+\infty}\frac{\mathscr{C}(n,k;\alpha)(tn^{\alpha})^{k}}{d_{n}(t)}f_{S_{\alpha,\theta}}(t)\mathrm{d}t\Big{|}$

with $d_{n}(t):=\sum_{j=1}^{n}\mathscr{C}(n,j;\alpha)(tn^{\alpha})^{j}$. Now,
let us define the following quantity:
$d^{\ast}_{n}(t):=e^{tn^{\alpha}}(n-1)!\frac{1}{t^{1/\alpha}}f_{\alpha}(\frac{1}{t^{1/\alpha}})$.
Accordingly, we can majorize the above right-hand side by means of the
following quantity

$\displaystyle\frac{1}{2}\sum_{k=1}^{n}\Big{|}\mathscr{C}(n,k;\alpha)\frac{\Gamma(k+\theta/\alpha)}{\alpha\Gamma(\theta/\alpha+1)}\frac{\Gamma(\theta+1)}{\Gamma(n+\theta)}-\int_{0}^{+\infty}\frac{\mathscr{C}(n,k;\alpha)(tn^{\alpha})^{k}}{d^{\ast}_{n}(t)}f_{S_{\alpha,\theta}}(t)\mathrm{d}t\Big{|}$
$\displaystyle+\frac{1}{2}\int_{0}^{+\infty}\frac{|d^{\ast}_{n}(t)-d_{n}(t)|}{d^{\ast}_{n}(t)}f_{S_{\alpha,\theta}}(t)\mathrm{d}t\
.$

Accordingly, by exploiting the identity
$\int_{0}^{+\infty}\frac{(tn^{\alpha})^{k}}{d^{\ast}_{n}(t)}f_{S_{\alpha,\theta}}(t)\mathrm{d}t=\frac{1}{(n-1)!}\frac{\Gamma(k+\theta/\alpha)}{n^{\theta}}\frac{\Gamma(\theta+1)}{\alpha\Gamma(\theta/\alpha+1)}$,
we can write that

$\displaystyle\sum_{k=1}^{n}\Big{|}\mathscr{C}(n,k;\alpha)\frac{\Gamma(k+\theta/\alpha)}{\alpha\Gamma(\theta/\alpha+1)}\frac{\Gamma(\theta+1)}{\Gamma(n+\theta)}-\int_{0}^{+\infty}\frac{\mathscr{C}(n,k;\alpha)(tn^{\alpha})^{k}}{d^{\ast}_{n}(t)}f_{S_{\alpha,\theta}}(t)\mathrm{d}t\Big{|}=\Big{|}1-\frac{\Gamma(n+\theta)}{\Gamma(n)n^{\theta}}\Big{|}$

which goes to zero as $n\rightarrow+\infty$ for any $\theta>-\alpha$, by a
direct application of Stirling’s approximation. To show that the integral
$\int_{0}^{+\infty}\frac{|d^{\ast}_{n}(t)-d_{n}(t)|}{d^{\ast}_{n}(t)}f_{S_{\alpha,\theta}}(t)\mathrm{d}t$
also goes to zero as $n\rightarrow+\infty$, we may resort to the identities
(13)–(14) of Dolera and Favaro (2020a), as well as Lemma 3 in Dolera and
Favaro (2020a). In particular, let
$\Delta\,:\,(0,+\infty)\rightarrow(0,+\infty)$ denote a suitable continuous
function independent of $n$, and such that $\Delta(z)=O(1)$ as $z\rightarrow
0$ and $\Delta(z)f_{\alpha}(1/z)=O(z^{-\infty})$ as $z\rightarrow+\infty$.
Then, we write that

$\displaystyle\int_{0}^{+\infty}\frac{|d^{\ast}_{n}(t)-d_{n}(t)|}{d^{\ast}_{n}(t)}f_{S_{\alpha,\theta}}(t)\mathrm{d}t$
$\displaystyle\leq\Big{|}\frac{(n/e)^{n}\sqrt{2\pi
n}}{n!}-1\Big{|}+\left(\frac{(n/e)^{n}\sqrt{2\pi
n}}{n!}\right)\frac{1}{n}\int_{0}^{+\infty}\Delta(t^{1/\alpha})f_{S_{\alpha,\theta}}(t)\mathrm{d}t\
.$

Since we have that
$\int_{0}^{+\infty}\Delta(t^{1/\alpha})f_{S_{\alpha,\theta}}(t)\mathrm{d}t<+\infty$
by Lemma 3 of Dolera and Favaro (2020a), both the summands on the above right-
hand side go to zero as $n\rightarrow+\infty$, again by a direct application
of Stirling’s approximation. Thus, the first summand on the right-hand side of
(25) goes to zero as $n\rightarrow+\infty$. As for the second summand on the
right-hand side of (25), it can be bounded by the quantity

$\int_{0}^{+\infty}\mathrm{d}_{TV}(K(\alpha,tn^{\alpha},n);1+P_{tn^{\alpha}})f_{S_{\alpha,\theta}}(t)\mathrm{d}t\
.$

By a dominated convergence argument, this quantity goes to zero as
$n\rightarrow+\infty$ as a consequence of (24). Finally, for the third summand
on the right-hand side of (25), we can resort to a conditioning argument in
order to reduce the problem to a direct application of the well-known law of
large numbers for renewal processes. See, e.g. (Grimmett and Stirzaker, 2001,
Section 10.2) and references therein. In particular, this leads to
$n^{-\alpha}P_{tn^{\alpha}}\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}t$
for any $t>0$, which entails that
$n^{-\alpha}P_{n^{\alpha}S_{\alpha,\theta}}\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}S_{\alpha,\theta}$ as $n\rightarrow+\infty$. Thus,
this third term also goes to zero as $n\rightarrow+\infty$, and (22) follows.

Now, we consider (23), showing that it arises by combining (21) with statement
ii) of Theorem 2. In particular, by an obvious conditioning argument, we can
write that as $n\rightarrow+\infty$ it holds true that

$\frac{K_{n}(\alpha,\tilde{X}_{\alpha,z,n}|\alpha|)}{\tilde{X}_{\alpha,z,n}}\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}1.$

At this stage, we consider the probability generating function of the random
variable $\tilde{X}_{\alpha,z,n}$, and therefore we immediately obtain
$\mathbb{E}[s^{\tilde{X}_{\alpha,z,n}}]:=B_{n}(-sz)/B_{n}(-z)$ for
$n\in\mathbb{N}$ and $s\in[0,1]$ with the same $B_{n}$ as in (13) and (14).
Therefore, the asymptotic expansion we already provided in (15) entails

$\frac{\tilde{X}_{\alpha,z,n}}{n^{\frac{-\alpha}{1-\alpha}}}\stackrel{{\scriptstyle
w}}{{\longrightarrow}}\frac{(\alpha z)^{\frac{1}{1-\alpha}}}{-\alpha}$ (26)

as $n\rightarrow+\infty$. In particular, (26) follows by applying exactly the
same arguments used to prove (8). Now, since

$\frac{K_{n}(\alpha,\tilde{X}_{\alpha,z,n}|\alpha|)}{n^{\frac{-\alpha}{1-\alpha}}}\stackrel{{\scriptstyle
d}}{{=}}\frac{K_{n}(\alpha,\tilde{X}_{\alpha,z,n}|\alpha|)}{\tilde{X}_{\alpha,z,n}}\frac{\tilde{X}_{\alpha,z,n}}{n^{\frac{-\alpha}{1-\alpha}}},$

the claim follows from a direct application of well-know Slutsky’s theorem.
This completes the proof. ∎

##  3 Discussion

The NB-CPSM is a compound Poisson sampling model generalizing the popular LS-
CMSM. In this paper, we introduced a compound Poisson perspective of the EP-SM
in terms of the NB-CPSM, thus extending the well-known compound Poisson
perspective of the E-SM in terms of the LS-CPSM. We conjecture that an
analogous perspective holds true for the class of $\alpha$-stable Poisson-
Kingman sampling models (Pitman, 2003, 2006), of which the EP-SM is a
noteworthy special case. That is, for $\alpha\in(0,1)$, we conjecture that an
$\alpha$-stable Poisson-Kingman sampling model admits a representation as a
randomized NB-CPSM with $\alpha\in(0,1)$ and $z>0$, where the randomization
acts on $z$ with respect a scale mixture between a Gamma and a suitable
transformation of the Mittag-Leffler distribution. We believe that such a
compound Poisson representation would be critical in order to introduce Berry-
Esseen type refinements of the large $n$ asymptotic behaviour of $K_{n}$ under
$\alpha$-stable Poisson-Kingman sampling models. See Pitman (2003, Section
6.1), and references therein. Such a line of research aims at extending
preliminary works of Dolera and Favaro (2020a, b) on Berry-Esseen type
theorems under the EP-SM. Work on this, and on the more general settings
induced by normalized random measures (Regazzini et al., 2003) and Poisson-
Kingman models (Pitman, 2003), is ongoing.

## Acknowledgement

Emanuele Dolera and Favaro received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme under grant agreement No 817257. Emanuele Dolera and Stefano Favaro
gratefully acknowledge the financial support from the Italian Ministry of
Education, University and Research (MIUR), “Dipartimenti di Eccellenza” grant
2018-2022.

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