A compound Poisson perspective of Ewens-Pitman sampling model

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.


Introduction
The Pitman-Yor process is a discrete random probability measure indexed by real parameters α and θ such that either α ∈ [0, 1) and θ > −α, or α < 0 and θ = −mα for some m ∈ N. See, e.g., the works of Perman et al. (1992), Pitman (1995) and Pitman and Yor (1997). Let {V i } i≥1 be independent random variables such that V i is distributed as a Beta distribution with parameter (1 − α, θ + iα), for i ≥ 1, with the convention for α < 0 that V m = 1 and V i is undefined for i > m. If P 1 := V 1 and P i := V i 1≤j≤i−1 (1 − V j ) for i ≥ 2, such that i≥1 P i = 1 almost surely, then the Pitman-Yor process is the random probability measurep α,θ on (N, 2 N ) such thatp α,θ ({i}) = P i for i ≥ 1. The Dirichlet process (Ferguson, 1973) arises for α = 0. Because of the discreteness of p α,θ , a random sample (X 1 , . . . , X n ) induces a random partition Π n of {1, . . . , n} by means of the equivalence i ∼ j ⇐⇒ X i = X j (Pitman, 2006). Let K n (α, θ) := K n (X 1 , . . . , X n ) ≤ n be the number of blocks of Π n and let M r,n (α, θ) := M r,n (X 1 , . . . , X n ), for r = 1, . . . , n, be the number of blocks with frequency r of Π n with 1≤r≤n M r,n = K n and 1≤r≤n rM r,n = n. Pitman (1995) showed that Pr[(M 1,n (α, θ), . . . , M n,n (α, θ)) = (x 1 , . . . , with (x) (n) being the ascending factorial of x of order n, i.e. (x) (n) := 0≤i≤n−1 (x + i). The distribution (1) is referred to as Ewens-Pitman sampling model (EP-SM), and for α = 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 λ = −z log(1 − q) for q ∈ (0, 1) and z > 0. For i ∈ 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 is the random number of N i 's equal to r such that r≥1 M r = K and r≥1 rM r = S. If (M 1 (z, n), . . . , M n (z, n)) denotes a random variable whose distribution coincides with the conditional distribution of (M 1 , . . . , M S ) given S = n, then (Charalambides, 2007, Section 3) it holds that 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 α = 0. Therefore, the distributions of K(z, n) = 1≤r≤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 w −→ denote the weak convergence for random variables. From the work of Korwar and Hollander (1973), K(z, n)/ log n w −→ z as n → +∞, whereas from Ewens (1972) it follows that M r (z, n) w −→ P z/r as n → +∞, 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 α and z such that either α ∈ (0, 1) and z > 0, or α < 0 and z < 0. The LS-CPSM is recovered by letting α → 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 α ∈ (0, 1), or α < 0. That is, we show that: i) for α ∈ (0, 1) the EP-SM (1) admits a representation as a randomized NB-CPSM with α ∈ (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 α < 0 the NB-CPSM admits a representation in terms of a randomized EP-SM with α < 0 and θ = −mα for some m ∈ 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 α-diversity (Pitman, 2006), namely the large n asymptotic behaviour of K n (α, θ) under the EP-SM.

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 λ = z[1 − (1 − q) α ] such that either q ∈ (0, 1), α ∈ (0, 1) and z > 0, or q ∈ (0, 1), α < 0 and z < 0. For i ∈ 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 for x ∈ N. Let S = 1≤i≤K N i and M r = 1≤i≤K ½ {N i =r} for r = 1, . . . , S, that is M r is the random number of N i 's equal to r such that r≥1 M r = K and r≥1 rM r = S. If (M 1 (α, z, n), . . . , M n (α, z, n)) is a random variable whose distribution coincides with the conditional distribution of (M 1 , . . . , M S ) given S = n, then it holds (Charalambides, 2007, Section 3) that where C (n, j; α) = 1 j! 0≤i≤j j i (−1) i (−iα) (n) is the generalized factorial coefficient (Charalambides, 2005), with the proviso C (n, 0, α) = 0 for all n ∈ N, C (n, j, α) = 0 for all j > n and C (0, 0, α) = 1. The distribution (5) is referred to as the NB-CPSM. In particular, as α → 0, it is easy to show that the distribution (4) reduces to the distribution (2). Accordingly, as α → 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(α, z, n) = 1≤r≤n M r (α, z, n) and M r (α, z, n) arising from the NB-CPSM.
Theorem 1. Let P λ denote a Poisson random variable with parameter λ > 0. As n → +∞ it holds i) for α ∈ (0, 1) and z > 0 ii) for α < 0 and z < 0 Proof. As regard the proof of the asymptotic behaviour in (6), we start by recalling that the probability generating function G(·; λ) of P λ is G(s; λ) = exp{−λ(s − 1)} for any s > 0. Now, let G(·; α, z, n) be the probability generating function of K(α, z, n). The distribution of K(α, z, n) follows by combining the NB-CPSM (5) with Theorem 2.15 of Charalambides (2005). In particular, it follows that Hereafter, we show that G(s; α, z, n) → s exp{z(s − 1)} as n → +∞, for any s > 0, which implies (6). In particular, by a direct application of the definition of C (n, k; α) we write the following identities where Γ(a, x) := +∞ x t a−1 e −t dt denotes the incomplete gamma function for a, x > 0 and Γ(a) := +∞ 0 t a−1 e −t dt denotes the Gamma function for a > 0. Accordingly, we can write the following identity . Now, since lim n→+∞ Γ(n,x) Γ(n) = 1 for any x > 0, the proof (6) is completed by showing that, for any t > 0, By the definition of ascending factorials and the reflection formula of the Gamma function, it holds true In particular, by means of the monotonicity of the function [1, +∞) ∋ z → Γ(z), we can write the identity 1 i!
for any n ∈ N such that n > 1/(1 − α), and i ∈ {2, . . . , n}. Note that Γ(n,x) Γ(n) ≤ 1. Then we apply (11) to get Now, by means of Stirling approximation it holds Γ(n−2α) Γ(n−α) ∼ 1 n α as n → +∞. Moreover, we have that where the finiteness of the integral follows, for any fixed t > 0, from the fact that tz α < 1 2 z if z > (2t) 1 1−α . 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 (α, 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] = 0≤i≤n−1 (a − i), for any a ∈ R + and n ∈ N 0 with the proviso (a) [0] = 1. Then, we write . Now, by means of the same argument applied in the proof of the statement in (6), it holds true that as n → +∞. Finally, the proof of the large n asymptotics (7) is completed by recalling that falling factorial moment of order s of P λ is E[(P λ ) [s] ] = λ s .
As regard the proof of the asymptotic behaviour in (8), let α = −σ for any σ > 0 and let z = −ζ for any ζ > 0. Then, by a direct application of Equation 2.27 of Charalambides (2005), we write the identity where S(v, j) is the Stirling number of that second type. Now, note that v 0≤j≤v ζ j S(v, j) is the moment of order v of a Poisson random variable with parameter ζ > 0. Then, we write the following identities That is, 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 σPw,1 , say µ σ,w is the following for t > 0. The discrete component of the distribution µ σ,w does not contribute in the expectation (13), so that we focus on the absolutely continuous component, whose density can be written as follows where W σ,τ (y) := j≥0 y j j!Γ(jσ+τ ) is the Wright function (Wright, 1935). In particular, for τ = 0 it holds If we split the integral as for any M > 0, the contribution of the latter integral is overwhelming with respect to the contribution of the former. Then W σ,0 can be equivalently replaced by the asymptotics W σ,0 (y) ∼ c(σ)y 1 2(1+σ) exp{σ −1 (σ + 1)(σy) 1 1+σ }, as y → +∞, for some constant c(σ) depending solely on σ. See Theorem 2 in Wright (1935). Hence, we can write the identity where A(w, σ) := σ+1 σ (σw) 1 1+σ . 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, In particular, observe that such a last asymptotic expansion leads directly to (8). Indeed let G(·; −σ, −ζ, n) be the probability generating function of the random variable K(−σ, −ζ, n), which reads as G(s; −σ, −ζ, n) = B n (sζ)/B n (ζ) for s > 0. Then, by means of (15), for any fixed s > 0 we write Note that (15) holds uniformly in w in a compact set. Accordingly, we consider the function G(s; −σ, −ζ, 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≥1 varies in a compact subset of [0, +∞). Thus, we can choose s n = s β(n) and β(n) = 1 n σ 1+σ and notice that β(n) → 0 as n → +∞. Thus, s n ≃ 1 + β(n) log s → 1 and we have that 1 1+σ σ as n → +∞. This completes the proof of (8). As regard the proof (9), let α = −σ for any σ > 0 and let z = −ζ for any ζ > 0. Similarly to the proof of (7), here we make use of the falling factorial moments of M r (−σ, −ζ, n). In particular, we can write 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 → +∞, it holds true that n−rs j=0 C (n − rs, j; −σ)(−ζ) j n j=0 C (n, j; −σ)(−ζ) j ∼ n −rs . Then, s follows from the fact that (n) [rs] ∼ n rs as n → +∞. Finally, the proof of the large n asymptotic behaviour in (9) is completed by recalling that falling factorial moment of order s of P λ is E[(P λ ) [s] ] = λ 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 wellknown compound Poisson perspective of the E-SM to the EP-SM for either α ∈ (0, 1), or α < 0. For α ∈ (0, 1) let f α denote the density function of a positive α-stable random variable X α , that is X α is a random variable for which the moment generating function is E[exp{−tX α }] = exp{−t α } for any t > 0. For α ∈ (0, 1) and θ > −α let S α,θ be a positive random variable with density function That is, the random variable S α,θ 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 α,θ . Then, for α ∈ (0, 1), θ > −α and n ∈ N we defineX α,θ,n d = G α θ+n,1 S α,θ .
Finally, for α < 0, z < 0 and n ∈ N letX α,z,n be a random variable on N whose distribution is a tilted Poisson distribution arising from the identity (12). Precisely, for any x ∈ N the distribution ofX α,z,n is In the next theorem, we make use of the random variablesX α,θ,n andX α,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.
As regard the proof of statement ii), for any α < 0, m ∈ N, k ≤ m and n ∈ N we start by defining the function m → A(m; k, α, n) = m!
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 α ∈ (0, 1) and θ > −α the EP-SM admits a representation in terms of the NB-CPSM with α ∈ (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 ofX α,θ,n . Statement ii) of Theorem 2 shows that for α < 0 and z < 0 the NB-CPSM admits a representation in terms of a randomized EP-SM with α < 0 and θ = −mα for some m ∈ 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 α < 0 and θ = −mα, for some m ∈ N, and then it assume that the parameter m is distributed according to an arbitrary distribution on 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.
as n → +∞. The random variable S α,θ is typically referred to as Pitman's α-diversity. For α < 0 and θ = −mα, for some m ∈ N, the large n asymptotic behaviour of K n (α, θ) is trivial, that is it holds as n → +∞. See Dolera and Favaro (2020a,b) for Berry-Esseen type refinements of the large n asymptotic behaviour (20), and to Favaro et al. (2009Favaro et al. ( , 2012 and  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 α-diversity.
ii) for α < 0 and z < 0 Proof. We show that (22) arises by combining (6) with statement i) of Theorem 2. For any pair of N-valued random variables U and V , let d T V (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 α ∈ (0, 1) and t > 0, we show that as n → +∞ d T V (K(α, tn α , n); 1 + P tn α ) → 0.
To show that the integral +∞ 0 |d * n (t)−dn(t)| d * n (t) f S α,θ (t)dt also goes to zero as n → +∞, we may resort to the identities (13)-(14) of Dolera and Favaro (2020a), as well as Lemma 3 in Dolera and Favaro LS-CPSM. We conjecture that an analogous perspective holds true for the class of α-stable Poisson-Kingman sampling models (Pitman, 2003(Pitman, , 2006, of which the EP-SM is a noteworthy special case. That is, for α ∈ (0, 1), we conjecture that an α-stable Poisson-Kingman sampling model admits a representation as a randomized NB-CPSM with α ∈ (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 α-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.