aa r X i v : . [ m a t h . P R ] S e p Evaluating moments of length of Pitman partition
Koji Tsukuda ∗ Abstract
Introduced in a seminal paper by Jim Pitman in 1995, the Pitman sampling formula has been intensively studiedas a distribution of random partitions. One of the objects of interest is the length K (= K n,θ,α ) of a randompartition that follows the Pitman sampling formula, where n ∈ N , α ∈ (0 , ∞ ) and θ > − α are parameters. Thispaper presents asymptotic evaluations of E [ K r ] ( r = 1 , , . . . ) under two asymptotic regimes. In particular, thegoals of this study are to provide a finer approximate evaluation of E [ K r ] as n → ∞ than has previously beendeveloped and to provide an approximate evaluation of E [ K r ] as the parameters n and θ simultaneously tendto infinity with θ/n →
0. The results presented in this paper will provide a more accurate understanding of theasymptotic behavior of K . Keywords : Pitman α -diversity; Pitman sampling formula; random partition. MSC2010 subject classifications : 60C05, 60F05, 62E20.
Let n be a positive integer, and let c i ( i = 1 , . . . , n ) be the component counts of an integer partition of n , which meansthat the partition has c i parts of size i . Note that c , . . . , c n satisfy n = P ni =1 ic i . Define a set P n = { ( c , . . . , c n ) ∈ ( N ∪ { } ) n ; P ni =1 ic i = n } . Let α ∈ [0 , θ ∈ ( − α, ∞ ). A P n -valued random variable C = ( C n,θ,α , . . . , C n,θ,αn ) iscomponent counts of a Pitman partition when its distribution is given by P ( C = ( c , . . . , c n )) = n ! ( θ ) k ; α ( θ ) n n Y i =1 (cid:18) (1 − α ) i − i ! (cid:19) c i c i ! (( c , . . . , c n ) ∈ P n ) , (1.1)where k = P ni =1 c i ,( x ) i ; y = ( i = 0) x ( x + y )( x + 2 y ) · · · ( x + ( i − y ) ( i = 1 , , . . . ) ( x > − y ; y ≥ , and ( x ) i = ( x ) i ;1 ( i = 0 , , , . . . ; x > − Pitman sampling formula , wasintroduced by Pitman [13]. A special case α = 0 of (1.1) is known as the Ewens sampling formula, and wasintroduced by Ewens [5] in the context of population genetics. Henceforth, unless otherwise mentioned, we considerthe case α = 0. The distribution of the length K (= K n,θ,α ) = P ni =1 C n,θ,αi of a Pitman partition is given by P ( K = k ) = c ( n, k, α ) α k ( θ ) k ; α ( θ ) n ( k = 1 , . . . , n ) , (1.2)where c ( n, k, α ) = ( − n − k C( n, k, α ) and C( n, k, α ) is the generalized Stirling number or the C-number of Char-alambides and Singh [2]; see, for example, Yamato, Sibuya and Nomachi [21]. This paper discusses an asymptoticproperty of K .Random partition models have been received considerable attention, not only because they are interesting asmathematical models, but also because they relate to broad scientific fields; see, for example, Crane [3] and Johnson,Kotz and Balakrishnan [11] (Chapter 41, its write-up was provided by S. Tavar´e and W.J. Ewens). As a typicaldistribution of random partiton models, the distribution (1.1) has been intensively studied. In particular, theproperties of (1.2) have been investigated by Dolera and Favaro [4], Favaro, Feng and Gao [6], Feng and Hoppe [9], ∗ Faculty of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka-shi, Fukuoka 819-0395, Japan. Kn α ⇒ S α,θ (1.3)as n → ∞ , where ⇒ denotes convergence in distribution and S α,θ is a continuous random variable whose distributionhas the density Γ( θ + 1)Γ( θα + 1) x θα g α ( x ) ( x > , where g α ( · ) is the function satisfying Z ∞ x p g α ( x ) dx = Γ( p + 1)Γ( pα + 1) ( p > − g α ( x ) can be written as g α ( x ) = 1 πα ∞ X i =1 ( − i +1 i ! Γ( iα + 1) x i − sin( πiα ) ( x > . The random variable S α,θ is called the Pitman α -diversity . A summary of the proof of (1.3) given by Yamato andSibuya [20] is as follows:1. It holds that E [ K r ] = r X i =0 ( − r − i (cid:18) θα (cid:19) i R (cid:18) r, i, θα (cid:19) Γ( θ + iα + n )Γ( θ + n ) Γ( θ + 1)Γ( θ + iα + 1) ( r = 1 , , . . . ) , (1.4)where R ( · , · , · ) is the weighted Stirling number of the second kind introduced by Carlitz [1].2. Using expression (1.4), the Stirling formula yields E (cid:20)(cid:18) Kn α (cid:19) r (cid:21) = (cid:18) θα (cid:19) r Γ( θ + 1)Γ( θ + rα + 1) + o (1) ( r = 1 , , . . . ) (1.5)as n → ∞ .3. Hence, (1.3) follows from the method of moments.In this paper, the moments E [ K r ] ( r = 1 , , . . . ) are investigated in more detail than (1.5). In particular, thereare two goals in this paper: the first is to provide an approximate evaluation of E [ K r ] as n → ∞ that is finer thanthat of (1.5); the second is to provide an approximate evaluation of E [ K r ] as the parameters n and θ simultaneouslytend to infinity with θ/n →
0. The phenomenon described by (1.3) is attractive, and our results will provide amore accurate understanding of (1.3).
Remark 1.
There are some known results that are stronger than (1.3) . For example, • There is a construction such that
K/n α → S α,θ a . s . holds. • A Berry–Esseen-type theorem holds: When α ∈ (0 , and θ > , sup x ≥ | P ( K/n α ≤ x ) − P ( S α,θ ≤ x ) | ≤ C ( α, θ ) /n α holds for n ∈ N , where C ( α, θ ) is a constant depending only on α and θ .For details, see Dolera and Favaro [4], Feng and Hoppe [9], and Pitman [14]. In this paper, we consider two asymptotic regimes. The first is n → ∞ with fixed θ , and the second is n → ∞ , θ → ∞ , θn → . (2.1)The former ( n → ∞ with fixed θ ) has been frequently considered. When α = 0, (2.1) has also been considered insome studies; see Remark 2 below. However, when α = 0, (2.1) has not been considered, although it also seemsnatural. Indeed, in the application of (1.1) to microdata risk assessment by Hoshino [10], the estimates of θ takelarge values (e.g., 523 , . . . , n = 27320); see Tables 3–6 of [10].2 emark 2. When α = 0 , the asymptotic regime in which n and θ simultaneously tend to infinity has been consideredby Feng [7] and Tsukuda [17, 18, 19]. In particular, (2.1) is Case D in Section 4 of [7]. Remark 3.
For the Pitman sampling formula, the asymptotic regime θ → ∞ with fixed n was considered byFeng [12]. Moreover, Feng [8] considered the asymptotic regime θ → ∞ in studying the Pitman–Yor process, whichis closely related to the Pitman sampling formula. For details of the Pitman–Yor process, see, for example, Pitmanand Yor [16]. Remark 4.
When considering the regime (2.1) , we assume that θ > without loss of generality. n → ∞ with fixed θ First, we provide an evaluation that is finer than that of (1.5) under the asymptotic regime n → ∞ with fixed θ . Theorem 3.1.
Suppose that α > . For r = 1 , , . . . , it holds that E (cid:20)(cid:18) Kn α (cid:19) r (cid:21) = (cid:18) θα (cid:19) r Γ( θ + 1)Γ( θ + rα + 1) (cid:20) − (cid:26) r ( r − α rθ (cid:27) Γ( θ + rα )Γ( θ + ( r − α + 1) 1 n α (cid:21) + O (cid:18) n α + 1 n (cid:19) as n → ∞ .Proof. It follows from (1.4) that E [ K r ] = r − X i =0 ( − r − i (cid:18) θα (cid:19) i R (cid:18) r, i, θα (cid:19) Γ( θ + iα + n )Γ( θ + n ) Γ( θ + 1)Γ( θ + iα + 1) − (cid:18) θα (cid:19) r − R (cid:18) r, r − , θα (cid:19) Γ( θ + ( r − α + n )Γ( θ + n ) Γ( θ + 1)Γ( θ + ( r − α + 1)+ (cid:18) θα (cid:19) r R (cid:18) r, r, θα (cid:19) Γ( θ + rα + n )Γ( θ + n ) Γ( θ + 1)Γ( θ + rα + 1) , (3.1)where P − i =0 a i = 0 for any sequence { a , a , . . . } . AsΓ( x ) = √ π e − x x x − / (cid:18) x + O (cid:18) x (cid:19)(cid:19) ( x → ∞ ) , (3.2)it holds that Γ( θ + iα + n )Γ( θ + n ) = n iα (cid:20) iα { n ( θ −
1) + θ } n ( n + θ ) + O (cid:18) n (cid:19)(cid:21) = n iα (cid:18) O (cid:18) n (cid:19)(cid:19) for i = 0 , , . . . . The first term on the right-hand side of (3.1) is O ( n ( r − α ). The second term on the right-handside of (3.1) is − (cid:18) θα (cid:19) r − (cid:26) r ( r − rθα (cid:27) Γ( θ + 1)Γ( θ + ( r − α + 1) n ( r − α + O (cid:16) n ( r − α − (cid:17) = − (cid:18) θα (cid:19) r αθ + αr (cid:26) r ( r − rθα (cid:27) Γ( θ + 1)Γ( θ + ( r − α + 1) n ( r − α + O (cid:16) n ( r − α − (cid:17) = − (cid:18) θα (cid:19) r Γ( θ + 1)Γ( θ + rα + 1) (cid:26) r ( r − α rθ (cid:27) Γ( θ + rα )Γ( θ + ( r − α + 1) n ( r − α + O (cid:16) n ( r − α − (cid:17) , because (3.2) of Carlitz [1] implies that R (cid:18) r, r − , θα (cid:19) = S ( r, r −
1) + rθα = r ( r − rθα , where S ( · , · ) is the Stirling number of the second kind. The third term on the right-hand side of (3.1) is (cid:18) θα (cid:19) r Γ( θ + 1)Γ( θ + rα + 1) n rα + O (cid:0) n rα − (cid:1) . α ∈ (0 , O ( n ( r − α ) + O (cid:16) n ( r − α − (cid:17) + O (cid:0) n rα − (cid:1) = O (cid:16) n ( r − α + n rα − (cid:17) . Therefore, E (cid:20)(cid:18) Kn α (cid:19) r (cid:21) = (cid:18) θα (cid:19) r Γ( θ + 1)Γ( θ + rα + 1) − (cid:18) θα (cid:19) r Γ( θ + 1)Γ( θ + rα + 1) (cid:26) r ( r − α rθ (cid:27) Γ( θ + rα )Γ( θ + ( r − α + 1) 1 n α + O (cid:18) n α + 1 n (cid:19) = (cid:18) θα (cid:19) r Γ( θ + 1)Γ( θ + rα + 1) (cid:20) − (cid:26) r ( r − α rθ (cid:27) Γ( θ + rα )Γ( θ + ( r − α + 1) 1 n α (cid:21) + O (cid:18) n α + 1 n (cid:19) . This completes the proof.
Remark 5.
When r > , as θ > − α , it holds that (cid:26) r ( r − α rθ (cid:27) Γ( θ + rα )Γ( θ + ( r − α + 1) 1 n α > . This means that almost all moments of
K/n α are smaller than those of S α,θ for large n . In particular, if θ > ,then all moments of K/n α are smaller than those of S α,θ for large n . In some cases, correcting some moments improves the quality of an approximation. Thus, a primitive applicationof Theorem 3.1 is correcting
K/n α in (1.3) to Kn α − θ Γ( θ + α )Γ( θ +1) whose expectation is E Kn α − θ Γ( θ + α )Γ( θ +1) = Γ( θ + 1) α Γ( θ + α ) + O (cid:18) n α + 1 n (cid:19) . When θ >
0, this correction enlarges
K/n α , and is consistent with Remark 5. (2.1) Next, under the asymptotic regime of (2.1), we provide a new evaluation.
Theorem 3.2.
Suppose that α > . For r = 1 , , . . . , it holds that E " αKθ (cid:8) ( n + θθ ) α − (cid:9) ! r = 1 + O (cid:18) θ α n α + θn + 1 θ (cid:19) under the asymptotic regime of (2.1) ; in particular, for r = 1 , , . . . , under the asymptotic regime of (2.1) , if θ α +1 n α → θ n → , then E " αKθ (cid:8) ( n + θθ ) α − (cid:9) ! r = 1 + r α (1 − α )2 θ + O (cid:18) θ α n α + θn + 1 θ (cid:18) θ α n α + 1 θ (cid:19)(cid:19) . Before presenting the proof of Theorem 3.2, let us prepare the following lemma.
Lemma 3.3.
Suppose that α > . For r = 1 , , . . . , it holds that E (cid:20)(cid:26) αKθ ( nθ ) α (cid:27) r (cid:21) = 1 − r (cid:18) θn (cid:19) α + r α (1 − α )2 θ + O (cid:18) θ α n α (cid:18) θ α n α + θ − α n − α (cid:19) + 1 θ (cid:18) θ α n α + 1 θ (cid:19)(cid:19) under the asymptotic regime of (2.1) . roof. It follows from (1.4) that E [ K r ] = r − X i =0 ( − r − i (cid:18) θα (cid:19) i R (cid:18) r, i, θα (cid:19) Γ( θ + iα + n )Γ( θ + 1)Γ( θ + n )Γ( θ + iα + 1) − (cid:18) θα (cid:19) r − (cid:26) r ( r − rθα (cid:27) Γ( θ + ( r − α + n )Γ( θ + 1)Γ( θ + n )Γ( θ + ( r − α + 1) + (cid:18) θα (cid:19) r Γ( θ + rα + n )Γ( θ + 1)Γ( θ + n )Γ( θ + rα + 1)= r − X i =0 ( − r − i R (cid:18) r, i, θα (cid:19) Γ (cid:0) θα + 1 + i (cid:1) Γ (cid:0) θα + 1 (cid:1) Γ( θ + n + iα )Γ( θ + n ) Γ( θ + 1)Γ( θ + 1 + iα ) − (cid:26) r ( r − rθα (cid:27) Γ (cid:0) θα + r (cid:1) Γ (cid:0) θα + 1 (cid:1) Γ( θ + n + ( r − α )Γ( θ + n ) Γ( θ + 1)Γ( θ + 1 + ( r − α )+ Γ (cid:0) θα + 1 + r (cid:1) Γ (cid:0) θα + 1 (cid:1) Γ( θ + n + rα )Γ( θ + n ) Γ( θ + 1)Γ( θ + 1 + rα ) . As R ( r, i, θ/α ) = O ( θ r − i ) for i = 0 , , . . . , r −
2, which follows from (3.2) of [1], according to Lemma 4.1, the firstterm is O θ (cid:18) θ n α θ α (cid:19) r − ! = O (cid:18)(cid:18) θ n α θ α (cid:19) r θ α n α (cid:19) that stems from the term of i = r −
2. Moreover, using Lemma 4.1 again, the second term is − (cid:18) O (1) + rθα (cid:19) (cid:18) θα n α θ α (cid:19) r − (cid:18) O (cid:18) θn + 1 θ (cid:19)(cid:19) = − (cid:18) θα n α θ α (cid:19) r (cid:26) r (cid:18) θn (cid:19) α + O (cid:18) θ α +1 n α +1 + 1 n α θ − α (cid:19)(cid:27) and the third term is (cid:18) θα n α θ α (cid:19) r (cid:26) rαθn + r α (1 − α )2 θ + O (cid:18) θ + θ n (cid:19)(cid:27) = (cid:18) θα n α θ α (cid:19) r (cid:26) r α (1 − α )2 θ + O (cid:18) θ + θn (cid:19)(cid:27) . These formulae yield E [ K r ] = (cid:18) θα n α θ α (cid:19) r (cid:26) − r (cid:18) θn (cid:19) α + r α (1 − α )2 θ + O (cid:18) θ α n α + 1 n α θ − α + 1 θ + θn (cid:19)(cid:27) , where O (cid:18) θ α n α + θ α +1 n α +1 + 1 n α θ − α + 1 θ + θn (cid:19) = O (cid:18) θ α n α + 1 n α θ − α + 1 θ + θn (cid:19) is used. Thus, it holds that E (cid:20)(cid:26) αKθ ( nθ ) α (cid:27) r (cid:21) = 1 − r (cid:18) θn (cid:19) α + r α (1 − α )2 θ + O (cid:18) θ α n α + 1 n α θ − α + 1 θ + θn (cid:19) . This completes the proof.Using Lemma 3.3, we prove Theorem 3.2.
Proof of Theorem 3.2.
It follows from( n + θθ ) α − nθ ) α = (cid:18) θn (cid:19) α − (cid:18) θn (cid:19) α = 1 − (cid:18) θn (cid:19) α + O (cid:18) θn (cid:19) that ( ( nθ ) α ( n + θθ ) α − ) r = 1 + r (cid:18) θn (cid:19) α + O (cid:18) θ α n α + θn (cid:19) . (3.3)5emma 3.3 and (3.3) yield E " αKθ (cid:8) ( n + θθ ) α − (cid:9) ! r = ( ( nθ ) α ( n + θθ ) α − ) r E (cid:20)(cid:26) αKθ ( nθ ) α (cid:27) r (cid:21) = (cid:26) r (cid:18) θn (cid:19) α + O (cid:18) θ α n α + θn (cid:19)(cid:27) (cid:26) − r (cid:18) θn (cid:19) α + r α (1 − α )2 θ + O (cid:18) θ α n α (cid:18) θ α n α + θ − α n − α (cid:19) + 1 θ (cid:18) θ α n α + 1 θ (cid:19)(cid:19)(cid:27) = 1 + r α (1 − α )2 θ + O (cid:18) θ α n α + θn + 1 θ (cid:18) θ α n α + 1 θ (cid:19)(cid:19) . It implies the desired conclusion. This completes the proof.
Corollary 3.4.
Suppose that α > . It holds that αKθ (cid:8) ( n + θθ ) α − (cid:9) → p under the asymptotic regime of (2.1) , where → p denotes convergence in probability.Proof. Using Theorem 3.2, the first and second moments of the left-hand side in (3.4) converge to 1. This completesthe proof.This corollary shows that, under the asymptotic regime of (2.1), K may asymptotically behave as if α was 0from the perspective of the following remark. Remark 6.
As for (3.4) , it holds that lim α → +0 αKθ (cid:8) ( n + θθ ) α − (cid:9) = Kθ log (cid:0) nθ (cid:1) . Moreover, when α = 0 , it is known that Kθ log (cid:0) nθ (cid:1) → p under the asymptotic regime of (2.1) ; see [7, 17]. Hence, we have lim α → +0 " p − lim (2.1) αKθ (cid:8) ( n + θθ ) α − (cid:9) = p − lim (2.1) " lim α → +0 αKθ (cid:8) ( n + θθ ) α − (cid:9) (= 1) . In this sense, (3.4) complements the previous result of (3.5) . In this section, we prove the following lemma, which was used in the proof of Lemma 3.3.
Lemma 4.1.
Under the asymptotic regime of (2.1) , it holds that Γ (cid:0) θα + 1 + i (cid:1) Γ (cid:0) θα + 1 (cid:1) Γ( θ + n + iα )Γ( θ + n ) Γ( θ + 1)Γ( θ + 1 + iα ) = (cid:18) θα n α θ α (cid:19) i (cid:26) iαθn + i α (1 − α )2 θ + O (cid:18) θ + θ n (cid:19)(cid:27) for i = 1 , , . . . . To prove this assertion, we first prove the following three lemmas.
Lemma 4.2.
Under the asymptotic regime (2.1) , it holds that Γ( θ + n + iα )Γ( θ + n ) = n iα (cid:26) iαθn + O (cid:18) n + θ n (cid:19)(cid:27) for i = 1 , , . . . . roof. It follows from (3.2) thatΓ( θ + n + iα )Γ( θ + n ) = e − ( θ + n + iα ) ( θ + n + iα ) θ + n + iα − / n θ + iα + n ) + O (cid:16) θ + n ) (cid:17)o e − ( θ + n ) ( θ + n ) θ + n − / n θ + n ) + O (cid:16) θ + n ) (cid:17)o = e − iα ( θ + n ) iα (cid:18) iαθ + n (cid:19) iα − / (cid:18) iαθ + n (cid:19) θ + n θ + iα + n ) + O (cid:16) θ + n ) (cid:17) θ + n ) + O (cid:16) θ + n ) (cid:17) . Hence, it holds thatΓ( θ + n + iα )Γ( θ + n ) = e − iα ( θ + n ) iα (cid:26) iαθ + n (cid:18) iα − (cid:19) + O (cid:18) θ + n ) (cid:19)(cid:27) e iα (cid:26) − i α θ + n ) + O (cid:18) θ + n ) (cid:19)(cid:27)(cid:26) θ + iα + n ) + O (cid:18) θ + n ) (cid:19)(cid:27) (cid:26) − θ + n ) + O (cid:18) θ + n ) (cid:19)(cid:27) = ( θ + n ) iα (cid:26) iαθ + n (cid:18) iα − (cid:19) + O (cid:18) θ + n ) (cid:19)(cid:27) (cid:26) − i α θ + n ) + O (cid:18) θ + n ) (cid:19)(cid:27)(cid:26) θ + iα + n ) − θ + n ) + O (cid:18) θ + n ) (cid:19)(cid:27) = ( θ + n ) iα (cid:26) iαθ + n (cid:18) iα − (cid:19) + O (cid:18) θ + n ) (cid:19)(cid:27) (cid:26) − i α θ + n ) + O (cid:18) θ + n ) (cid:19)(cid:27)(cid:26) O (cid:18) θ + n ) (cid:19)(cid:27) = ( θ + n ) iα (cid:26) iα ( iα − θ + n ) + O (cid:18) θ + n ) (cid:19)(cid:27) . Equation (2.1) assumes that θ/n →
0, so it holds thatΓ( θ + n + iα )Γ( θ + n ) = n iα (cid:18) θn (cid:19) iα ( iα ( iα − n (cid:18) θn (cid:19) − + O (cid:18) θ + n ) (cid:19)) = n iα (cid:26) iαθn + O (cid:18) θ n (cid:19)(cid:27) (cid:26) iα ( iα − n (cid:18) − θn + O (cid:18) θ n (cid:19)(cid:19) + O (cid:18) n (cid:19)(cid:27) = n iα (cid:26) iαθn + iα ( iα − n + O (cid:18) θ n (cid:19)(cid:27) = n iα (cid:26) iαθn + O (cid:18) n + θ n (cid:19)(cid:27) . This completes the proof.
Lemma 4.3. As θ → ∞ , Γ( θ + 1)Γ( θ + 1 + iα ) = θ − iα (cid:26) − iα ( iα + 1)2 θ + O (cid:18) θ (cid:19)(cid:27) for i = 1 , , . . . .Proof. Using a similar argument as in the proof of Lemma 4.2, we haveΓ( θ + 1)Γ( θ + 1 + iα ) = ( θ + 1) − iα (cid:26) − iα ( iα − θ + 1) + O (cid:18) θ (cid:19)(cid:27) . Hence, it holds that Γ( θ + 1)Γ( θ + 1 + iα ) = θ − iα (cid:18) θ (cid:19) − iα ( − iα ( iα − θ (cid:18) θ (cid:19) − + O (cid:18) θ (cid:19)) = θ − iα (cid:18) − iαθ + O (cid:18) θ (cid:19)(cid:19) (cid:26) − iα ( iα − θ + O (cid:18) θ (cid:19)(cid:27) = θ − iα (cid:26) − iα ( iα + 1)2 θ + O (cid:18) θ (cid:19)(cid:27) . Lemma 4.4. As θ → ∞ , Γ (cid:0) θα + 1 + i (cid:1) Γ (cid:0) θα + 1 (cid:1) = (cid:18) θα (cid:19) i (cid:26) i ( i + 1) α θ + O (cid:18) θ (cid:19)(cid:27) for i = 1 , , . . . .Proof. Using a similar argument as in the proof of Lemma 4.2, we haveΓ( θα + 1 + i )Γ( θα + 1) = (cid:18) θα + 1 (cid:19) i ( i ( i − θα + 1) + O (cid:18) θ (cid:19)) . Hence, it holds thatΓ( θα + 1 + i )Γ( θα + 1) = (cid:18) θα (cid:19) i (cid:16) αθ (cid:17) i (cid:26) i ( i − α θ (cid:16) αθ (cid:17) − + O (cid:18) θ (cid:19)(cid:27) = (cid:18) θα (cid:19) i (cid:18) iαθ + O (cid:18) θ (cid:19)(cid:19) (cid:26) i ( i − α θ + O (cid:18) θ (cid:19)(cid:27) = (cid:18) θα (cid:19) i (cid:26) i ( i + 1) α θ + O (cid:18) θ (cid:19)(cid:27) . This completes the proof.
Proof of Lemma 4.1.
From Lemmas 4.2, 4.3, and 4.4, it follows thatΓ (cid:0) θα + 1 + i (cid:1) Γ (cid:0) θα + 1 (cid:1) Γ( θ + n + iα )Γ( θ + n ) Γ( θ + 1)Γ( θ + 1 + iα )= (cid:18) θα n α θ α (cid:19) i (cid:26) iαθn + O (cid:18) n + θ n (cid:19)(cid:27) (cid:26) − iα ( iα + 1)2 θ + O (cid:18) θ (cid:19)(cid:27) (cid:26) i ( i + 1) α θ + O (cid:18) θ (cid:19)(cid:27) = (cid:18) θα n α θ α (cid:19) i (cid:26) iαθn + i α (1 − α )2 θ + O (cid:18) θ + θ n (cid:19)(cid:27) , where O (cid:18) θ + 1 n + θ n (cid:19) = O (cid:18) θ + θ n (cid:19) is used. This completes the proof. Under the asymptotic regime n → ∞ , θ → ∞ , θ α +1 n α → , θ n → , (5.1)Theorem 3.2 yields E [ Z ] → E (cid:2) Z (cid:3) → α (1 − α ), where Z = √ θ " αKθ (cid:8) ( n + θθ ) α − (cid:9) − . We thus expect that Z (or asymptotically equivalent quantities to Z ) has a non-degenerate limit distribution under(5.1). Deriving asymptotic properties of Z under (5.1) is a possible future direction. Acknowledgments
The author was supported in part by Japan Society for the Promotion of Science KAKENHI Grant Number18K13454. This study was partly carried out when the author was a member of Graduate School of Arts andSciences, the University of Tokyo. 8 eferences [1] Carlitz, L. (1980). Weighted Stirling numbers of the first and second kind. I.
Fibonacci Quart.
18, no. 2,147–162.[2] Charalambides, Ch.A.; Singh, J. (1988). A review of the Stirling numbers, their generalizations and statisticalapplications.
Comm. Statist. Theory Methods
17, no. 8, 2533–2595.[3] Crane, H. (2016). The ubiquitous Ewens sampling formula.
Statist. Sci.
31, no.1, 1–19.[4] Dolera, E.; Favaro, S. (2020). A Berry–Esseen theorem for Pitman’s α -diversity. Ann. Appl. Probab.
30, no. 2,847–869.[5] Ewens, W.J. (1972). The sampling theory of selectively neutral alleles.
Theoret. Population Biology
3, 87–112;erratum, ibid. 3 (1972), 240; erratum, ibid. 3 (1972), 376.[6] Favaro, S.; Feng, S.; Gao, F. (2018). Moderate deviations for Ewens–Pitman sampling models.
Sankhya
A 80,no. 2, 330–341.[7] Feng, S. (2007a). Large deviations associated with Poisson–Dirichlet distribution and Ewens sampling formula.
Ann. Appl. Probab.
17, no. 5–6, 1570–1595.[8] Feng, S. (2007b). Large deviations for Dirichlet processes and Poisson–Dirichlet distribution with two param-eters.
Electron. J. Probab.
12, no. 27, 787–807.[9] Feng, S; Hoppe, F.M. (1998). Large deviation principles for some random combinatorial structures in populationgenetics and Brownian motion.
Ann. Appl. Probab.
8, no. 4, 975–994.[10] Hoshino, N. (2001). Applying Pitman’s sampling formula to microdata disclosure risk assessment.
J. OfficialStatist. , 17, no. 4, 499–520.[11] Johnson, N.L., Kotz, S. and Balakrishnan, N. (1997).
Discrete Multivariate Distributions . Wiley Series inProbability and Statistics: Applied Probability and Statistics. A Wiley-Interscience Publication. John Wiley& Sons, Inc., New York.[12] Kerov, S. (2005). Coherent random allocations, and the Ewens–Pitman formula. With comments by AlexanderGnedin.
Zap. Nauchn. Sem. S. -Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 325 (2005), Teor. Predst. Din.Sist. Komb. i Algoritm. Metody. 12, 127–145, 246; reprinted in
J. Math. Sci. (N.Y.) 138 (2006), no. 3, 5699–5710.[13] Pitman, J. (1995). Exchangeable and partially exchangeable random partitions.
Probab. Theory Related Fields
Bernoulli
Electron. J. Probab.
4, no. 11, 1–33.[16] Pitman, J.; Yor, M. (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordi-nator.
Ann. Probab.
25, no. 2, 855–900.[17] Tsukuda, K. (2017). Estimating the large mutation parameter of the Ewens sampling formula.
J. Appl. Probab.
54, no. 1, 42–54; correction, ibid. 55 (2018), no. 3, 998–999.[18] Tsukuda, K. (2019). On Poisson approximations for the Ewens sampling formula when the mutation parametergrows with the sample size.
Ann. Appl. Probab.
29, no. 2, 1188–1232.[19] Tsukuda, K. Error bounds for the normal approximation to the length of a Ewens partition. To appear in
Pioneering Works on Distribution Theory: In Honor of Masaaki Sibuya .[20] Yamato, H.; Sibuya, M. (2000). Moments of some statistics of Pitman sampling formula.
Bull. Inform. Cybernet.
32, no. 1, 1–10.[21] Yamato, H.; Sibuya, M.; Nomachi, T. (2001). Ordered sample from two-parameter GEM distribution.