# A martingale approach to Gaussian fluctuations and laws of iterated
logarithm for Ewens-Pitman model

Bernard Bercu<EMAIL_ADDRESS>Institut de Mathématiques de
Bordeaux, Université de Bordeaux, France Stefano Favaro
<EMAIL_ADDRESS>Department of Economics and Statistics, University of
Torino and Collegio Carlo Alberto, Italy

###### Abstract

The Ewens-Pitman model refers to a distribution for exchangeable random
partitions of $[n]=\\{1,\ldots,n\\}$, which is indexed by a pair of parameters
$\alpha\in[0,1)$ and $\theta>-\alpha$, with $\alpha=0$ corresponding to the
celebrated Ewens model in population genetics. The large $n$ asymptotic
properties of the Ewens-Pitman model have been the subject of numerous
studies, with the focus being on the number $K_{n}$ of partition sets and the
number $K_{r,n}$ of partition sets that appear $r$ times, for $r=1,\ldots,n$.
While for $\alpha=0$ asymptotic results have been obtained in terms of almost-
sure convergence and Gaussian fluctuations, for $\alpha\in(0,1)$ only almost-
sure convergences are available, with the proof for $K_{r,m}$ being given only
as a sketch. In this paper, we make use of martingales to develop a unified
and comprehensive treatment of the large $n$ asymptotic behaviours of $K_{n}$
and $K_{r,n}$ for $\alpha\in(0,1)$, providing alternative, and rigorous,
proofs of the almost-sure convergences of $K_{n}$ and $K_{r,n}$, and covering
the gap of Gaussian fluctuations. We also obtain new laws of the iterated
logarithm for $K_{n}$ and $K_{r,n}$.

Keywords: Almost-sure limit; Ewens-Pitman model; exchangeable random
partition; Gaussian fluctuation; law of iterated logarithm; martingale;
Mittag-Leffler distribution

##  1 Introduction

The Ewens-Pitman (EP) model refers to a distribution for exchangeable random
partitions, introduced by Pitman (1995) as a generalization of the Ewens model
in population genetics (Ewens, 1972). For $n\in\mathbb{N}$, consider a random
partition of $[n]=\\{1,\ldots,n\\}$ into $K_{n}\in\\{1,\ldots,n\\}$ partition
sets of which $K_{r,n}\in\\{0,1,\ldots,n\\}$ appear $r$ times, for
$r=1,\ldots,n$, such that

$n=\sum_{r=1}^{n}rK_{r,n}\quad\text{and}\quad K_{n}=\sum_{r=1}^{n}K_{r,n}.$

For $\alpha\in[0,1)$ and $\theta>-\alpha$, the EP model assigns to
$\mathbf{K}_{n}=(K_{1,n},\ldots,K_{n,n})$ the probability

$\mathbb{P}(\mathbf{K}_{n}=(k_{1},\ldots,k_{n}))=n!\frac{\left(\frac{\theta}{\alpha}\right)^{(\sum_{i=1}^{n}k_{i})}}{(\theta)^{(n)}}\prod_{i=1}^{n}\left(\frac{\alpha(1-\alpha)^{(i-1)}}{i!}\right)^{k_{i}}\frac{1}{k_{i}!},$
(1)

where, for any $a\in\mathbb{R}$, $(a)^{(n)}$ stands for the rising factorial
of $a$ of order $n$, i.e. $(a)^{(n)}=a(a+1)\cdots(a+n-1)$. The Ewens model is
recovered from (1) by setting $\alpha=0$. The distribution (1) admits several
constructions, with the most common being a sequential or generative
construction through the Chinese restaurant process (Pitman, 1995; Feng and
Hoppe, 1998; Zabell, 2005) and a Poisson process construction through random
sampling from the Pitman-Yor random probability measure (Perman et al., 1992;
Pitman, 1995; Pitman and Yor, 1997). The EP model plays a critical role in a
variety of research fields, such as population genetics, Bayesian
nonparametric statistics, excursion theory, combinatorics, machine learning
and statistical physics. We refer to Pitman (2006, Chapter 3 and Chapter 4)
and Crane (2016) for a comprehensive account on the EP model and
generalizations thereof.

Under the EP model, there have been several studies on the large $n$
asymptotic behaviour of $K_{n}$ and the $K_{r,n}$’s (Pitman, 2006, Chapter 3).
In particular, for $\alpha=0$, Korwar and Hollander (1973, Theorem 2.3)
exploited the fact that $K_{n}$ is a sum of $n$ independent Bernoulli random
variables, and showed that

$\lim_{n\rightarrow+\infty}\frac{K_{n}}{\log n}=\theta\qquad\text{a.s.}$ (2)

Furthermore, it follows directly from Lindeberg-Lévy central limit theorem
that

$\frac{K_{n}-\theta\log n}{\sqrt{\log
n}}\underset{n\rightarrow+\infty}{\overset{\mathcal{L}}{\longrightarrow}}\sqrt{\theta}\,\mathcal{N}(0,1),$
(3)

with $\mathcal{N}(0,1)$ being a standard Gaussian random variable. See Feng
and Hoppe (1998) and references therein for functional versions of both (2)
and (3). As regard to the $K_{r,n}$’s, from Arratia et al. (1992, Theorem 1)
it follows that

$K_{r,n}\underset{n\rightarrow+\infty}{\overset{\mathcal{L}}{\longrightarrow}}Z_{r}$
(4)

and

$(K_{1,n},K_{2,n},\ldots)\underset{n\rightarrow+\infty}{\overset{\mathcal{L}}{\longrightarrow}}(Z_{1},Z_{2},\ldots),$
(5)

where the $Z_{r}$’s are independent Poisson random variables with
$\mathds{E}[Z_{r}]=\theta/r$, for all $r\geq 1$. We refer to Arratia et al.
(2003) for various generalizations and refinements of these asymptotic
results, and their interplay with combinatorics.

For $\alpha\in(0,1)$, $K_{n}$ is no longer a sum of independent Bernoulli
random variables. However, Pitman (2006, Theorem 3.8) used a martingale
construction for $K_{n}$ based on a likelihood ratio defined through the
distribution (1), and applied the strong law of large numbers for martingales
to show that prove that

$\lim_{n\rightarrow+\infty}\frac{K_{n}}{n^{\alpha}}=S_{\alpha,\theta}\qquad\text{a.s.},$
(6)

where $S_{\alpha,\theta}$ is a positive and almost surely finite random
variable. If $f_{\alpha}$ is the positive $\alpha$-Stable density function,
then the distribution of $S_{\alpha,\theta}$ has density function

$f_{S_{\alpha,\theta}}(s)=\frac{\Gamma(\theta+1)}{\alpha\Gamma(\theta/\alpha+1)}s^{\frac{\theta-1}{\alpha}-1}f_{\alpha}(s^{-1/\alpha}),$
(7)

where $\Gamma$ stands for the Euler Gamma function. Precisely, (7) is a
generalization of the Mittag-Leffler density function, which is recovered for
$\theta=0$ (Zolotarev, 1986). As regard to the $K_{r,n}$’s, from Pitman (2006,
Lemma 3.11) it follows that

$\lim_{n\rightarrow+\infty}\frac{K_{r,n}}{n^{\alpha}}=p_{\alpha}(r)S_{\alpha,\theta}\qquad\text{a.s.},$
(8)

where

$p_{\alpha}(r)=\frac{\alpha(1-\alpha)^{(r-1)}}{r!}.$

The almost-sure convergences (6) and (8) are the natural counterparts of (2)
and (4), respectively, though at different scales. Instead, it is still an
open problem of obtaining a counterpart of (3), as well as Gaussian
fluctuations for $K_{r,n}$.

###  1.1 Our contributions

For $\alpha\in(0,1)$ and $\theta>-\alpha$, we make use of martingales to
develop a unified and comprehensive treatment of the large $n$ asymptotic
behaviours of $K_{n}$ and $K_{r,n}$, providing alternative, and rigorous,
proofs of the almost-sure limits (6) and (8), and covering the gap of Gaussian
fluctuations. In particular, we propose a new martingale construction for
$K_{n}$, simpler than that of Pitman (2006, Theorem 3.8), which leads to an
alternative proof of (6), still relying on the strong law of large numbers for
martingales. Further, by exploiting our martingale construction, as well as
(6), we show that

$\sqrt{n^{\alpha}}\left(\frac{K_{n}}{n^{\alpha}}-S_{\alpha,\theta}\right)\underset{n\rightarrow+\infty}{\overset{\mathcal{L}}{\longrightarrow}}\sqrt{S_{\alpha,\theta}^{\prime}}\,\mathcal{N}(0,1),$
(9)

where $S^{\prime}_{\alpha,\theta}$ is random variable independent of
$\mathcal{N}(0,1)$ and sharing the same distribution as $S_{\alpha,\theta}$.
We also prove the law of the iterated logarithm for $K_{n}$, i.e.,

$\limsup_{n\rightarrow\infty}\frac{\big{(}K_{n}-n^{\alpha}S_{\alpha,\theta}\big{)}^{2}}{2n^{\alpha}\log\log
n}=S_{\alpha,\theta}\qquad\text{a.s.}$ (10)

Then, we extend our analysis to the $K_{r,n}$’s, for which only a sketch of
the proof of (8) is given in Pitman (2006, Lemma 3.11). In particular, we
introduce a martingale construction for $K_{r,n}$, providing a rigorous proof
of (8) by the strong law of large numbers for martingales. Further, by
exploiting our martingale construction and (8), we establish a Gaussian
fluctuation as well as the law of iterated logarithm for $K_{r,n}$. Critical
for our study is the work of Heyde (1977), which provides sufficient
conditions to achieve Gaussian fluctuations and laws of iterated logarithm
from the martingale convergence theorem.

###  1.2 Organization of the paper

The paper is structured as follows. In Section 2, we present the martingale
construction for $K_{n}$ and the proofs of the almost-sure convergence, as
well as a corresponding $\mathbb{L}^{p}$ convergence, the Gaussian
fluctuation, and the law of iterated logarithm. Section 3 contains an
analogous asymptotic analysis for $K_{r,n}$. In Section 4, we conclude by
discussing some directions of future research. The Appendix concerns an
alternative proof, without relying on martingales, of the $\mathbb{L}^{p}$
convergence of $K_{n}$ and $K_{r,n}$.

##  2 Asymptotic results for the number of partition sets

We start by introducing the keystone martingale construction for $K_{n}$. This
is a critical tool at the basis of all the results in this section. From the
sequential or generative construction of the EP model (Pitman, 1995,
Proposition 9), for all $n\geq 1$,

$\vspace{-2ex}\mathbb{P}(K_{n+1}=K_{n}+k\,|\,K_{n})=\left\\{\begin{array}[]{ccc}{\displaystyle\frac{\alpha
K_{n}+\theta}{n+\theta}}&\text{ if }&k=1,\vspace{1ex}\\\\[11.38092pt]
{\displaystyle\frac{n-\alpha K_{n}}{n+\theta}}&\text{ if
}&k=0.\end{array}\right.$

It simply means that

$K_{n+1}=K_{n}+\xi_{n+1}$ (11)

where the conditional distribution of $\xi_{n+1}$, given
$\mathcal{F}_{n}=\sigma(K_{1},\ldots,K_{n})$, is the Bernoulli
$\mathcal{B}(p_{n})$ distribution with parameter

$p_{n}=\frac{\alpha K_{n}+\theta}{n+\theta}.$ (12)

According to the above definition of the sequence $(\xi_{n})$, as
$\mathbb{E}[\xi_{n+1}\,|\,\mathcal{F}_{n}]=p_{n}$,

$\mathbb{E}[K_{n+1}\,|\,\mathcal{F}_{n}]=\mathbb{E}[K_{n}+\xi_{n+1}\,|\,\mathcal{F}_{n}]=K_{n}+p_{n}=\beta_{n}K_{n}+\frac{\theta}{n+\theta}\qquad\text{a.s.},$
(13)

where

$\beta_{n}=1+\frac{\alpha}{n+\theta}.$ (14)

Let $(b_{n})$ be the sequence defined by $b_{1}=1$ and for all $n\geq 2$,

$b_{n}\\!=\\!\prod_{k=1}^{n-1}\beta_{k}^{-1}\\!=\\!\prod_{k=1}^{n-1}\left(\frac{k+\theta}{k+\alpha+\theta}\right)\\!=\\!\left\\{\\!\\!\begin{array}[]{ccc}{\displaystyle\\!\\!\Big{(}\frac{\alpha+\theta}{\theta}\Big{)}\frac{(\theta)^{(n)}}{(\alpha+\theta)^{(n)}}}&\text{
if }&\theta\neq 0,\vspace{1ex}\\\\[11.38092pt]
{\displaystyle\frac{\alpha(n-1)!}{(\alpha)^{(n)}}}&\text{ if
}&\theta=0.\end{array}\right.$ (15)

Herafter, we introduce the sequence of random variables $(M_{n})$ that is
defined, for all $n\geq 1$, by

$M_{n}=b_{n}K_{n}+\left(\frac{\alpha+\theta}{\alpha}\right)\frac{(\theta)^{(n)}}{(\alpha+\theta)^{(n)}}=b_{n}\left(K_{n}+\frac{\theta}{\alpha}\right).$
(16)

Since $b_{n}=\beta_{n}b_{n+1}$, we have from (13) and (14) that for all $n\geq
1$,

$\displaystyle\mathbb{E}[M_{n+1}\,|\,\mathcal{F}_{n}]$
$\displaystyle=b_{n+1}\mathbb{E}\left[K_{n+1}+\frac{\theta}{\alpha}\,|\,\mathcal{F}_{n}\right]$
$\displaystyle=b_{n+1}\left(\mathbb{E}[K_{n+1}\,|\,\mathcal{F}_{n}]+\frac{\theta}{\alpha}\right)$
$\displaystyle=b_{n+1}\left(\beta_{n}K_{n}+\frac{\theta}{n+\theta}+\frac{\theta}{\alpha}\right)$
$\displaystyle=b_{n+1}\left(\beta_{n}K_{n}+\frac{\theta}{\alpha}\Big{(}1+\frac{\alpha}{n+\theta}\Big{)}\right)$
$\displaystyle=b_{n+1}\beta_{n}\left(K_{n}+\frac{\theta}{\alpha}\right)$
$\displaystyle=b_{n}\left(K_{n}+\frac{\theta}{\alpha}\right)$
$\displaystyle=M_{n}$

almost surely. Moreover, $(M_{n})$ is square integrable as $K_{n}\leq n$.
Consequently, $(M_{n})$ is a locally square integrable martingale. Since
$\mathcal{L}(\xi_{n+1}\,|\,\mathcal{F}_{n})=\mathcal{B}(p_{n})$, it is not
hard to compute high-order conditional moments of $(M_{n})$. For example, as

$M_{n+1}-M_{n}=b_{n+1}\big{(}K_{n+1}-\mathbb{E}[K_{n+1}\,|\,\mathcal{F}_{n}]\big{)},$

we clearly have from (11) and (13) that

$\displaystyle\mathbb{E}\big{[}(M_{n+1}-M_{n})^{2}\,|\,\mathcal{F}_{n}\big{]}$
$\displaystyle=b_{n+1}^{2}\big{(}\mathbb{E}\big{[}K_{n+1}^{2}\,|\,\mathcal{F}_{n}\big{]}-(\mathbb{E}[K_{n+1}\,|\,\mathcal{F}_{n}])^{2}\big{)}$
$\displaystyle=b_{n+1}^{2}\big{(}\mathbb{E}\big{[}(K_{n}+\xi_{n+1})^{2}\,|\,\mathcal{F}_{n}\big{]}-\big{(}K_{n}+p_{n}\big{)}^{2}\big{)}$
$\displaystyle=b_{n+1}^{2}\big{(}K_{n}^{2}+2K_{n}p_{n}+p_{n}-\big{(}K_{n}+p_{n}\big{)}^{2}\big{)}$
$\displaystyle=b_{n+1}^{2}p_{n}(1-p_{n})$

almost surely, which leads, via (12), to

$\mathbb{E}\big{[}(M_{n+1}-M_{n})^{2}\,|\,\mathcal{F}_{n}\big{]}=b_{n+1}^{2}\left(\frac{(\theta+\alpha
K_{n})(n-\alpha K_{n})}{(n+\theta)^{2}}\right)\qquad\text{a.s.}$ (17)

###  2.1 An alternative approach for the almost-sure convergence

Based on the above martingale construction, we present an alternative proof of
(2), by relying on the strong law of large numbers for martingale. Our proof
is more natural and intuitive than that of Pitman (2006, Theorem 3.8), which
makes use of the exchangeable partition probability function induced by the EP
model.

######  Theorem 2.1.

Let $K_{n}$ be the number of partition sets in a random partition of $[n]$
distributed as the EP model with $\alpha\in(0,1)$ and $\theta>-\alpha$. Then,
we have

$\lim_{n\rightarrow+\infty}\frac{K_{n}}{n^{\alpha}}=S_{\alpha,\theta}\qquad\text{a.s.}$
(18)

where $S_{\alpha,\theta}$ is a positive and almost surely finite random
variable whose distribution has probability density function (7). This
convergence holds in $\mathbb{L}^{p}$ for any integer $p\geq 1$,

$\lim_{n\rightarrow+\infty}\mathbb{E}\left[\left|\frac{K_{n}}{n^{\alpha}}-S_{\alpha,\theta}\right|^{p}\right]=0.$
(19)

###### Proof.

We already saw that $(M_{n})$ is a locally square integrable martingale.
Denote by $\langle M\rangle_{n}$ its predictable quadratic variation,

$\langle
M\rangle_{n}=\sum_{k=0}^{n-1}\mathbb{E}[(M_{k+1}-M_{k})^{2}\,|\,\mathcal{F}_{k}].$

We have from (17) that

$\langle
M\rangle_{n}=\left(\frac{\alpha+\theta}{\alpha}\right)^{2}+\sum_{k=1}^{n-1}b_{k+1}^{2}\left(\frac{(\theta+\alpha
K_{k})(k-\alpha K_{k})}{(k+\theta)^{2}}\right)\qquad\text{a.s.}$ (20)

Taking the expectation on both sides of (20), we obtain that

$\mathbb{E}\big{[}\langle
M\rangle_{n}\big{]}=\left(\frac{\alpha+\theta}{\alpha}\right)^{2}+\sum_{k=1}^{n-1}b_{k+1}^{2}\left(\frac{\theta
k+\alpha(k-\theta)\mathbb{E}[K_{k}]-\alpha^{2}\mathbb{E}[K_{k}^{2}]}{(k+\theta)^{2}}\right).$

Hence, it exists some constant $C>0$ such that for $n$ large enough,

$\mathbb{E}\big{[}\langle M\rangle_{n}\big{]}\leq
C\sum_{k=1}^{n}\frac{b_{k}^{2}\mathbb{E}[K_{k}]}{k}.$ (21)

Moreover, we have from (13) and (15) that if $\theta\neq 0$,

$\mathbb{E}[K_{n}]=b_{n}^{-1}\sum_{k=0}^{n-1}\frac{(\theta)^{(k)}}{(\alpha+\theta+1)^{(k)}}=\frac{\theta}{\alpha}\left(\frac{(\alpha+\theta)^{(n)}-(\theta)^{(n)}}{(\theta)^{(n)}}\right),$
(22)

whereas if $\theta=0$,

$\mathbb{E}[K_{n}]=\prod_{k=1}^{n-1}\left(1+\frac{\alpha}{k}\right)=\frac{(\alpha)^{(n)}}{\alpha(n-1)!}.$
(23)

In addition, it follows from standard results on the asymptotic behavior of
the Euler Gamma function that for $\theta\neq 0$,

$\lim_{n\rightarrow+\infty}\frac{1}{n^{\alpha}}\frac{(\alpha+\theta)^{(n)}}{(\theta)^{(n)}}=\lim_{n\rightarrow+\infty}\frac{\Gamma(\theta+1)\Gamma(n+\alpha+\theta)}{n^{\alpha}\theta\Gamma(\alpha+\theta)\Gamma(n+\theta)}=\frac{\Gamma(\theta+1)}{\theta\Gamma(\alpha+\theta)}.$
(24)

Hence, we obtain from (22), (23) and (24) that whatever the value of
$\theta>-\alpha$,

$\lim_{n\rightarrow+\infty}\frac{1}{n^{\alpha}}\mathbb{E}[K_{n}]=\frac{\Gamma(\theta+1)}{\alpha\Gamma(\alpha+\theta)}.$

Consequently, we deduce from (21), (15) and (24) that

$\sup_{n\geq 1}\mathbb{E}\big{[}\langle M\rangle_{n}\big{]}<+\infty.$ (25)

Therefore, we deduce from (25) that $(M_{n})$ is bounded in $\mathbb{L}^{2}$.
Then, it follows from Doob’s martingale convergence theorem (Hall and Heyde,
1980, Corollary 2.2), that the sequence $(M_{n})$ converges a.s. and in
$\mathbb{L}^{2}$ to a square integrable random variable $M_{\alpha,\theta}$.
In fact, we shall prove below that $(M_{n})$ is bounded in $\mathbb{L}^{p}$
for any integer $p\geq 1$ which means that $(M_{n})$ converges a.s. and in
$\mathbb{L}^{p}$ to $M_{\alpha,\theta}$ for any integer $p\geq 1$. Hereafter,
we obtain from (16) and (24) that

$\lim_{n\rightarrow+\infty}\frac{K_{n}}{n^{\alpha}}=S_{\alpha,\theta}\qquad\text{a.s.},$

where the random variables $M_{\alpha,\theta}$ and $S_{\alpha,\theta}$ are
tightly related by means of the identity

$M_{\alpha,\theta}=\Big{(}\frac{\Gamma(\alpha+\theta+1)}{\Gamma(\theta+1)}\Big{)}S_{\alpha,\theta},$

which completes the proof of (18). We shall now proceed to the proof of the
convergence in $\mathbb{L}^{p}$ for any integer $p\geq 1$. In particular, we
are going to compute the falling factorial moments $\mathbb{E}[(K_{n})_{(p)}]$
of $K_{n}$ where, for any $a\in\mathbb{R}$, $(a)_{(p)}=a(a-1)\cdots(a-p+1)$
with $(a)_{(0)}=1$. By an application of Vandermonde identity for the falling
factorial together with (11), for any integer $p\geq 1$,

$(K_{n+1})_{(p)}=(K_{n}+\xi_{n+1})_{(p)}=\sum_{k=0}^{p}{p\choose
k}(K_{n})_{(k)}(\xi_{n+1})_{(p-k)}.$

By taking the conditional expectation on both sides of this identity, we
obtain that

$\mathbb{E}[(K_{n+1})_{(p)}\,|\,\mathcal{F}_{n}]=\sum_{k=0}^{p}{p\choose
k}(K_{n})_{(k)}\mathbb{E}[(\xi_{n+1})_{(p-k)}\,|\,\mathcal{F}_{n}]\qquad\text{a.s.},$
(26)

where
$\mathbb{E}[(\xi_{n+1})_{(1)}\,|\,\mathcal{F}_{n}]=\mathbb{E}[\xi_{n+1}\,|\,\mathcal{F}_{n}]=p_{n}$,
and $\mathbb{E}[(\xi_{n+1})_{(k)}\,|\,\mathcal{F}_{n}]=0$ for all $k\geq 2$.
Accordingly, we can deduce from (12) and (26) that for any integer $p\geq 1$,

$\displaystyle\mathbb{E}[(K_{n+1})_{(p)}\,|\,\mathcal{F}_{n}]$
$\displaystyle=(K_{n})_{(p)}+p(K_{n})_{(p-1)}\Big{(}\frac{\alpha
K_{n}+\theta}{n+\theta}\Big{)}\qquad\text{a.s.}$
$\displaystyle=(K_{n})_{(p)}+\frac{\alpha
p}{n+\theta}(K_{n})_{(p-1)}(K_{n}-p+1)+L_{n}(p)\qquad\text{a.s.}$
$\displaystyle=\Big{(}1+\frac{\alpha
p}{n+\theta}\Big{)}(K_{n})_{(p)}+L_{n}(p)\qquad\text{a.s.}$ (27)

where

$L_{n}(p)=\Big{(}\frac{p(\alpha(p-1)+\theta)}{n+\theta}\Big{)}(K_{n})_{(p-1)}.$

Denote $b_{1}(p)=1$ and for all $n\geq 2$,

$b_{n}(p)=\prod_{k=1}^{n-1}\left(\frac{k+\alpha
p+\theta}{k+\theta}\right)=\left\\{\begin{array}[]{ccc}{\displaystyle\Big{(}\frac{\theta}{\alpha
p+\theta}\Big{)}\frac{(\alpha p+\theta)^{(n)}}{(\theta)^{(n)}}}&\text{ if
}&\theta\neq 0,\vspace{1ex}\\\\[11.38092pt] {\displaystyle\frac{(\alpha
p)^{(n)}}{\alpha p(n-1)!}}&\text{ if }&\theta=0.\end{array}\right.$ (28)

It follows from (27) and (28) that for all $n\geq 2$ and for any integer
$p\geq 2$,

$\mathbb{E}[(K_{n})_{(p)}]=b_{n}(p)\sum_{k=1}^{n-1}\big{(}b_{k+1}(p)\big{)}^{-1}\mathbb{E}[L_{k}(p)]$

leading, for $\theta\neq 0$, to

$\mathbb{E}[(K_{n})_{(p)}]=\Big{(}\frac{p(\alpha(p-1)+\theta)}{\alpha
p+\theta}\Big{)}\Big{(}\frac{(\alpha
p+\theta)^{(n)}}{(\theta)^{(n)}}\Big{)}\sum_{k=1}^{n-1}\Big{(}\frac{(\theta)^{(k)}\mathbb{E}[(K_{k})_{(p-1)}]}{(\alpha
p+\theta+1)^{(k)}}\Big{)}$ (29)

while, for $\theta=0$, to

$\mathbb{E}[(K_{n})_{(p)}]=\Big{(}\frac{(p-1)(\alpha
p)^{(n)}}{(n-1)!}\Big{)}\sum_{k=1}^{n-1}\Big{(}\frac{(k-1)!\mathbb{E}[(K_{k})_{(p-1)}]}{(\alpha
p+1)^{(k)}}\Big{)}.$ (30)

We deduce from (22) and (29) that for $\theta\neq 0$,

$\mathbb{E}[(K_{n})_{(1)}]=\frac{\theta}{\alpha(\theta)^{(n)}}\big{(}(\alpha+\theta)^{(n)}-(\theta)^{(n)}\big{)},$
$\mathbb{E}[(K_{n})_{(2)}]=\frac{\theta(\theta+\alpha)}{\alpha^{2}(\theta)^{(n)}}\big{(}(2\alpha+\theta)^{(n)}-2(\alpha+\theta)^{(n)}+(\theta)^{(n)}\big{)},$

and more generally, for all $n\geq 1$ and for any integer $p\geq 1$,

$\displaystyle\mathbb{E}[(K_{n})_{(p)}]$
$\displaystyle=\frac{\prod_{k=0}^{p-1}(k\alpha+\theta)}{\alpha^{p}(\theta)^{(n)}}\sum_{k=0}^{p}(-1)^{p-k}{p\choose
k}(k\alpha+\theta)^{(n)},$
$\displaystyle=\frac{\big{(}\frac{\theta}{\alpha}\big{)}^{(p)}}{(\theta)^{(n)}}\sum_{k=0}^{p}(-1)^{p-k}{p\choose
k}(k\alpha+\theta)^{(n)}.$ (31)

In addition, we have from (23) and (30) that for $\theta=0$,

$\mathbb{E}[(K_{n})_{(1)}]=\frac{(\alpha)^{(n)}}{\alpha(n-1)!},$
$\mathbb{E}[(K_{n})_{(2)}]=\frac{1}{\alpha(n-1)!}\big{(}(2\alpha)^{(n)}-2(\alpha)^{(n)}\big{)},$

and more generally, for all $n\geq 1$ and for any integer $p\geq 1$,

$\mathbb{E}[(K_{n})_{(p)}]=\frac{(p-1)!}{\alpha(n-1)!}\sum_{k=1}^{p}(-1)^{p-k}{p\choose
k}(k\alpha)^{(n)}.$ (32)

We obtain from (28) together with (31) and (32) that it exists a positive
constant $C(p,\alpha,\theta)$ such that for all $n\geq 1$ and $p\geq 1$,

$\mathbb{E}[(K_{n})_{(p)}]\leq C(p,\alpha,\theta)b_{n}(p).$ (33)

Moreover, we also have the elementary inequality

$\mathbb{E}[K_{n}^{p}]\leq
p^{p}+p!\mathbb{E}[(K_{n})_{(p)}\mathrm{I}_{K_{n}\geq p}].$ (34)

Consequently, we find from (16), (33) and (34) that it exists a positive
constant $D(p,\alpha,\theta)$ such that for all $n\geq 1$ and $p\geq 1$,

$\mathbb{E}[M_{n}^{p}]\leq D(p,\alpha,\theta)b_{n}(p)b_{n}^{p}.$ (35)

Furthermore, it is easy to see from (15) and (28) that

$b_{n}(p)b_{n}^{p}=\prod_{k=1}^{n-1}\left(1+\frac{\alpha
p}{k+\theta}\right)\left(1+\frac{\alpha}{k+\theta}\right)^{\\!\\!-p}\leq 1.$
(36)

Hence, we deduce from (35) and (36) that

$\sup_{n\geq 1}\mathbb{E}[M_{n}^{p}]<+\infty,$

which means that the martingale $(M_{n})$ is bounded in $\mathbb{L}^{p}$ for
any $p\geq 1$. Therefore, it follows from Doob’s martingale convergence
theorem (Hall and Heyde, 1980, Corollary 2.2), the sequence $(M_{n})$
converges almost surely and in $\mathbb{L}^{p}$ to a finite random variable
$M_{\alpha,\theta}$. Our goal is now to compute all the moments of
$M_{\alpha,\theta}$. We shall only carry out the proof for $\theta\neq 0$
inasmuch as the proof for $\theta=0$ follows exactly the same arguments. First
of all, as in (24),

$\lim_{n\rightarrow+\infty}\frac{(\alpha p+\theta)^{(n)}}{n^{\alpha
p}(\theta)^{(n)}}=\frac{\Gamma(\theta+1)}{\theta\Gamma(\alpha p+\theta)}.$

Hence, we obtain once again from (31) that for any integer $p\geq 1$,

$\lim_{n\rightarrow+\infty}\frac{1}{n^{\alpha
p}}\mathbb{E}[(K_{n})_{(p)}]=\frac{\Gamma(\theta+1)}{\theta\Gamma(\alpha
p+\theta)}\Big{(}\frac{\theta}{\alpha}\Big{)}^{\\!(p)\\!}.$ (37)

However, we have for all $p\geq 1$,

$\mathbb{E}[K_{n}^{p}]=\sum_{k=0}^{p}\left\\{\begin{matrix}p\\\
k\end{matrix}\right\\}\mathbb{E}[(K_{n})_{(k)}],$ (38)

where the curly brackets are the Stirling numbers of the second kind given by

$\left\\{\begin{matrix}p\\\
k\end{matrix}\right\\}=\frac{1}{k!}\sum_{i=0}^{k}(-1)^{k-i}{k\choose i}i^{p}.$

Since, $\Big{\\{}\begin{matrix}p\\\ p\end{matrix}\Big{\\}}=1$, we obtain from
(37) and (38) that for all $p\geq 1$,

$\lim_{n\rightarrow+\infty}\frac{1}{n^{\alpha
p}}\mathbb{E}[K_{n}^{p}]=\frac{\Gamma(\theta+1)}{\theta\Gamma(\alpha
p+\theta)}\Big{(}\frac{\theta}{\alpha}\Big{)}^{\\!(p)\\!}.$ (39)

We also recall from (15) that

$\lim_{n\rightarrow+\infty}n^{\alpha}b_{n}=\lim_{n\rightarrow+\infty}\frac{n^{\alpha}(\theta+1)_{(n)}}{(\alpha+\theta+1)_{(n)}}=\frac{\Gamma(\alpha+\theta+1)}{\Gamma(\theta+1)}.$
(40)

Consequently, it follows from the conjunction of (16), (39) and (40) that

$\lim_{n\rightarrow+\infty}\mathbb{E}[M_{n}^{p}]=\lim_{n\rightarrow+\infty}b_{n}^{\,p}\mathbb{E}\Big{[}\Big{(}K_{n}+\frac{\theta}{\alpha}\Big{)}^{p}\Big{]}=\lim_{n\rightarrow+\infty}b_{n}^{\,p}\sum_{k=0}^{p}{p\choose
k}\mathbb{E}[K_{n}^{k}]\left(\frac{\theta}{\alpha}\right)^{p-k},$

which drastically reduces to

$\lim_{n\rightarrow+\infty}\mathbb{E}[M_{n}^{p}]=\lim_{n\rightarrow+\infty}b_{n}^{\,p}\mathbb{E}[K_{n}^{p}]\\!=\\!\left(\frac{\Gamma(\alpha+\theta+1)}{\Gamma(\theta+1)}\right)^{p}\\!\\!\frac{\Gamma(\theta+1)}{\theta\Gamma(\alpha
p+\theta)}\\!\left(\frac{\theta}{\alpha}\right)^{(p)}\\!\\!\\!\\!\\!\\!\\!.$
(41)

It immediately leads, for all $p\geq 1$, to

$\mathbb{E}[M_{\alpha,\theta}^{p}]=\left(\frac{\Gamma(\alpha+\theta+1)}{\Gamma(\theta+1)}\right)^{p}\frac{\Gamma(\theta+1)}{\theta\Gamma(\alpha
p+\theta)}\left(\frac{\theta}{\alpha}\right)^{(p)},$

as well as

$\mathbb{E}[S_{\alpha,\theta}^{p}]=\frac{\Gamma(\theta+1)}{\theta\Gamma(\alpha
p+\theta)}\left(\frac{\theta}{\alpha}\right)^{(p)}.$

The distribution of the random variable $S_{\alpha,\theta}$ has probability
density function given by (7). Finally, (19) follows from (18) and (39)
together with Riesz-Scheffé theorem, which completes the proof of Theorem 2.1.
∎

######  Remark 2.2.

An alternative proof of the convergence in $\mathbb{L}^{2}$ given by (19),
which follows from the Poisson process construction of the EP model (Pitman,
2003, Proposition 9) can be found in Appendix A. See also Pitman (2006,
Chapter 4).

###  2.2 The Gaussian fluctuation

Our martingale approach for $K_{n}$ suggests that, in addition to the almost-
sure convergence, it is possible to establish a Gaussian fluctuations of
$K_{n}$, properly normalized, around its almost-sure limit
$S_{\alpha,\theta}$.

######  Theorem 2.3.

Let $K_{n}$ be the number of partition sets in a random partition of $[n]$
distributed as the EP model with $\alpha\in(0,1)$ and $\theta>-\alpha$. Then,
we have

$\frac{K_{n}-n^{\alpha}S_{\alpha,\theta}}{\sqrt{K_{n}}}\underset{n\rightarrow+\infty}{\overset{\mathcal{L}}{\longrightarrow}}\mathcal{N}(0,1),$
(42)

where $\mathcal{N}(0,1)$ denotes a standard Gaussian random variable.
Moreover, we also have

$\sqrt{n^{\alpha}}\left(\frac{K_{n}}{n^{\alpha}}-S_{\alpha,\theta}\right)\underset{n\rightarrow+\infty}{\overset{\mathcal{L}}{\longrightarrow}}\sqrt{S_{\alpha,\theta}^{\prime}}\,\mathcal{N}(0,1),$
(43)

where $S_{\alpha,\theta}^{\prime}$ is a random variable independent of
$\mathcal{N}(0,1)$ and sharing the same distribution as $S_{\alpha,\theta}$.

###### Proof.

The proof relies on a beautiful result due to Heyde (1977) concerning with
Gaussian fluctuations for martingales. For all $n\geq 1$, denote $\Delta
M_{n+1}=M_{n+1}-M_{n}$ and

$\Lambda_{n}=\sum_{k=n}^{\infty}\mathbb{E}\big{[}\Delta
M_{k+1}^{2}|\mathcal{F}_{k}\big{]}.$

We immediately have from (17) that

$\Lambda_{n}=\sum_{k=n}^{\infty}b_{k+1}^{2}\left(\frac{(\theta+\alpha
K_{k})(k-\alpha K_{k})}{(k+\theta)^{2}}\right).$ (44)

On the one hand, we know from (18) that

$\lim_{n\rightarrow+\infty}\frac{K_{n}}{n^{\alpha}}=S_{\alpha,\theta}\qquad\text{a.s.}$
(45)

On the other hand, we already saw from (40) that

$\lim_{n\rightarrow+\infty}n^{\alpha}b_{n}=\frac{\Gamma(\alpha+\theta+1)}{\Gamma(\theta+1)}.$
(46)

Consequently, we deduce from (44), (45) and (46) that

$\lim_{n\rightarrow+\infty}n^{\alpha}\Lambda_{n}=\Big{(}\frac{\Gamma(\alpha+\theta+1)}{\Gamma(\theta+1)}\Big{)}^{2}S_{\alpha,\theta}\qquad\text{a.s.}$
(47)

By the same token, if

$s_{n}^{2}=\mathbb{E}[\Lambda_{n}]=\sum_{k=n}^{\infty}\mathbb{E}\big{[}\Delta
M_{k+1}^{2}\big{]},$

we also have

$s_{n}^{2}=\sum_{k=n}^{\infty}b_{k+1}^{2}\left(\frac{\theta
k+\alpha(k-\theta)\mathbb{E}[K_{k}]-\alpha^{2}\mathbb{E}[K_{k}^{2}]}{(k+\theta)^{2}}\right).$

However, we already saw from (39) that

$\lim_{n\rightarrow+\infty}\frac{1}{n^{\alpha}}\mathbb{E}[K_{n}]=\frac{\Gamma(\theta+1)}{\alpha\Gamma(\alpha+\theta)}$
(48)

and

$\lim_{n\rightarrow+\infty}\frac{1}{n^{2\alpha}}\mathbb{E}[K_{n}^{2}]=\frac{(\alpha+\theta)\Gamma(\theta+1)}{\alpha^{2}\Gamma(2\alpha+\theta)}.$
(49)

It ensures that

$\lim_{n\rightarrow+\infty}n^{\alpha}s_{n}^{2}=\Big{(}\frac{\alpha+\theta}{\alpha}\Big{)}\frac{\Gamma(\alpha+\theta+1)}{\Gamma(\theta+1)}.$

Hereafter, we claim that for any $\eta>0$,

$\lim_{n\rightarrow+\infty}n^{\alpha}\sum_{k=n}^{\infty}\mathbb{E}\big{[}\Delta
M_{k+1}^{2}\mathrm{I}_{\\{|\Delta
M_{k+1}|>\eta\sqrt{n^{-\alpha}}\\}}\big{]}=0.$ (50)

As a matter of fact, we clearly have for any $\eta>0$,

$n^{\alpha}\sum_{k=n}^{\infty}\mathbb{E}\big{[}\Delta
M_{k+1}^{2}\mathrm{I}_{\\{|\Delta
M_{k+1}|>\eta\sqrt{n^{-\alpha}}\\}}\big{]}\leq\frac{n^{2\alpha}}{\eta^{2}}\sum_{k=n}^{\infty}\mathbb{E}\big{[}\Delta
M_{k+1}^{4}\big{]}.$

Moreover, it follows from tedious but straightforward calculations that for
all $n\geq 1$,

$\mathbb{E}\big{[}\Delta
M_{n+1}^{4}|\mathcal{F}_{n}\big{]}=b_{n+1}^{4}\sum_{p=0}^{4}e_{n}(p)K_{n}^{p}$
(51)

where

$\displaystyle e_{n}(0)$
$\displaystyle=\frac{n\theta}{(n+\theta)^{4}}(n^{2}-n\theta+\theta^{2}),$
$\displaystyle e_{n}(1)$
$\displaystyle=\frac{\alpha(n-\theta)}{(n+\theta)^{4}}(n^{2}-4n\theta+\theta^{2}),$
$\displaystyle e_{n}(2)$
$\displaystyle=\frac{-2\alpha^{2}(2n-\theta)(n-2\theta)}{(n+\theta)^{4}},$
$\displaystyle e_{n}(3)$
$\displaystyle=\frac{6\alpha^{3}(n-\theta)}{(n+\theta)^{4}},$ $\displaystyle
e_{n}(4)$ $\displaystyle=-\frac{3\alpha^{4}}{n^{4}}.$

Taking the expectation on both sides of (51), we obtain that for all $n\geq
1$,

$\mathbb{E}\big{[}\Delta
M_{n+1}^{4}\big{]}=b_{n+1}^{4}\sum_{p=0}^{4}e_{n}(p)\mathbb{E}[K_{n}^{p}]$
(52)

Furthermore, we already saw from (39) that

$\lim_{n\rightarrow+\infty}\frac{1}{n^{3\alpha}}\mathbb{E}[K_{n}^{3}]=\frac{(\alpha+\theta)(2\alpha+\theta)\Gamma(\theta+1)}{\alpha^{3}\Gamma(3\alpha+\theta)}$
(53)

and

$\lim_{n\rightarrow+\infty}\frac{1}{n^{4\alpha}}\mathbb{E}[K_{n}^{4}]=\frac{(\alpha+\theta)(2\alpha+\theta)(3\alpha+\theta)\Gamma(\theta+1)}{\alpha^{4}\Gamma(4\alpha+\theta)}.$
(54)

Hence, we deduce from (46) and (52) together with (48), (49), (53) and (54)
that

$\lim_{n\rightarrow+\infty}n^{1+3\alpha}\mathbb{E}\big{[}\Delta
M_{n+1}^{4}\big{]}=(\alpha+\theta)\Big{(}\frac{\Gamma(\alpha+\theta+1)}{\Gamma(\theta+1)}\Big{)}^{3}.$
(55)

Consequently, we obtain from (55) that

$\lim_{n\rightarrow+\infty}n^{3\alpha}\sum_{k=n}^{\infty}\mathbb{E}\big{[}\Delta
M_{k+1}^{4}\big{]}=(\alpha+\theta)\Big{(}\frac{\Gamma(\alpha+\theta+1)}{\Gamma(\theta+1)}\Big{)}^{3}$

which clearly leads to (50). Finally, let $(P_{n})$ be the martingale defined
by

$P_{n+1}=\sum_{k=1}^{n}k^{\alpha}\big{(}\Delta M_{k+1}^{2}-\mathbb{E}[\Delta
M_{k+1}^{2}\,|\,\mathcal{F}_{k}]\big{)}.$ (56)

Its predictable quadratic variation is given by

$\langle P\rangle_{n+1}=\sum_{k=1}^{n}k^{2\alpha}\big{(}\mathbb{E}[\Delta
M_{k+1}^{4}\,|\,\mathcal{F}_{k}]-(\mathbb{E}[\Delta
M_{k+1}^{2}\,|\,\mathcal{F}_{k}])^{2}\big{)}.$

Therefore, we find from (51) that

$\langle
P\rangle_{n+1}\leq\sum_{p=0}^{4}\sum_{k=1}^{n}k^{2\alpha}b_{k+1}^{4}e_{k}(p)K_{k}^{p}.$

One can observe that we always have $e_{n}(4)<0$. For $n$ large enough,
$e_{n}(2)<0$. Furthermore, it exists some constant $C>0$ such that for $n$
large enough, $e_{n}(0)\leq C/n$, $e_{n}(1)\leq C/n$ and $e_{n}(3)\leq
C/n^{3}$. Consequently, we get that for $n$ large enough,

$\langle P\rangle_{n}\leq
C\left(\sum_{k=1}^{n}\frac{k^{2\alpha}b_{k}^{4}}{k}+\sum_{k=1}^{n}\frac{k^{2\alpha}b_{k}^{4}K_{k}}{k}+\sum_{k=1}^{n}\frac{k^{2\alpha}b_{k}^{4}K_{k}^{3}}{k^{3}}\right).$
(57)

Hence, we obtain from (45), (46) and (57) that $\langle P\rangle_{n}$
converges a.s. to a finite random variable. Then, we deduce from the strong
law of large numbers for martingales given by the first part of (Duflo, 1997,
Theorem 1.3.15) that $(P_{n})$ converges a.s. to a finite random variable. All
the conditions of the second part of Theorem 1 and Corollaries 1 and 2 in
Heyde (1977) are satisfied, which leads to

$\frac{M_{n}-M_{\alpha,\theta}}{\sqrt{\Lambda_{n}}}\stackrel{{\scriptstyle\mathcal{L}}}{{\longrightarrow}}\mathcal{N}(0,1)$
(58)

as well as to

$\sqrt{n^{\alpha}}(M_{n}-M_{\alpha,\theta})\stackrel{{\scriptstyle\mathcal{L}}}{{\longrightarrow}}\frac{\Gamma(\alpha+\theta+1)}{\Gamma(\theta+1)}\sqrt{S_{\alpha,\theta}^{\prime}}\mathcal{N}(0,1),$
(59)

where $M_{\alpha,\theta}$ is the almost-sure limit of the martingale $(M_{n})$
and $S_{\alpha,\theta}^{\prime}$ is independent of the Gaussian
$\mathcal{N}(0,1)$ random variable and $S_{\alpha,\theta}^{\prime}$ shares the
same distribution as $S_{\alpha,\theta}$. However, the random variables
$M_{\alpha,\theta}$ and $S_{\alpha,\theta}$ are tightly related through

$M_{\alpha,\theta}=\Big{(}\frac{\Gamma(\alpha+\theta+1)}{\Gamma(\theta+1)}\Big{)}S_{\alpha,\theta}.$

In addition, we obtain from the asymptotic behavior of the ratio of two Gamma
functions that

$b_{n}=\frac{\Gamma(\alpha+\theta+1)}{\Gamma(\theta+1)n^{\alpha}}\left(1-\frac{\alpha(2\theta+\alpha-1)}{2n}+O\Big{(}\frac{1}{n^{2}}\Big{)}\right).$
(60)

Finally, the Gaussian fluctuations (42) and (43) follow from (58) and (59)
together with the almost-sure convergences (18), (47), (60) and Slutsky’s
lemma, which achieves the proof of Theorem 2.3. ∎

###  2.3 The law of the iterated logarithm

The law of the iterated logarithm for $K_{n}$ around its limiting random
variable $S_{\alpha,\theta}$ is as follows.

######  Theorem 2.4.

Let $K_{n}$ be the number of partition sets in a random partition of $[n]$
distributed as the EP model with $\alpha\in(0,1)$ and $\theta>-\alpha$. Then,
we have

$\limsup_{n\rightarrow\infty}\left(\frac{K_{n}-n^{\alpha}S_{\alpha,\theta}}{\sqrt{2K_{n}\log\log
n}}\right)=-\liminf_{n\rightarrow\infty}\left(\frac{K_{n}-n^{\alpha}S_{\alpha,\theta}}{\sqrt{2K_{n}\log\log
n}}\right)=1\qquad\text{a.s.}$ (61)

Moreover, we also have

$\limsup_{n\rightarrow\infty}\left(\frac{K_{n}-n^{\alpha}S_{\alpha,\theta}}{\sqrt{2n^{\alpha}\log\log
n}}\right)=-\liminf_{n\rightarrow\infty}\left(\frac{K_{n}-n^{\alpha}S_{\alpha,\theta}}{\sqrt{2n^{\alpha}\log\log
n}}\right)=\sqrt{S_{\alpha,\theta}}\qquad\text{a.s.}$

In particular,

$\limsup_{n\rightarrow\infty}\frac{\big{(}K_{n}-n^{\alpha}S_{\alpha,\theta}\big{)}^{2}}{2n^{\alpha}\log\log
n}=S_{\alpha,\theta}\qquad\text{a.s.}$

###### Proof.

The proof is a direct application of Heyde (1977). Using the same notation as
in the proof of Theorem 2.3, we have for any $\eta>0$,

$\sum_{n=1}^{\infty}n^{\alpha/2}\mathbb{E}\big{[}|\Delta
M_{n}|\mathrm{I}_{\\{|\Delta
M_{n}|>\eta\sqrt{n^{-\alpha}}\\}}\big{]}\leq\frac{1}{\eta^{3}}\sum_{n=1}^{\infty}n^{2\alpha}\mathbb{E}\big{[}\Delta
M_{n}^{4}\big{]}.$ (62)

Hence, we deduce from (55) and (62) that it exists some constant $C>0$ such
that

$\sum_{n=1}^{\infty}n^{\alpha/2}\mathbb{E}\big{[}|\Delta
M_{n}|\mathrm{I}_{\\{|\Delta M_{n}|>\eta\sqrt{n^{-\alpha}}\\}}\big{]}\leq
C\sum_{n=1}^{\infty}\frac{1}{n^{1+\alpha}}<\infty.$

Moreover, we also have for any $\delta>0$,

$\sum_{n=1}^{\infty}n^{2\alpha}\mathbb{E}\big{[}\Delta
M_{n}^{4}\mathrm{I}_{\\{|\Delta
M_{n}|\leq\delta\sqrt{n^{-\alpha}}\\}}\big{]}\leq\sum_{n=1}^{\infty}n^{2\alpha}\mathbb{E}\big{[}\Delta
M_{n}^{4}\big{]}<\infty.$

Furthermore, we already saw that the martingale $(P_{n})$, defined by (56),
converges a.s. to a finite random variable. Consequently, all the conditions
of the second part of Theorem 1 and Corollary 2 in Heyde (1977) are satisfied,
which ensures that

$\limsup_{n\rightarrow\infty}\left(\frac{M_{n}-M_{\alpha,\theta}}{\sqrt{2\Lambda_{n}\log\log
n}}\right)=-\liminf_{n\rightarrow\infty}\left(\frac{M_{n}-M_{\alpha,\theta}}{\sqrt{2\Lambda_{n}\log\log
n}}\right)=1\qquad\text{a.s.}$ (63)

where $M_{\alpha,\theta}$ is the almost-sure limit of the martingale
$(M_{n})$. Finally, the law of iterated logarithm (61) follows from (63)
together with the almost-sure convergences (18) and (47), which completes the
proof of Theorem 2.4. ∎

##  3 Asymptotic results the partition sets with frequency $r$

We start by introducing the keystone martingale construction for $K_{r,n}$.
Our approach is quite different from that developed in Section 2, since we are
going to build a martingale that will not converge almost surely to a finite
random variable. First, we consider the case $r=1$. From the sequential
construction of the EP model (Pitman, 1995, Proposition 9), we have for all
$n\geq 1$,

$K_{1,n+1}=K_{1,n}+\xi_{1,n+1}$ (64)

where the conditional distribution of $\xi_{1,n+1}$, given
$\mathcal{F}_{n}=\sigma(K_{1},\ldots,K_{n},K_{1,1},\ldots,K_{1,n})$, is such
that

$\mathbb{P}(\xi_{1,n+1}=k\,|\,\mathcal{F}_{n})=\left\\{\begin{array}[]{ccc}{p_{1,n}}&\text{
if }&k=1,\vspace{1ex}\\\\[11.38092pt] {q_{1,n}}&\text{ if
}&k=-1,\vspace{1ex}\\\\[11.38092pt] {1-p_{1,n}-q_{1,n}}&\text{ if
}&k=0,\end{array}\right.$ (65)

where

$p_{1,n}=\frac{\alpha K_{n}+\theta}{n+\theta}\qquad\text{and}\qquad
q_{1,n}=\frac{(1-\alpha)K_{1,n}}{n+\theta}.$ (66)

According to the above definition of the sequence $(\xi_{1,n})$, we clearly
have $\mathds{E}[\xi_{1,n+1}\,|\,\mathcal{F}_{n}]=p_{1,n}-q_{1,n}$.
Consequently, we obtain from (64) that

$\mathds{E}[K_{1,n+1}\,|\,\mathcal{F}_{n}]=\mathds{E}[K_{1,n}+\xi_{1,n+1}\,|\,\mathcal{F}_{n}]=K_{1,n}+p_{1,n}-q_{1,n}\qquad\text{a.s},$

which leads to

$\mathds{E}[K_{1,n+1}\,|\,\mathcal{F}_{n}]=\beta_{1,n}K_{1,n}+p_{1,n}\qquad\text{a.s.},$
(67)

where

$\beta_{1,n}=1-\frac{1-\alpha}{n+\theta}=\frac{n-1+\theta+\alpha}{n+\theta}.$
(68)

Let $(b_{1,n})$ be the sequence defined by $b_{1,1}=1$ and for all $n\geq 2$,

$b_{1,n}=\prod_{k=1}^{n-1}\beta_{1,k}^{-1}=\prod_{k=1}^{n-1}\left(\frac{k+\theta}{k-1+\theta+\alpha}\right)=\frac{(\theta+1)^{(n-1)}}{(\alpha+\theta)^{(n-1)}}.$
(69)

Now, we introduce the sequence of random variables $(M_{1,n})$ that is
defined, for all $n\geq 1$, by

$M_{1,n}=b_{1,n}K_{1,n}-A_{1,n},$ (70)

where

$A_{1,n}=\sum_{k=1}^{n-1}b_{1,k+1}\left(\frac{\alpha
K_{k}+\theta}{k+\theta}\right).$ (71)

One can observe that $A_{1,n+1}=A_{1,n}+p_{1,n}b_{1,n+1}$ and
$b_{1,n}=\beta_{1,n}b_{1,n+1}$. Hence, we have from (67) and (68) that for all
$n\geq 1$,

$\displaystyle\mathds{E}[M_{1,n+1}\,|\,\mathcal{F}_{n}]$
$\displaystyle=b_{1,n+1}\mathds{E}[K_{1,n+1}\,|\,\mathcal{F}_{n}]-A_{1,n+1}$
$\displaystyle=b_{1,n+1}(\beta_{1,n}K_{1,n}+p_{1,n})-A_{1,n}-p_{1,n}b_{1,n+1}$
$\displaystyle=\beta_{1,n}b_{1,n+1}K_{1,n}-A_{1,n}$
$\displaystyle=b_{1,n}K_{1,n}-A_{1,n}$ $\displaystyle=M_{1,n}$

almost surely. Moreover, $(M_{1,n})$ is square integrable as $K_{1,n}\leq
K_{n}\leq n$. Consequently, $(M_{1,n})$ is a locally square integrable
martingale. Because of (65), it is not hard to compute all the higher order
conditional moments of $(M_{1,n})$. For example, as

$\Delta
M_{1,n+1}=M_{1,n+1}-M_{1,n}=b_{1,n+1}(K_{1,n+1}-\mathds{E}[K_{1,n+1}\,|\,\mathcal{F}_{n}])$

we clearly have from (64) and (67) that

$\mathds{E}[(\Delta
M_{1,n+1})^{2}\,|\,\mathcal{F}_{n}]=b_{1,n+1}^{2}(\mathds{E}[K_{1,n+1}^{2}\,|\,\mathcal{F}_{n}]-(\mathds{E}[K_{1,n+1}\,|\,\mathcal{F}_{n}])^{2})$

with

$\mathds{E}[K_{1,n+1}^{2}\,|\,\mathcal{F}_{n}]=K_{1,n}^{2}+2K_{1,n}(p_{1,n}-q_{1,n})+p_{1,n}+q_{1,n}$

and

$(\mathds{E}[K_{1,n+1}\,|\,\mathcal{F}_{n}])^{2}=(K_{1,n}+p_{1,n}-q_{1,n})^{2}$

almost surely, which reduces to

$\mathds{E}[(\Delta
M_{1,n+1})^{2}\,|\,\mathcal{F}_{n}]=b_{1,n+1}^{2}\big{(}p_{1,n}+q_{1,n}-(p_{1,n}-q_{1,n})^{2}\big{)}\hskip
28.45274pt\text{a.s.}$ (72)

The above calculations extend easily to the case $r\geq 2$. From the
sequential construction of the EP model (Pitman, 1995, Proposition 9), we have
for all $n\geq 1$,

$K_{r,n+1}=K_{r,n}+\xi_{r,n+1}$ (73)

where the conditional distribution of $\xi_{r,n+1}$, given
$\mathcal{F}_{n}=\sigma(K_{r-1,1},\ldots,K_{r-1,n},K_{r,1},\ldots,K_{r,n})$,
is such that

$\mathbb{P}(\xi_{r,n+1}=k\,|\,\mathcal{F}_{n})=\left\\{\begin{array}[]{ccc}{p_{r,n}}&\text{
if }&k=1,\vspace{1ex}\\\\[11.38092pt] {q_{r,n}}&\text{ if
}&k=-1,\vspace{1ex}\\\\[11.38092pt] {1-p_{r,n}-q_{r,n}}&\text{ if
}&k=0,\end{array}\right.$ (74)

where

$p_{r,n}=\frac{(r-1-\alpha)K_{r-1,n}}{n+\theta}\qquad\text{and}\qquad
q_{r,n}=\frac{(r-\alpha)K_{r,n}}{n+\theta}.$ (75)

As before, we clearly have
$\mathds{E}[\xi_{r,n+1}\,|\,\mathcal{F}_{n}]=p_{r,n}-q_{r,n}$, which implies
that

$\mathds{E}[K_{r,n+1}\,|\,\mathcal{F}_{n}]=\beta_{r,n}K_{r,n}+p_{r,n}\qquad\text{a.s.},$
(76)

where

$\beta_{r,n}=1-\frac{r-\alpha}{n+\theta}=\frac{n-r+\theta+\alpha}{n+\theta}.$
(77)

Let $(b_{r,n})$ be the sequence defined by $b_{r,1}=1$ and for all $n\geq 2$,

$b_{r,n}=\prod_{k=r}^{n-1}\beta_{r,k}^{-1}=\prod_{k=r}^{n-1}\left(\frac{k+\theta}{k-r+\theta+\alpha}\right)=\frac{(\theta+1)^{(n-1)}}{(\alpha+\theta)^{(n-r)}}.$
(78)

Hereafter, we introduce the sequence of random variables $(M_{r,n})$ that is
defined, for all $n\geq 1$, by

$M_{r,n}=b_{r,n}K_{r,n}-A_{r,n},$ (79)

where

$A_{r,n}=\sum_{k=1}^{n-1}b_{r,k+1}\left(\frac{r-1-\alpha}{k+\theta}\right)K_{r-1,k}.$
(80)

Since $A_{r,n+1}=A_{r,n}+p_{r,n}b_{r,n+1}$ and $b_{r,n}=\beta_{r,n}b_{r,n+1}$,
we have from (76) and (77) that for all $n\geq 1$,

$\displaystyle\mathds{E}[M_{r,n+1}\,|\,\mathcal{F}_{n}]$
$\displaystyle=b_{r,n+1}\mathds{E}[K_{r,n+1}\,|\,\mathcal{F}_{n}]-A_{r,n+1}$
$\displaystyle=b_{r,n+1}(\beta_{r,n}K_{r,n}+p_{r,n})-A_{r,n}-p_{r,n}b_{r,n+1}$
$\displaystyle=\beta_{r,n}b_{r,n+1}K_{r,n}-A_{r,n}$
$\displaystyle=b_{r,n}K_{r,n}-A_{r,n}$ $\displaystyle=M_{r,n}$

almost surely. In addition, $(M_{r,n})$ is square integrable as $K_{r,n}\leq
K_{n}\leq n$. Consequently, $(M_{r,n})$ is a locally square integrable
martingale. Because of (74), it is not hard to compute all the higher order
conditional moments of $(M_{r,n})$, as it was previously done for $r=1$.

###  3.1 An alternative approach for the almost-sure convergence

Based on the above martingale construction, we present an alternative proof of
(8), relying on the strong law of large numbers for martingale. Our proof is
more rigorous than Pitman (2006, Lemma 3.11), which contains only a sketch for
the proof of (8).

######  Theorem 3.1.

Let $K_{r,n}$ be the number of partition sets that appear exactly $r$ times in
a random partition of $[n]$ distributed as the EP model with $\alpha\in(0,1)$
and $\theta>-\alpha$. Then, we have for all $r\geq 1$

$\lim_{n\rightarrow+\infty}\frac{K_{r,n}}{n^{\alpha}}=p_{\alpha}(r)S_{\alpha,\theta}\qquad\text{a.s.}$
(81)

where

$p_{\alpha}(r)=\frac{\alpha(1-\alpha)^{(r-1)}}{r!}$

and $S_{\alpha,\theta}$ is a positive and almost surely finite random variable
whose distribution has probability density function (7). This convergence
holds in $\mathbb{L}^{p}$ for any integer $p\geq 1$,

$\lim_{n\rightarrow+\infty}\mathbb{E}\left[\left|\frac{K_{r,n}}{n^{\alpha}}-p_{\alpha}(r)S_{\alpha,\theta}\right|^{p}\right]=0.$
(82)

###### Proof.

The proof is by induction on $r\geq 1$. In particular, we start by considering
the case $r=1$. We already saw that $(M_{1,n})$ is a locally square integrable
martingale. We deduce from (72) that its predictable quadratic variation is
given by

$\langle
M_{1}\rangle_{n}=\sum_{k=1}^{n-1}b_{1,k+1}^{2}\big{(}p_{1,k}+q_{1,k}-(p_{1,k}-q_{1,k})^{2}\big{)}.$

Consequently,

$\langle
M_{1}\rangle_{n}\leq\sum_{k=1}^{n-1}b_{1,k+1}^{2}\big{(}p_{1,k}+q_{1,k}\big{)}.$

On the one hand, it follows directly from the definition of $p_{1,n}$ in (66)
together with the almost-sure convergence (18) that

$\lim_{n\rightarrow+\infty}n^{1-\alpha}p_{1,n}=\alpha
S_{\alpha,\theta}\qquad\text{a.s.}$ (83)

On the other hand, we have from (69) that

$\lim_{n\rightarrow+\infty}\frac{b_{1,n}}{n^{1-\alpha}}=\frac{\Gamma(\alpha+\theta)}{\Gamma(\theta+1)}.$
(84)

Accordingly, as $(n+\theta)q_{1,n}\leq K_{n}$, we deduce from (83) and (84)
through a direct application of Toeplitz lemma that

$\langle
M_{1}\rangle_{n}=O\left(\sum_{k=1}^{n}k^{1-\alpha}\right)=O(n^{2-\alpha})\qquad\text{a.s.}$

Therefore, it follows from the strong law of large numbers for martingales
given by (Duflo, 1997, Theorem 1.3.24) that

$(M_{1,n})^{2}=O(n^{2-\alpha}\log n)\qquad\text{a.s.},$

which leads to

$(b_{1,n}K_{1,n}-A_{1,n})^{2}=O(n^{2-\alpha}\log n)\qquad\text{a.s.}$

It clearly implies that

$\lim_{n\rightarrow+\infty}\frac{b_{1,n}K_{1,n}-A_{1,n}}{n}=0\qquad\text{a.s.}$
(85)

Furthermore, by combining (83) and (84) together with (71), we find that

$\lim_{n\rightarrow+\infty}\frac{A_{1,n}}{n}=\frac{\alpha\Gamma(\alpha+\theta)}{\Gamma(\theta+1)}S_{\alpha,\theta}\qquad\text{a.s.}$
(86)

Consequently, we obtain from (85) and (86) that

$\lim_{n\rightarrow+\infty}\frac{b_{1,n}K_{1,n}}{n}=\lim_{n\rightarrow+\infty}\frac{A_{1,n}}{n}=\frac{\alpha\Gamma(\alpha+\theta)}{\Gamma(\theta+1)}S_{\alpha,\theta}\qquad\text{a.s.}$

Then, we deduce from (84) that

$\lim_{n\rightarrow+\infty}\frac{K_{1,n}}{n^{\alpha}}=\alpha
S_{\alpha,\theta}\qquad\text{a.s.}$

This completes the proof for $r=1$. For $r\geq 2$, we shall proceed by
induction, assuming that

$\lim_{n\rightarrow+\infty}\frac{K_{r-1,n}}{n^{\alpha}}=\frac{\alpha(1-\alpha)^{(r-2)}}{(r-1)!}S_{\alpha,\theta}\qquad\text{a.s.}$
(87)

We already saw that $(M_{r,n})$ is a locally square integrable martingale. As
in the case $r=1$, one can easily see that its predictable quadratic variation
is given by

$\langle
M_{r}\rangle_{n}=\sum_{k=1}^{n-1}b_{r,k+1}^{2}\big{(}p_{r,k}+q_{r,k}-(p_{r,k}-q_{r,k})^{2}\big{)}.$
(88)

On the one hand, it follows from the definition of $p_{r,n}$ in (75) together
with the induction hypothesis (87) that

$\displaystyle\lim_{n\rightarrow+\infty}n^{1-\alpha}p_{r,n}$ $\displaystyle=$
$\displaystyle\frac{(r-1-\alpha)\alpha(1-\alpha)^{(r-2)}}{(r-1)!}S_{\alpha,\theta}\qquad\text{a.s.}$
(89) $\displaystyle=$
$\displaystyle\frac{\alpha(1-\alpha)^{(r-1)}}{(r-1)!}S_{\alpha,\theta}\qquad\text{a.s.}$

On the other hand, we have from (78) that

$\lim_{n\rightarrow+\infty}\frac{b_{r,n}}{n^{r-\alpha}}=\frac{\Gamma(\alpha+\theta)}{\Gamma(\theta+1)}.$
(90)

Accordingly, as $(n+\theta)q_{r,n}\leq rK_{n}$, we deduce from (88), (89) and
(90) together with Toeplitz lemma that

$\langle
M_{r}\rangle_{n}=O\left(\sum_{k=1}^{n}k^{2r-1-\alpha}\right)=O(n^{2r-\alpha})\qquad\text{a.s.}$

Therefore, it follows from the strong law of large numbers for martingales
that

$(M_{r,n})^{2}=O(n^{2r-\alpha}\log n)\qquad\text{a.s.},$

which leads to

$(b_{r,n}K_{r,n}-A_{r,n})^{2}=O(n^{2r-\alpha}\log n)\qquad\text{a.s.}$

As before, it implies that

$\lim_{n\rightarrow+\infty}\frac{b_{r,n}K_{r,n}-A_{r,n}}{n^{r}}=0\qquad\text{a.s.}$
(91)

Hereafter, by combining (89) and (90) together with (80), we obtain that

$\lim_{n\rightarrow+\infty}\frac{A_{r,n}}{n^{r}}=\frac{\Gamma(\alpha+\theta)}{\Gamma(\theta+1)}\frac{\alpha(1-\alpha)^{(r-1)}}{r!}S_{\alpha,\theta}\qquad\text{a.s.}$
(92)

Finally, we find from (91) and (92) that

$\lim_{n\rightarrow+\infty}\frac{b_{r,n}K_{r,n}}{n^{r}}=\lim_{n\rightarrow+\infty}\frac{A_{r,n}}{n^{r}}=\frac{\Gamma(\alpha+\theta)}{\Gamma(\theta+1)}\frac{\alpha(1-\alpha)^{(r-1)}}{r!}S_{\alpha,\theta}\qquad\text{a.s.}$

which ensures via (84) that

$\lim_{n\rightarrow+\infty}\frac{K_{r,n}}{n^{\alpha}}=\frac{\alpha(1-\alpha)^{(r-1)}}{r!}S_{\alpha,\theta}\qquad\text{a.s.}$

It only remains to prove the convergence in $\mathbb{L}^{p}$ given by (82). We
have from Favaro et al. (2013, Proposition 1) that for any integer $p\geq 1$,

$\mathds{E}[(K_{r,n})_{(p)}]=\big{(}p_{r}(\alpha)\big{)}^{p}\frac{n!}{(n-rp)!}\left(\frac{\theta}{\alpha}\right)^{(p)}\frac{(\theta+\alpha
p)^{(n-rp)}}{(\theta)^{(n)}}.$

Consequently,

$\lim_{n\rightarrow+\infty}\frac{1}{n^{\alpha
p}}\mathds{E}[(K_{r,n})_{(p)}]=\big{(}p_{r}(\alpha)\big{)}^{p}\Big{(}\frac{\theta}{\alpha}\Big{)}^{\\!(p)\\!}\frac{\Gamma(\theta+1)}{\theta\Gamma(\alpha
p+\theta)}\Big{(}\frac{\theta}{\alpha}\Big{)}^{\\!(p)\\!}$ (93)

However, as in identity (38), we have

$\mathbb{E}[K_{r,n}^{p}]=\sum_{k=0}^{p}\left\\{\begin{matrix}p\\\
k\end{matrix}\right\\}\mathbb{E}[(K_{r,n})_{(k)}].$ (94)

Then, we deduce from (93) and (94) that

$\lim_{n\rightarrow+\infty}\frac{1}{n^{\alpha
p}}\mathbb{E}[K_{r,n}^{p}]=\big{(}p_{r}(\alpha)\big{)}^{p}\frac{\Gamma(\theta+1)}{\theta\Gamma(\alpha
p+\theta)}\Big{(}\frac{\theta}{\alpha}\Big{)}^{\\!(p)\\!}=\mathds{E}\big{[}\big{(}p_{r}(\alpha)S_{\alpha,\theta}\big{)}^{p}\big{]}.$
(95)

Finally, (82) follows from (81) and (95) together with Riesz-Scheffé theorem,
which completes the proof of Theorem 3.1.

∎

######  Remark 3.2.

An alternative proof of the convergence in $\mathbb{L}^{2}$ given by (3.1),
which follows from the Poisson process construction of the EP model (Pitman,
2003, Proposition 9) can be found in Appendix A. See also Pitman (2006,
Chapter 4).

###  3.2 The Gaussian fluctuation

We shall now focus on a Gaussian fluctuation of $K_{r,n}$, properly
normalized, around an estimator of its almost-sure limit
$p_{\alpha}(r)S_{\alpha,\theta}$.

######  Theorem 3.3.

Let $K_{r,n}$ be the number of partition sets that appear exactly $r$ times in
a random partition of $[n]$ distributed as the EP model with $\alpha\in(0,1)$
and $\theta>-\alpha$. Then, we have for all $r\geq 1$

$\frac{1}{\sqrt{K_{r,n}}}\left(K_{r,n}-\frac{A_{r,n}}{b_{r,n}}\right)\underset{n\rightarrow+\infty}{\overset{\mathcal{L}}{\longrightarrow}}\mathcal{N}(0,1),$
(96)

where $\mathcal{N}(0,1)$ denotes a standard Gaussian random variable.
Moreover, we also have for all $r\geq 1$

$\sqrt{n^{\alpha}}\left(\frac{K_{r,n}}{n^{\alpha}}-\frac{A_{r,n}}{b_{r,n}n^{\alpha}}\right)\underset{n\rightarrow+\infty}{\overset{\mathcal{L}}{\longrightarrow}}\sqrt{p_{\alpha}(r)S_{\alpha,\theta}^{\prime}}\,\mathcal{N}(0,1),$
(97)

where $S_{\alpha,\theta}^{\prime}$ is a random variable independent of
$\mathcal{N}(0,1)$ and sharing the same distribution as $S_{\alpha,\theta}$.

###### Proof.

We are going to carry out the proof for any $r\geq 1$. It follows from (81),
(88), (89) and (90) together with Toeplitz lemma that the predictable
quadratic variation $\langle M_{r}\rangle_{n}$ satisfies

$\lim_{n\rightarrow+\infty}\frac{\langle
M_{r}\rangle_{n}}{n^{2r-\alpha}}=p_{\alpha}(r)\left(\frac{\Gamma(\alpha+\theta)}{\Gamma(\theta+1)}\right)^{2}S_{\alpha,\theta}\hskip
28.45274pt\text{a.s.}$ (98)

Moreover, we have from (82) with $p=1$ and $p=2$ that

$\lim_{n\rightarrow+\infty}n^{1-\alpha}\mathbb{E}[p_{r,n}]=(r-1-\alpha)p_{\alpha}(r-1)\mathbb{E}[S_{\alpha,\theta}]$
(99)

and

$\lim_{n\rightarrow+\infty}n^{2-2\alpha}\mathbb{E}[p_{r,n}^{2}]=(r-1-\alpha)^{2}p^{2}_{\alpha}(r-1)\mathbb{E}[S_{\alpha,\theta}^{2}].$
(100)

In addition, we also have from (82) with $p=1$ and $p=2$ that

$\lim_{n\rightarrow+\infty}n^{1-\alpha}\mathbb{E}[q_{r,n}]=(r-\alpha)p_{\alpha}(r)\mathbb{E}[S_{\alpha,\theta}]$
(101)

and

$\lim_{n\rightarrow+\infty}n^{2-2\alpha}\mathbb{E}[q_{r,n}^{2}]=(r-\alpha)^{2}p^{2}_{\alpha}(r)\mathbb{E}[S_{\alpha,\theta}^{2}].$
(102)

Therefore, denote $s_{r,n}^{2}=\mathbb{E}[\langle M_{r}\rangle_{n}]$. We
deduce from (88) and (90) together with (99), (100), (101) and (102) that

$\displaystyle\lim_{n\rightarrow+\infty}\frac{s_{r,n}^{2}}{n^{2r-\alpha}}$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow+\infty}\frac{1}{n^{2r-\alpha}}\sum_{k=1}^{n-1}b_{r,k+1}^{2}\big{(}\mathbb{E}[p_{r,k}]+\mathbb{E}[q_{r,k}]\big{)},$
(103) $\displaystyle=$
$\displaystyle\frac{p_{\alpha}(r)}{\alpha}\left(\frac{\Gamma(\alpha+\theta)}{\Gamma(\theta+1)}\right).$

Hence, we obtain from (98) and (103) that

$\lim_{n\rightarrow+\infty}\frac{\langle
M_{r}\rangle_{n}}{s_{r,n}^{2}}=\alpha\left(\frac{\Gamma(\alpha+\theta)}{\Gamma(\theta+1)}\right)S_{\alpha,\theta}\hskip
28.45274pt\text{a.s.}$ (104)

We are now going to show that for any $\eta>0$,

$\lim_{n\rightarrow+\infty}\frac{1}{n^{2r-\alpha}}\sum_{k=1}^{n}\mathbb{E}\big{[}\Delta
M_{r,k}^{2}\mathrm{I}_{\\{|\Delta M_{r,k}|>\eta s_{r,n}\\}}\big{]}=0.$ (105)

We clearly have for any $\eta>0$,

$\sum_{k=1}^{n}\mathbb{E}\big{[}\Delta M_{r,k}^{2}\mathrm{I}_{\\{|\Delta
M_{r,k}|>\eta
s_{r,n}\\}}\big{]}\leq\frac{1}{\eta^{2}s_{r,n}^{2}}\sum_{k=1}^{n}\mathbb{E}\big{[}\Delta
M_{r,k}^{4}\big{]}.$

According to (103), in order to prove (105), it is only necessary to show that

$\lim_{n\rightarrow+\infty}\frac{1}{n^{4r-2\alpha}}\sum_{k=1}^{n}\mathbb{E}\big{[}\Delta
M_{r,k}^{4}\big{]}=0.$ (106)

As in the proof of (51), it follows from tedious but straightforward
calculations that for all $n\geq 1$,

$\mathbb{E}\big{[}\Delta
M_{r,n+1}^{4}|\mathcal{F}_{n}\big{]}=b_{r,n+1}^{4}\sum_{\ell=1}^{4}e_{r,n}(\ell)$
(107)

where

$\displaystyle e_{r,n}(1)$ $\displaystyle=(p_{r,n}+q_{r,n}),$ $\displaystyle
e_{r,n}(2)$ $\displaystyle=-4(p_{r,n}-q_{r,n})^{2},$ $\displaystyle
e_{r,n}(3)$ $\displaystyle=6(p_{r,n}+q_{r,n})(p_{r,n}-q_{r,n})^{2},$
$\displaystyle e_{r,n}(4)$ $\displaystyle=-3(p_{r,n}-q_{r,n})^{4}.$

Then, it follows from (107) that it exists some constant $C>0$ such that for
$n$ large enough,

$\mathbb{E}\big{[}\Delta M_{r,n+1}^{4}|\mathcal{F}_{n}\big{]}\leq
Cb_{r,n+1}^{4}\big{(}p_{r,n}+q_{r,n}\big{)}.$ (108)

Taking the expectation on both sides of (108), we obtain that for $n$ large
enough,

$\mathbb{E}\big{[}\Delta M_{r,n+1}^{4}\big{]}\leq
Cb_{r,n+1}^{4}\big{(}\mathbb{E}[p_{r,n}]+\mathbb{E}[q_{r,n}]\big{)}.$ (109)

Consequently, we deduce from (90), (99), (101) and (109) that

$\sum_{k=1}^{n}\mathbb{E}\big{[}\Delta
M_{r,k}^{4}\big{]}=O\left(\sum_{k=1}^{n}k^{4r-1-3\alpha}\right)=O(n^{4r-3\alpha})$

which leads to (106). Hence, we find from Corollary 1 in Heyde (1977) that

$\frac{M_{r,n}}{\sqrt{\langle
M_{r}\rangle_{n}}}\stackrel{{\scriptstyle\mathcal{L}}}{{\longrightarrow}}\mathcal{N}(0,1),$
(110)

and

$\frac{M_{r,n}}{s_{r,n}}\underset{n\rightarrow+\infty}{\overset{\mathcal{L}}{\longrightarrow}}\sqrt{\frac{\alpha\Gamma(\alpha+\theta)}{\Gamma(\theta+1)}}\,\sqrt{S_{\alpha,\theta}^{\prime}}\mathcal{N}(0,1)$
(111)

where $S_{\alpha,\theta}^{\prime}$ stands for a random variable independent of
$\mathcal{N}(0,1)$ and sharing the same distribution as $S_{\alpha,\theta}$.
Finally, as $M_{r,n}=b_{r,n}K_{r,n}-A_{r,n}$, the Gaussian fluctuations (96)
and (97) follow from (90), (110) and (111) together with the almost-sure
convergences (81), (98), (104) and Slutsky’s lemma, which achieves the proof
of Theorem 3.3. ∎

###  3.3 The law of the iterated logarithm

The law of the iterated logarithm for $K_{r,n}$ is as follows.

######  Theorem 3.4.

Let $K_{r,n}$ be the number of partition sets that appear exactly $r$ times in
a random partition of $[n]$ distributed as the EP model with $\alpha\in(0,1)$
and $\theta>-\alpha$. Then, we have for all $r\geq 1$

$\displaystyle\limsup_{n\rightarrow\infty}\sqrt{\frac{n^{\alpha}}{2\log\log
n}}\left(\frac{K_{r,n}}{n^{\alpha}}-\frac{A_{r,n}}{b_{r,n}n^{\alpha}}\right)$
$\displaystyle=$
$\displaystyle-\liminf_{n\rightarrow\infty}\sqrt{\frac{n^{\alpha}}{2\log\log
n}}\left(\frac{K_{r,n}}{n^{\alpha}}-\frac{A_{r,n}}{b_{r,n}n^{\alpha}}\right)$
$\displaystyle=$
$\displaystyle\sqrt{p_{\alpha}(r)S_{\alpha,\theta}}\qquad\text{a.s.}$

In particular,

$\limsup_{n\rightarrow\infty}\frac{n^{\alpha}}{2\log\log
n}\left(\frac{K_{r,n}}{n^{\alpha}}-\frac{A_{r,n}}{b_{r,n}n^{\alpha}}\right)^{2}=p_{\alpha}(r)S_{\alpha,\theta}\qquad\text{a.s.}$

###### Proof.

As in Section 2, the proof is a direct application of Heyde (1977). We have
for any $\eta>0$,

$\sum_{n=1}^{\infty}\frac{n^{\alpha/2}}{n^{r}}\mathbb{E}\big{[}|\Delta
M_{r,n}|\mathrm{I}_{\\{\sqrt{n^{\alpha}}|\Delta M_{r,n}|>\eta
n^{r}\\}}\big{]}\leq\frac{1}{\eta^{3}}\sum_{n=1}^{\infty}\frac{n^{2\alpha}}{n^{4r}}\mathbb{E}\big{[}\Delta
M_{r,n}^{4}\big{]}.$ (113)

Hence, we deduce from (109) and (113) together with (90), (99) and (101) that
it exists some constant $C>0$ such that

$\sum_{n=1}^{\infty}\frac{n^{\alpha/2}}{n^{r}}\mathbb{E}\big{[}|\Delta
M_{r,n}|\mathrm{I}_{\\{\sqrt{n^{\alpha}}|\Delta M_{r,n}|>\eta
n^{r}\\}}\big{]}\leq C\sum_{n=1}^{\infty}\frac{1}{n^{1+\alpha}}<\infty.$

Moreover, we also have for any $\delta>0$,

$\sum_{n=1}^{\infty}\frac{n^{2\alpha}}{n^{4r}}\mathbb{E}\big{[}\Delta
M_{r,n}^{4}\mathrm{I}_{\\{\sqrt{n^{\alpha}}|\Delta M_{r,n}|\leq\delta
n^{r}\\}}\big{]}\leq\sum_{n=1}^{\infty}\frac{n^{2\alpha}}{n^{4r}}\mathbb{E}\big{[}\Delta
M_{r,n}^{4}\big{]}<\infty.$

Hereafter, let $(P_{r,n})$ be the martingale defined by

$P_{r,n+1}=\sum_{k=1}^{n}\frac{k^{\alpha}}{k^{2r}}\big{(}\Delta
M_{r,k+1}^{2}-\mathbb{E}[\Delta M_{r,k+1}^{2}\,|\,\mathcal{F}_{k}]\big{)}.$
(114)

Its predictable quadratic variation is given by

$\langle
P_{r}\rangle_{n+1}=\sum_{k=1}^{n}\frac{k^{2\alpha}}{k^{4r}}\big{(}\mathbb{E}[\Delta
M_{k+1}^{4}\,|\,\mathcal{F}_{k}]-(\mathbb{E}[\Delta
M_{k+1}^{2}\,|\,\mathcal{F}_{k}])^{2}\big{)}.$

Hence, we obtain from (108) that it exists some constant $C>0$ such that for
$n$ large enough,

$\langle P_{r}\rangle_{n}\leq
C\sum_{k=1}^{n}\frac{k^{2\alpha}}{k^{4r}}b_{r,k+1}^{4}\big{(}p_{r,k}+q_{r,k}\big{)}.$
(115)

Consequently, we deduce from (88), (89) and (115) that $\langle
P_{r}\rangle_{n}$ converges a.s. to a finite random variable. Then, it follows
from the strong law of large numbers for martingales that $(P_{r,n})$
converges a.s. to a finite random variable. Therefore, all the conditions of
the second part of Theorem 1 and Corollary 2 in Heyde (1977) are satisfied,
which ensures that

$\limsup_{n\rightarrow\infty}\frac{M_{r,n}}{\sqrt{2\langle
M_{r}\rangle_{n}\log\log
n}}=-\liminf_{n\rightarrow\infty}\frac{M_{r,n}}{\sqrt{2\langle
M_{r}\rangle_{n}\log\log n}}=1\quad\text{a.s.}$ (116)

Finally, as $M_{r,n}=b_{r,n}K_{r,n}-A_{r,n}$, the law of iterated logarithm
(3.4) follows from (90) together with the almost-sure convergence (98) and
(116), which completes the proof of Theorem 3.4. ∎

##  4 Discussion

In this paper, we presented a unified and comprehensive treatment of the large
$n$ asymptotic behaviour of $K_{n}$ and $K_{r,m}$ in the EP model with
$\alpha\in(0,1)$ and $\theta>-\alpha$. By means of a novel martingale
construction for $K_{n}$ and $K_{r,m}$, we obtained alternative, and rigorous,
proofs of the almost-sure convergence of $K_{n}$ and $K_{r,n}$, and also
covered the gap of Gaussian fluctuations. We argue that our martingale
approach may be further investigated to refine Theorem 2.1 and Theorem 3.1 in
terms of Berry–Esseen theorems, as well as sharp large deviations and
concentration inequalities for $K_{n}$ and $K_{r,n}$. We refer to Feng and
Hoppe (1998), Favaro et al. (2015), Favaro et al. (2018), Dolera and Favaro
(2020) and Oliveira et al. (2022) for early results along these lines of
research. Investigating the large $n$ asymptotic behaviour of functions that
involve both $K_{n}$ and the $K_{r,n}$’s is also a promising direction for
future research. In particular, our almost-sure limits imply

$\lim_{n\rightarrow+\infty}\frac{K_{1,n}}{K_{n}}=\alpha\qquad\text{a.s.},$

meaning that $K_{1,n}/K_{n}$ is a consistent estimator for the parameter
$\alpha$. Establishing a Gaussian fluctuation for $K_{1,n}/K_{n}$ is an
interesting open problem, with potential applications in Bayesian
nonparametrics (Balocchi et al., 2023). A further problem of interest is to
establish an almost-sure limit and a Gaussian fluctuation for
$(K_{1,n},K_{2,n},\ldots)$, thus providing a counterpart of (5).

## Appendix A. Alternative proofs of the convergence in $\mathbb{L}^{2}$

###  A.1 Convergence in $\mathbb{L}^{2}$ for the number of partition sets

We propose an alternative proof of the convergence in $\mathbb{L}^{2}$,

$\lim_{n\rightarrow+\infty}\mathbb{E}\left[\left(\frac{K_{n}}{n^{\alpha}}-S_{\alpha,\theta}\right)^{2}\right]=0,$
(A.1)

which follows from the Poisson process construction of the EP model (Pitman,
2003, Proposition 9). See also Pitman (2006, Chapter 4). From Pitman (2006,
Equation 3.11),

$\mathds{P}(K_{n}=k)=\frac{\left(\frac{\theta}{\alpha}\right)^{(k)}}{(\theta)^{(n)}}\mathscr{C}(n,k;\alpha),$
(A.2)

where $\mathscr{C}(n,k;\alpha)$ is the generalized factorial coefficient

$\mathscr{C}(n,k;\alpha)=\frac{1}{k!}\sum_{j=1}^{k}(-1)^{j}{k\choose
j}(-j\alpha)^{(n)}.$

Denote $T_{\alpha,\theta}=S_{\alpha,\theta}^{-1/\alpha}$. By combining Pitman
(2003, Equation 66) with Charalambides (2005, Equation 2.61), we have

$\mathds{P}(K_{n}=k\,|\,T_{\alpha,\theta}=t)=V_{n,k}(t)\frac{\mathscr{C}(n,k;\alpha)}{\alpha^{k}},$
(A.3)

where

$V_{n,k}(t)=\frac{1}{\Gamma(n-k\alpha)}\left(\frac{\alpha}{t^{\alpha}}\right)^{k}\int_{0}^{1}v^{n-1-k\alpha}\frac{f_{\alpha}((1-v)t)}{f_{\alpha}(t)}\mathrm{d}v$

with $f_{\alpha}$ being the positive $\alpha$-Stable density function. Then,
it follows from (7) and (A.3), as well as the tower property of the
conditional expectation, that

$\displaystyle\mathds{E}[K_{n}S_{\alpha,\theta}]$
$\displaystyle=\int_{0}^{\infty}\mathds{E}[K_{n}S_{\alpha,\theta}|S_{\alpha,\theta}=s]f_{S_{\alpha,\theta}}(s)\mathrm{d}s,$
$\displaystyle=\frac{\Gamma(\theta+1)}{\Gamma\left(\theta/\alpha+1\right)}\int_{0}^{+\infty}\mathds{E}[K_{n}|T_{\alpha,\theta}=t]t^{-\alpha-\theta}f_{\alpha}(t)\mathrm{d}t,$
$\displaystyle=\frac{\Gamma(\theta+1)}{\Gamma\left(\theta/\alpha+1\right)}\sum_{k=1}^{n}k\int_{0}^{+\infty}\mathds{P}(K_{n}=k\,|\,T_{\alpha,\theta}=t)t^{-\alpha-\theta}f_{\alpha}(t)\mathrm{d}t,$
$\displaystyle=\frac{\Gamma(\theta+1)}{\Gamma\left(\theta/\alpha+1\right)}\sum_{k=1}^{n}\frac{k\mathscr{C}(n,k;\alpha)}{\Gamma(n-k\alpha)}I_{n,k}(t),$
(A.4)

where

$\displaystyle I_{n,k}(t)$
$\displaystyle=\int_{0}^{+\infty}t^{-k\alpha-\alpha-\theta}\int_{0}^{1}v^{n-1-k\alpha}f_{\alpha}((1-v)t)\mathrm{d}v\mathrm{d}t,$
$\displaystyle=\int_{0}^{+\infty}z^{-k\alpha-\alpha-\theta}\int_{0}^{1}v^{n-1-k\alpha}(1-v)^{k\alpha+\alpha+\theta-1}f_{\alpha}(z)\mathrm{d}v\mathrm{d}z.$

Since

$\int_{0}^{+\infty}z^{-k\alpha-\alpha-\theta}f_{\alpha}(z)\mathrm{d}z=\frac{\Gamma\left(\theta/\alpha+k+2\right)}{\Gamma(\alpha+\theta+k\alpha+1)},$

and

$\int_{0}^{1}v^{n-k\alpha-1}(1-v)^{k\alpha+\alpha+\theta-1}\mathrm{d}v=\frac{\Gamma(n-k\alpha)\Gamma(k\alpha+\alpha+\theta)}{\Gamma(n+\alpha+\theta)},$

the expectation (A.4) reduces to

$\displaystyle\mathds{E}[K_{n}S_{\alpha,\theta}]$
$\displaystyle=\frac{\Gamma(\theta+1)}{\Gamma\left(\theta/\alpha+1\right)\Gamma(n+\alpha+\theta)}\sum_{k=1}^{n}\frac{k\mathscr{C}(n,k;\alpha)}{(\alpha+\theta+k\alpha)}\Gamma\left(\theta/\alpha+k+2\right),$
$\displaystyle=\frac{\Gamma(\theta+n)}{\Gamma(n+\alpha+\theta)}\sum_{k=1}^{n}k\left(\frac{\theta}{\alpha}+k\right)\frac{\left(\frac{\theta}{\alpha}\right)^{(k)}}{(\theta)^{(n)}}\mathscr{C}(n,k;\alpha).$
(A.5)

Hence, it follows from (A.2) and (A.5) that

$\mathds{E}[K_{n}S_{\alpha,\theta}]=\frac{\Gamma(\theta+n)}{\Gamma(n+\alpha+\theta)}\left(\frac{\theta}{\alpha}\mathds{E}[K_{n}]+\mathds{E}[K^{2}_{n}]\right).$
(A.6)

We are now in the position to prove (A.1). We get from (39) with $p=1$ and
$p=2$ that

$\lim_{n\rightarrow+\infty}\frac{1}{n^{\alpha}}\mathbb{E}[K_{n}]=\mathds{E}[S_{\alpha,\theta}]\hskip
22.76228pt\text{and}\hskip
22.76228pt\lim_{n\rightarrow+\infty}\frac{1}{n^{2\alpha}}\mathbb{E}[K_{n}^{2}]=\mathds{E}[S_{\alpha,\theta}^{2}],$
(A.7)

leading to

$\lim_{n\rightarrow+\infty}\frac{1}{n^{\alpha}}\mathbb{E}[K_{n}S_{\alpha,\theta}]=\mathds{E}[S_{\alpha,\theta}^{2}].$
(A.8)

Therefore, we deduce from (A.7) and (A.8) that

$\displaystyle\lim_{n\rightarrow+\infty}\mathds{E}\left[\left(\frac{K_{n}}{n^{\alpha}}-S_{\alpha,\theta}\right)^{2}\right]$
$\displaystyle=\lim_{n\rightarrow+\infty}\frac{1}{n^{2\alpha}}\mathbb{E}[K_{n}^{2}]-\frac{2}{n^{\alpha}}\mathbb{E}[K_{n}S_{\alpha,\theta}]+\mathds{E}[S_{\alpha,\theta}^{2}],$
$\displaystyle=\mathds{E}[S_{\alpha,\theta}^{2}]-2\mathds{E}[S_{\alpha,\theta}^{2}]+\mathds{E}[S_{\alpha,\theta}^{2}]=0.$

Similar arguments can be applied in order to show the convergence in
$\mathbb{L}^{p}$ of $n^{-\alpha}K_{n}$ to $S_{\alpha,\theta}$, for any integer
$p\geq 1$, thus providing an alternative proof of (19).

###  A.2 Convergence in $\mathbb{L}^{2}$ for partition sets with frequency $r$

We still rely on the Poisson process construction of the EP model (Pitman,
2003, Proposition 9) to propose an alternative proof of the convergence in
$\mathbb{L}^{2}$,

$\lim_{n\rightarrow+\infty}\mathbb{E}\left[\left(\frac{K_{r,n}}{n^{\alpha}}-p_{\alpha}(r)S_{\alpha,\theta}\right)^{2}\right]=0.$
(A.9)

Denote $T_{\alpha,\theta}=S_{\alpha,\theta}^{-1/\alpha}$. By combining Pitman
(2006, Equation 66) with Favaro et al. (2013, Proposition 1), we have

$\mathds{P}(K_{r,n}=k\,|\,T_{\alpha,\theta}=t)=p_{\alpha}(r)(n)_{(r)}V_{n,k}(t)\frac{\mathscr{C}(n-r,k;\alpha)}{\alpha^{k}}$
(A.10)

where

$V_{n,k}(t)=\frac{1}{\Gamma(n-k\alpha)}\left(\frac{\alpha}{t^{\alpha}}\right)^{k}\int_{0}^{1}v^{n-1-k\alpha}\frac{f_{\alpha}((1-v)t)}{f_{\alpha}(t)}\mathrm{d}v$

with $f_{\alpha}$ being the positive $\alpha$-Stable density function.
Consequently, as

$\mathds{E}[K_{r,n}S_{\alpha,\theta}]=\frac{\Gamma(\theta+1)}{\Gamma\left(\theta/\alpha+1\right)}\int_{0}^{+\infty}\mathds{E}[K_{r,n}|T_{\alpha,\theta}=t]t^{-\alpha-\theta}f_{\alpha}(t)\mathrm{d}t,$

we obtain along lines similar to the computation of
$\mathds{E}[K_{n}S_{\alpha,\theta}]$ in (A.5) that

$\displaystyle\mathds{E}[K_{r,n}S_{\alpha,\theta}]$
$\displaystyle=\frac{\Gamma(\theta+1)}{\alpha\Gamma(n+\alpha+\theta)}p_{\alpha}(r)(n)_{(r)}\sum_{k=1}^{n}\left(\frac{\theta}{\alpha}+1\right)^{(k)}\mathscr{C}(n-r,k-1;\alpha)$
$\displaystyle=\frac{\Gamma(\theta+1)}{\alpha\Gamma(n+\alpha+\theta)}p_{\alpha}(r)(n)_{(r)}\frac{\Gamma\left(\theta/\alpha+2\right)}{\Gamma\left(\theta/\alpha+1\right)}\sum_{k=0}^{n-r}\left(\frac{\theta}{\alpha}+2\right)^{(k)}\mathscr{C}(n-r,k;\alpha),$

which leads to

$\mathds{E}[K_{r,n}S_{\alpha,\theta}]=\frac{\Gamma(\theta+1)}{\Gamma(n+\alpha+\theta)}p_{\alpha}(r)(n)_{(r)}\left(\frac{\alpha+\theta}{\alpha^{2}}\right)(\theta+2\alpha)^{(n-r)}.$
(A.11)

We are now in position to prove (A.9). We find from (95) with $p=2$ that

$\lim_{n\rightarrow+\infty}\frac{1}{n^{2\alpha}}\mathbb{E}[K_{r,n}^{2}]=\big{(}p_{r}(\alpha)\big{)}^{2}\mathds{E}[\big{(}S_{\alpha,\theta}\big{)}^{2}].$
(A.12)

In addition, we deduce from (A.11) that

$\lim_{n\rightarrow+\infty}\frac{1}{n^{\alpha}}\mathds{E}[K_{r,n}S_{\alpha,\theta}]=p_{r}(\alpha)\left(\frac{\alpha+\theta}{\alpha^{2}}\right)\frac{\Gamma(\theta+1)}{\Gamma(2\alpha+\theta)}=p_{r}(\alpha)\mathds{E}[S_{\alpha,\theta}^{2}].$
(A.13)

Therefore, it follows from (A.12) and (A.13) that

$\displaystyle\lim_{n\rightarrow+\infty}\mathds{E}\left[\left(\frac{K_{r,n}}{n^{\alpha}}-p_{r}(\alpha)S_{\alpha,\theta}\right)^{2}\right]$
$\displaystyle\quad=\lim_{n\rightarrow+\infty}\frac{1}{n^{2\alpha}}\mathbb{E}[K_{r,n}^{2}]-\frac{2p_{r}(\alpha)}{n^{\alpha}}\mathbb{E}[K_{r,n}S_{\alpha,\theta}]+\big{(}p_{r}(\alpha)\big{)}^{2}\mathds{E}[\big{(}S_{\alpha,\theta}\big{)}^{2}],$
$\displaystyle\quad=\big{(}p_{r}(\alpha)\big{)}^{2}\mathds{E}[S_{\alpha,\theta}^{2}]-2\big{(}p_{r}(\alpha)\big{)}^{2}\mathds{E}[S_{\alpha,\theta}^{2}]+\big{(}p_{r}(\alpha)\big{)}^{2}\mathds{E}[\big{(}S_{\alpha,\theta}\big{)}^{2}]=0.$

Similar arguments can be applied to show the convergence in $\mathbb{L}^{p}$
of $n^{-\alpha}K_{r,n}$ to $p_{\alpha}(r)S_{\alpha,\theta}$, for any integer
$p\geq 1$, thus leading to an alternative proof of (82).

## Acknowledgement

Stefano 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. Stefano Favaro acknowledge the financial support
from the Italian Ministry of Education, University and Research (MIUR),
“Dipartimenti di Eccellenza” grant 2023-2027.

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