arXiv:0705.3496v3 [math.PR] 12 Jan 2010 tetnsteoeprmtrfml fdsrbtoskonas known distributions of family one-parameter tributions the extends It eateto ahmtc,Sg nvriy aa840-8502 Saga University, E-mail: Saga Mathematics, of Department HANDA KENJI process point Poisson–Dirichlet two-parameter The DOI: Bernoulli 1350-7265 [ Yor and PD( Pitman denoted information. distribution, Poisson–Dirichlet bibliographic parameter and topics related func of view Keywords: of point the distribution. Markov–Kr from a Poisson–Dirichlet The parameter discussed are case. we is Yor, one-parameter process and the Dirichlet Pitman for to known due previously approach Poisson–Diri transform two-parameter Laplace t the the in the of points apply we structure as this, Using mathematical masses functions. the correlation the regarding of associat terms by the obtained consider of process hence characterization point and A 1 distribution. sum discrete with sequences pro decreasing a positive is distribution Poisson–Dirichlet two-parameter The ie 0 Given dis probability of family two-parameter a simplex with infinite-dimensional concerned is paper This Introduction 1. distribution hsi neetoi ern fteoiia ril publis 1082–1116 detail. article 4, typographic original No. the 15, of Vol. reprint 2009, electronic an is This 10.3150/08-BEJ180 [email protected] ≤ hc eeitoue yKnmn[ Kingman by introduced were which , 15 orlto ucin akvKeniett;pitproce point identity; Markov–Krein function; correlation
< α c 4,20,1082–1116 2009, (4), 09ISI/BS 2009 V and 1 1 ∇ = ∞ Y = 1 > θ , ( − v i α ( = ) V en eune( sequence a define , ∇ i hsrpitdffr rmteoiia npgnto and pagination in original the from differs reprint This . (1 = ∞ v 1 fnnngtv eraigsqecswt u 1, sum with sequences decreasing non-negative of v , − 2 . . . , Y 1 ) : ) v (1 1 ≥ − v Y V 2 i i 32 ≥ ≥ − frno aibe by variables random of ) 1 dpitpoes(hti,terandom the is, (that process point ed ;se o xml,[ example, for see, ]; ) e ythe by hed aiiydsrbto ntettlt of totality the on distribution bability Y hoyo on rcse orva the reveal to processes point of theory i he itiuin lo developing Also, distribution. chlet ,θ α, Japan. , inlaayi ae ntetwo- the on based analysis tional dt oenmse farandom a of masses govern to ed 0 epstv elln)i ie in given is line) real positive he i dniyfrtegeneralized the for identity ein , ,i h olwn manner. following the in ), ( i l oetn eea results several extend to ble 2 = s Poisson–Dirichlet ss; v ISI/BS i oso–iihe dis- Poisson–Dirichlet , 1 = 45 3 . . . , endtetwo- the defined ] in . rbtoso the on tributions ) , Bernoulli 1 , 44 , 50 , (1.1) for ] The two-parameter Poisson–Dirichlet point process 1083 where Y , Y ,... are independent and each Y is Beta(1 α, θ + iα)-distributed. Let (V ) 1 2 i − i be the decreasing order statistics of (V ), namely V V are the ranked values of i 1 ≥ 2 ≥ (Vi), and define PD(α, θ) to be the law of (Vi) on . There is some background to the study of these distributions, as explained in [45] and∇∞ [44]. In particular, a number of results concerning PD(α, θ) were obtained in [45]. The proofs there use auxiliary random variables and related processes (such as stable subordinators and gamma processes) and require deep insight into them. As for the original Poisson–Dirichlet distributions, which form a one-parameter family PD(0,θ): θ> 0 and correspond to the gamma processes, certain independence property{ often makes the} analysis relatively easier. See [50] and [51] for extensive discussions. One purpose of this article is to provide another approach to study the two-parameter family PD(α, θ):0 α< 1,θ> α , based mainly on conventional arguments in the theory{ of point processes;≤ see, for− example,} [7] for general accounts of the theory. This means that a random element (Vi) governed by PD(α, θ) is studied through the ran- dom point process ξ := δVi , which we call the (two-parameter) Poisson–Dirichlet point process with parameters (α, θ), or simply the PD(α, θ) process. Note that although the above ξ is a ‘non-labeled’ object, it is sufficient to recover the law of the ranked sequence (V ). For example, we have ξ([t, )) = 0 precisely when V
2. The correlation function of the PD(α, θ) process
The main subject of this section is the correlation function. We begin with some defi- nitions and concepts from the theory of point processes [7]. (See also Section 2 in [27] for a comprehensive exposition of this material.) Let ξ be a point process taking values in R. For simplicity, suppose that ξ is expressed as ξ = δXi for some random vari- ables X1,X2,.... Throughout what follows, we assume that ξ is simple, in the sense that Xi = Xj (i = j) a.s. For any positive integer n, the nth correlation measure (also called the nth factorial moment measure) of ξ, if it exists, is defined to be a σ-finite measure The two-parameter Poisson–Dirichlet point process 1085
n µn such that for non-negative measurable functions f on R ,
E f(Xi1 ,...,Xin ) = f(x1,...,xn)µn(dx1 dxn), (2.1) Rn i1,...,in(=) where the sum is taken over n-tuples of distinct indices. In particular, µ1 is the mean measure of ξ and it follows that µ (dx dx ) is necessarily symmetric in x ,...,x . n 1 n 1 n If µn has a density with respect to the n-dimensional Lebesgue measure, the density is called the nth correlation function of ξ. In what follows, α and θ are such that 0 α< 1 and θ> α, unless otherwise men- ≤ − tioned. Notice that the PD(α, θ) process ξ = δ = δVi , with (Vi) and (Vi) defined Vei as in the Introduction, is simple. Denote by cn,α,θ the product n n θ , α = 0, Γ(θ +1+(i 1)α) n 1 − = Γ(θ + 1)Γ(θ/α + n)α − Γ(1 α)Γ(θ + iα) , 0 <α< 1. i=1 − Γ(θ + αn)Γ(θ/α + 1)Γ(1 α)n − In view of the left-hand side, it is clear that
c = c c (m,n 0, 1, 2,... ) (2.2) m+n,α,θ m,α,θ n,α,θ+αm ∈{ } with the convention that c0,α,θ = 1. We also need the following notation for the n- dimensional unit simplex:
∆ = (v ,...,v ): v 0,...,v 0, v + + v 1 . n { 1 n 1 ≥ n ≥ 1 n ≤ }
In general, the indicator function of a set (or an event) E is denoted by 1E . The main result of this section generalizes Watterson’s formula [54] ((18), page 644) for the corre- lation functions of the PD(0,θ) process to the PD(α, θ) process.
Theorem 2.1. For each n =1, 2,..., the nth correlation function of the PD(α, θ) process is given by
θ+αn 1 n n − (α+1) q (v ,...,v ) := c v− 1 v 1 (v ,...,v ). (2.3) n,α,θ 1 n n,α,θ i − j ∆n 1 n i=1 j=1 We will give a proof based on a known property of the size-biased permutation for PD(α, θ). For the one-parameter family PD(0,θ): θ> 0 , such an idea to derive (2.3) is suggested in [39], Corollary 7.4. From the{ definition of size-biased} permutation (see [45] or Section 4 of [3]), we can make the following observation. If a sequence (Vi) of random variables such that Vi > 0 (i =1, 2,...) and Vi = 1 is given, then, for each n =1, 2,..., the nth correlation measure µn of δV exists and i n i 1 − 1 ♯ µ (dv dv )= v− 1 v 1 (v ,...,v )µ (dv dv ), (2.4) n 1 n i − j ∆n 1 n n 1 n i=1 j=1 1086 K. Handa
♯ ♯ ♯ where µn is the joint law of (V1 ,...,Vn ), the first n components of the size-biased permu- tation (V ♯) of(V ). In (2.4) and throughout, we adopt the convention that 0 = 0. i i j=1 Proof of Theorem 2.1. Let (Vi) have PD(α, θ) distribution. A remarkable result ob- ♯ tained in [37, 41, 43] then tells us that the law of (Vi ) coincides with the law of (Vi) in (1.1). Hence for any non-negative measurable function f on Rn, the expectation ♯ ♯ E[f(V1 ,...,Vn )] is given by
n α θ+iα 1 dy1 dyn f(v1,...,vn)cn,α,θ yi− (1 yi) − , (2.5) n { − } [0,1] i=1 i 1 0 where v = v (y ,...,y )= y − (1 y ), with the convention that =1. Note i i 1 n i j=1 − j j=1 that the mapping (y ,...,y ) =: y (v ,...,v ) =: v from [0, 1]n to ∆ has the inverse 1 n → 1 n n
i 1 1 − − y = y (v)= v 1 v (i =1,...,n). (2.6) i i i − j j=1 Therefore, (2.5) becomes
n ∂y α θ+iα 1 c dv dv f(v) y− (1 y ) − . (2.7) n,α,θ 1 n ∂v { i − i } ∆n i=1 Observing from (2.6) that
θ+αn 1 n n n − α θ+iα 1 α y− (1 y ) − = v− 1 v , (2.8) { i − i } i − j i=1 i=1 j=1 we have, by (2.5) and (2.7),
n i 1 1 − − ♯ ♯ E[f(V1 ,...,Vn )] = dv1 dvn f(v)qn,α,θ(v) vi 1 vj , (2.9) Rn − i=1 j=1 where qn,α,θ is defined by (2.3). With the help of (2.4), this proves Theorem 2.1.
It is known that correlation functions appear in the expansion of the ‘probability generating function’ of a random point process δXi ; see Section 5 in [7] for general accounts. This functional is defined to be the expectation of an infinite product of the form g(Xi). For the sake of clarity, we provide the following definition of infinite The two-parameter Poisson–Dirichlet point process 1087 products. Given a sequence t of complex numbers, define { i} n lim (1 + ti) [1, ], if ti 0 (i =1, 2,...), n ∞ →∞ ∈ ∞ ≥ (1 + t )= i=1 i n i=1 lim (1 + ti) C, if the limit exists. n ∈ →∞ i=1 In general, it holds that
∞ ∞ ∞ 1 (1 + t )=1+ t t =1+ t t . (2.10) | i| | i1 | | in | n! | i1 | | in | i=1 n=1 i1< Also, ∞ (1 + t ) admits the corresponding expansion whenever ∞ (1 + t ) is finite. i=1 i i=1 | i| In what follows, E [ ] denotes the expectation with respect to PD(α, θ). α,θ Corollary 2.2. For any measurable function φ : (0, 1] C, → ∞ E (1 + φ(V ) ) α,θ | i | i=1 (2.11) n n θ+αn 1 ∞ c φ(v ) − =1+ n,α,θ | i | 1 v dv dv . n! α+1 − j 1 n n=1 ∆n i=1 vi j=1 If the above series converges, then ∞ Eα,θ (1 + φ(Vi)) i=1 (2.12) n n θ+αn 1 ∞ c φ(v ) − =1+ n,α,θ i 1 v dv dv . n! α+1 − j 1 n n=1 ∆n i=1 vi j=1 Proof. (2.11) is immediate from (2.10), (2.1) and (2.3) together. Convergence of the series in (2.11) allows us to show (2.12) by the dominated convergence theorem. For instance, if φ : (0, 1] C is a measurable function with support contained in [ε, 1] → for some 0 <ε< 1, the series in (2.11) is easily verified to converge. Roughly speaking, the assertion of Corollary 2.2 is equivalent to that of Theorem 2.1 (cf. Propositions 2.2 and 2.3 in [27]). 1088 K. Handa 3. The use of Laplace transforms This section is intended to provide a general tool (Lemma 3.1) and to exploit it in the study of PD(α, θ) processes. It contains a certain inversion formula for Laplace trans- forms. Interestingly, in spite of the generality of the formulation, the formula involves Dirichlet measures. This seems to show an aspect of the analytic importance of such measures. Lemma 3.1. Let β 0 and δ β. Let R be a function defined at least on a neigh- ≥ ≥− borhood of 0 in which R is expressed as an absolutely convergent series of the form R(u)= r u + r u2 + . 1 2 (i) Suppose that F and G are complex-valued measurable functions on (0, ) such δ 1 λ0s (β+1) λ0z ∞ that ∞ dss − e− F (s) < and ∞ dzz− e− G(z) < for some λ > 0. If 0 | | ∞ 0 | | ∞ 0 F and G are connected with each other in such a way that λz δ ∞ δ 1 λs β ∞ e− λ dss − e− F (s)= R λ− dz G(z) (3.1) zβ+1 0 0 for sufficiently large λ> 0, then F (s) coincides with n n δ+βn 1 ∞ r G(sv ) − n i 1 v dv dv (β + δ> 0) (3.2) Γ(δ + βn) β+1 − j 1 n n=1 ∆n i=1 vi j=1 or n 1 n 1 ∞ − G(svi) G(s(1 j=1− vj )) r1G(s)+ rn − dv1 dvn 1 n 1 − ∆ −1 vi 1 − v n=2 n i=1 j=1 j − (3.3) (β =0= δ) for a.e. s> 0, where the series and integrals converge absolutely for a.e. s> 0. (ii) Suppose that G is a complex-valued measurable function on (0, ) such that (β+1) λ0z ∞ ∞ dzz− e− G(z) < for some λ > 0. Then the series (3.2)or(3.3) with G( ) 0 | | ∞ 0 | | and rn in place of G( ) and rn, respectively, converges for a.e. s> 0 and F (s) defined by | | δ 1 λ0s the expression (3.2) or (3.3) satisfies ∞ dss − e− F (s) < . Moreover, the relation 0 | | ∞ (3.1) holds for sufficiently large λ> 0. Proof. We consider only the case where β + δ> 0 since the other case can be handled quite similarly. To prove assertion (i), we first show that n n δ+βn 1 ∞ r G(sv ) − G (s) := | n| | i | 1 v dv dv 0 Γ(δ + βn) β+1 − j 1 n n=1 ∆n i=1 vi j=1 The two-parameter Poisson–Dirichlet point process 1089 is finite for a.e. s> 0. Setting, for each s> 0 and n =1, 2,..., ∆ (s)= (z ,...,z ): z 0,...,z 0,z + + z s , (3.4) n { 1 n 1 ≥ n ≥ 1 n ≤ } we observe, by Fubini’s theorem, that δ+βn 1 n n − ∞ δ 1 λs G(svi) dv1 dvn dss − e− | | 1 v β − j v v 0 ∆n i=1 vi j=1 1 n δ+βn 1 n n − ∞ λs G(zi) dz1 dzn = ds e− | | s z β − j z z 0 ∆n(s) i=1 zi j=1 1 n δ+βn 1 n n − G(zi) ∞ λs = dz1 dzn | β+1 | ds e− s zj n n − (0, ) i=1 zi Pj=1 zj j=1 ∞ λz n Γ(δ + βn) β ∞ e− = λ− dz G(z) . λδ zβ+1 | | 0 Therefore, term-by-term integration yields n ∞ λz δ ∞ δ 1 λs β ∞ e− λ dss − e− G (s)= r λ− dz G(z) . 0 | n| zβ+1 | | 0 n=1 0 Since the above series converges for sufficiently large λ, G (s) < for a.e. s> 0 and thus 0 ∞ the desired absolute convergence follows. Calculations needed for the proof of (i) have almost been done. Indeed, denoting by G1(s) the sum (3.2), one can see by obvious modification of the above calculations that (3.1) with F replaced by G1 holds for sufficiently large λ> 0. We thus conclude, by virtue of the uniqueness of Laplace transforms, that F (s)= G1(s) for a.e. s> 0. This proves assertion (i). The proof of assertion (ii) is essentially contained in the above and is therefore omitted. At least formally, (3.3) is obtained as the ‘degenerate limit’ of (3.2), that is, by first setting β = 0 in (3.2) and then taking the limit as δ 0. Although our inversion formula ↓ given in Lemma 3.1(i) requires the Laplace transform to have a certain special form, one advantage is that it is described in the ‘real world’, that is, we do not need any complex integrals. Considered as prototypes of the inversion formula are integral representations of the Dickman function and the Buchstab function (cf. [1], page 22). Both functions appeared in asymptotic number theory [4, 12] and are related to PD(0, 1). On the other hand, assertion (ii) will be used below to compute the Laplace transform of a probability generating function of the PD(α, θ) process discussed in [45]. 1090 K. Handa In such applications, we will employ Γ(θ)(eθu 1), α = 0, θ> 0, − θ/α 1 Rα,θ(u) := Γ(θ + 1) (1 Cαu)− 1 θ− , 0 <α< 1, θ = 0, (3.5) 1 { − − } α− log(1 C u), 0 <α< 1, θ = 0, − − α where Cα = α/Γ(1 α). The precise meanings of the power and logarithm in (3.5) as complex functions are− as follows. Given p R and t C ( , 0], define tp = exp(p log t), where log t = log t +arg t, with arg t being∈ chosen∈ in ( \π−∞, π). As mentioned in Exercise | | − 1.2.7 of [44], the function Rα,θ itself appears in connection with generalized Stirling numbers. Note that three expressions in the right-hand side of (3.5) vary continuously in (α, θ), even in the limit as θ 0 or α 0. This fact is consistent with the continuity → ↓ of the two-parameter Poisson–Dirichlet family [52], [50]. More importantly, Rα,θ admits an expansion of the form ∞ Γ(θ + αn)c R (u)= n,α,θ un (3.6) α,θ n! n=1 as long as u < 1/Cα(= for α = 0 by definition). Combining Corollary 2.2 and Lemma 3.1, we prove| | the following∞ formula, a refinement of Corollaries 49 and 50 in the afore- mentioned paper [45] by Pitman and Yor. We will use the convention that inf ∅ = so that, for a subset X of R, inf X< means X = ∅. ∞ ∞ Theorem 3.2. Suppose that g : (0, ) C is a measurable function such that λα(g) := (α+1) λz ∞ → inf λ> 0 : ∞ dzz− e− g(z) 1 < < . Put { 0 | − | ∞} ∞ λz Cα ∞ e− λ∗ (g) = inf λ′ > λ (g): dz (g(z) 1) / [1, ) for all λ λ′ α α λα zα+1 − ∈ ∞ ≥ 0 so that, in particular, λ0∗(g)= λ0(g). Then for a.e. s> 0, ∞ E (1+ g(sV ) 1 ) < (3.7) α,θ | i − | ∞ i=1 and for λ > λα∗ (g), θ λ ∞ θ 1 λs ∞ dss − e− E g(sV ) 1 Γ(θ + 1) α,θ i − 0 i=1 (3.8) λz 1 ∞ e 1 exp θ dz − (g(z) 1) , α =0, θ> 0, θ z − − θ 0 λz θ/α 1 C ∞ e − 1 = α − 1 α dz α+1 (g(z) 1) , 0 <α< 1, θ =0, θ − λ 0 z − − θ λz 1 Cα ∞ e− log 1 α dz α+1 (g(z) 1) , 0 <α< 1, θ =0. −α − λ 0 z − The two-parameter Poisson–Dirichlet point process 1091 Proof. Consider ∞ F (s) := E (1 + g(sV ) 1 ) 1, 0 α,θ | i − | − i=1 which admits a series expansion due to (2.11). Noting (3.6), we apply the first half of Lemma 3.1(ii) with β = α, δ = θ and R = R to show that F (s) < for a.e. s> 0. α,θ 0 ∞ Also, the last half of Lemma 3.1(ii) can be applied to the series expression of F (s) := E [ ∞ g(sV )] 1 due to (2.12) and we obtain α,θ i=1 i − λz θ ∞ θ 1 λs α ∞ e− λ dss − e− F (s)= R λ− dz (g(z) 1) α,θ zα+1 − 0 0 for sufficiently large λ. This extends to all λ > λα∗ (g) by analytic continuation, showing (3.8). The proof of Theorem 3.2 is thus complete. This result will be exploited later in a variety of ways by taking 1(0,1)(s), in Section 4, g(s)= exp( sp), in Section 6.2, ψ ( −s), in Section 7, ν ± where p> 0 and ψν is the characteristic function of a distribution ν. We conclude this section with the observation that the probability generating function of the PD(α, θ) process can also be characterized by an integral equation. Proposition 3.3. Suppose that g : (0, ) C is as in Theorem 3.2 and set φ(s)= ∞ → g(s) 1. Then Π (s) := E [ ∞ g(sV )] solves the following integral equation for a.e. − g α,θ i=1 i s> 0: (i) for α =0 and θ> 0, s θ θ 1 s Π (s) θ g(s t)t − Π (t) dt = 0; (3.9) g − − g 0 (ii) for 0 <α< 1 and θ =0, s s α θ α θ dt (s t)− t Π (t) dt (s t)− t g(s t) − g − − − 0 0 s u (α+1) θ α du dt (u t)− φ(u t)t Π (t) (3.10) − − − g 0 0 s u α θ 1 θ du dt (u t)− g(u t)t − (Π (t) 1) = 0; − − − g − 0 0 1092 K. Handa (iii) for 0 <α< 1 and θ =0, s s α α dt (s t)− Πg(t) t− g(t) dt 0 − − 0 (3.11) s u (α+1) α du dt (u t)− φ(u t)Π (t)=0. − − − g 0 0 Proof. Given a measurable function f on (0, ), let f denote the Laplace transform of f and introduce the temporary notation f (z)=∞ f(z)/zβ for β R. All equalities below β ∈ involving λ hold at least for sufficiently large λ. θ θ 1 (i) Set f(s)= s Πg(s) so that f1(s)= s − Πg(s). We start with a simplified version θ of (3.8) for α = 0, that is, λ f1(λ)=Γ(θ) exp(θφ1(λ)). Taking the logarithmic derivative of both sides with respect to λ, we get θ f(λ) θ = θφ(λ) = θg(λ)+ λ − f1(λ) − − λ and hence f(λ)= θg(λ)f1(λ). This suffices to prove (3.9). (ii) For notational simplicity, let Φ(s) = Π (s) 1 and G(s) = Π (s). In the case where g − g 0 <α< 1 and θ = 0, the equality (3.8) reads θ θ/α θλ Cα − Φ1 θ(λ)= 1 φα+1(λ) 1. Γ(1 + θ) − − λα − Setting β = θ(<α< 1), we convert the above identity to − (Γ(1 β)λβ βΦ (λ))1/β = (Γ(1 α)λα αφ (λ))1/α. − − β+1 − − α+1 By taking the logarithmic derivative, β 1 α 1 Φ (λ)+Γ(1 β)λ − φ (λ)+Γ(1 α)λ − β − = α − . β α Γ(1 β)λ βΦβ+1(β) Γ(1 α)λ αφα+1(λ) − − − − γ Here, considering the function 1γ( z) := 1/z for every γ< 1, note that the numerator of the left-hand side equals Φβ(λ)+ 1β(λ)= Gβ(λ) and, similarly, α 1 φα(λ)+Γ(1 α)λ − = gα(λ). (3.12) − Consequently, we have α β 1 (λ)G (λ) 1 (λ)g (λ) φ (λ)G (λ)+ g (λ)Φ (λ)=0. (3.13) α β − β α − λ α+1 β λ α β+1 This proves (3.10 ). The two-parameter Poisson–Dirichlet point process 1093 (iii) Differentiating (3.8) with θ = 0 and again using (3.12), we get 1 α 1 (λ)Π (λ) g (λ) φ (λ)Π (λ)=0, (3.14) α g − λ α − λ α+1 g which shows (3.11). The proof is thus completed. In fact, the generality of the assumptions in Proposition 3.3 makes the resulting equa- tions (3.10) and (3.11) rather complicated. Under an additional hypothesis which ensures that both g and Πg are of bounded variation on each finite interval, the following equa- tions are derived instead. For 0 <α< 1 and θ = 0, s s α θ θ α (s t)− t dΠg(t) (s t) t− dg(t) 0 − − 0 − (3.15) s α θ 1 (α+1) θ dt (s t)− θt − + α(s t)− t φ(s t)(Π (t) 1)=0. − { − − } − g − 0 For 0 <α< 1 and θ = 0, s s α α (α+1) (s t)− dΠ (t) s− φ(s) α dt (s t)− φ(s t)Π (t)=0. (3.16) − g − − − − g 0 0 The proof changes only after multiplying (3.13) and (3.14) by λ and then uses ∞ λs ∞ λs 1 (λ)λG (λ) 1 (λ)λg (λ)= 1 (λ) e− dΦ (s) 1 (λ) e− dφ (s) α β − β α α β − β α 0 0 ∞ λs and λΠg(λ)=1+ 0 e− dΠg(s), respectively. The details are omitted. 4. The two-parameter generalization of the Dickman function One of fundamental ‘observables’ in a point process on R is the ‘position of the last particle’ (cf. [27]), if any. As for the PD(α, θ) process, this is nothing but V1, the first component of a PD(α, θ)-distributed random element (Vi). For special values of α and θ, the law of V1 was found in various contexts much earlier than Kingman’s discovery of the Poisson–Dirichlet limit. For example, the Dickman function [12], usually denoted ρ( ), is identified with ρ(s)= P0,1(sV1 < 1); see, for example, Section III 5.3 of [48] for related discussions in asymptotic number theory and Section 1.1 of [1]. Here and in what follows, Pα,θ is a probability distribution under which (Vi) is PD(α, θ)-distributed. Also, a distribution function found independently in [8] and [35] can be identified with 1 Pα,0(V1− s). See also [32] ((77), page 14). It is natural to introduce the two-parameter version of≤ the Dickman function by ρα,θ(s)= Pα,θ(sV1 < 1). 1094 K. Handa Clearly, ρα,θ(s) = 1 for all s 1. The one-parameter family ρ0,θ : θ> 0 has been studied in connection with both population≤ genetics [18, 54] and{ the asymptotic} theory of the symmetric group [22]. For details, we refer the reader to the identity (4.13) in [3] (resp., θ 1 Lemma 4.7 in [1]), where gθ(s) (resp., pθ(s)) is identical with s − ρ0,θ(s) up to some multiplicative constant. It also appears in a natural extension [21] of Dickman’s result in number theory. Another interesting context in which the one-parameter family arises is the identification of limit distributions associated with random minimal directed spanning trees [40]. The aim of this section is to describe consequences for ρα,θ’s which follow from the results in Sections 2 and 3. First, a choice of φ in (2.12) to give an expression for ρα,θ is φ(v)= 1[1, )(sv), with s> 0 being given. With this choice, (2.12) now reads − ∞ ∞ ( 1)nc ρ (s)= − n,α,θ I (s), (4.1) α,θ n! n,α,θ n=0 where I ( ) 1 and where, for n =1, 2,..., 0,α,θ ≡ θ+αn 1 n n − 1[1, )(svi) I (s)= ∞ 1 v dv dv . (4.2) n,α,θ α+1 − j 1 n ∆n i=1 vi j=1 The reader is cautioned that our notation In,α,θ is in conflict with that of [45]. We observe that In,α,θ(s) = 0 whenever n>s. So, the right-hand side of (4.1) is in fact a finite sum taken over n with 0 n s. In the case (α, θ)=(0, 1), we recover a well-known formula for the Dickman function≤ ≤ (cf. (1.35) in [1]), while the above identity with θ = 0 extends 1 the formula for Pα,0(V1− s) =: Hα(s), obtained in [35] ((3.7), page 730), to all values of s 1. As a two-parameter≤ example, we find in [34] (Theorem 3.3.1, page 164) this ≥ type of expression for ρ1/2,1/2, which determines the limit distribution of the maximal size of trees associated with a random mapping. It is proved in Propositions 19 and 20 of [45] that for all s> 0, −1 min s ,1 θ+α 1 Γ(θ + 1) { } (1 v) − 1 v ρ (s)= dv − ρ − (4.3) α,θ Γ(θ + α)Γ(1 α) vα+1 α,θ+α v − 0 and hence the probability density P (V dv)/dv is given by α,θ 1 ∈ θ+α 1 Γ(θ + 1) (1 v) − 1 v f (v) := − ρ − 1 (v). (4.4) α,θ Γ(θ + α)Γ(1 α) vα+1 α,θ+α v (0,1) − For later convenience, we remark that underlying (4.3) are ‘termwise equalities’ shown in the next lemma. For every positive integer n and s> 0, let (s)= (v ,...,v ): v v 0, v + + v s . ∇n { 1 n 1 ≥ ≥ n ≥ 1 n ≤ } In particular, (1) is simply denoted by . ∇n ∇n The two-parameter Poisson–Dirichlet point process 1095 Lemma 4.1. For all n =1, 2,... and s> 0, it holds that 1 θ+α 1 (1 v) − 1 v In,α,θ(s)= n dv − α+1 In 1,α,θ+α − . (4.5) min s−1,1 v − v { } Proof. Obviously, we may assume that s 1. Also, (4.5) with n = 1 is clear from the ≥ definition (4.2). For n 2, by symmetry of the integrand in (4.2), I (s) equals ≥ n,α,θ θ+αn 1 1 n − dvn dv1 dvn 1 n! 1 svn 1 − 1 vn−1 vn 1 vj . (4.6) α+1 { ≥ } α+1 α+1 { ≥ } − 0 vn n−1(1 vn) v1 vn 1 j=1 ∇ − − The change of variables v = (1 v )u converts the inner integral to i − n i θ+αn 1 n 1 − θ+α 1 du1 dun 1 − − (1 vn) − 1 (1 vn)un−1 vn 1 uj − α+1 α+1 { − ≥ } − n−1 u1 un 1 j=1 ∇ − 1 θ+α 1 1 vn = (1 vn) − In 1,α,θ+α − . (n 1)! − − v − n Substituting this into (4.6), we get (4.5). It should be noted that (4.3) does not give a closed equation for ρα,θ unless α =0. To derive such equations for the general case, we shall apply some results from the previous section. Setting g = 1(0,1) in (3.8) and observing that λα∗ (g)=0, we see from Theorem 3.2 that for all λ> 0, λz θ ∞ θ 1 λs α ∞ e− λ dss − e− (ρ (s) 1)= R λ− dz . (4.7) α,θ − α,θ − zα+1 0 1 For θ> 0, this relation is rewritten in a slightly simpler form: −λz ∞ e θ exp θ 1 dz z , α = 0, λ ∞ θ 1 λs − dss − e− ρα,θ(s)= θ/α (4.8) Γ(θ) 0 −λz − Cα ∞ e 1+ λα 1 dz zα+1 , 0 <α< 1. Integral equations satisfied by ρα,θ will be derived as the ones equivalent to (4.7), in other words, as consequences of Proposition 3.3, its variant (3.15) and (3.16). Preliminary observations concerning this are that Πg(s)= ρα,θ(s) for g = 1(0,1) and that (3.9), (3.15) and (3.16) hold for all s 1 because of continuity. As will be discussed, the equation ≥ below is known in the cases α = 0 and θ = 0. For this reason, the proof of the next theorem will be concerned with the remaining case only. 1096 K. Handa Theorem 4.2. ρα,θ solves the equation s θ α θ (s 1) ρα,θ(s 1)+ (s t)− t dρα,θ(t)=0 (s> 1). (4.9) − − s 1 − − Proof. The case where 0 <α< 1 and θ =0. (3.15) now reads, for all s> 1, as s α θ θ (s t)− t dρ (t) + (s 1) − α,θ − 0 s 1 − α θ 1 (α+1) θ + dt (s t)− θt − + α(s t)− t (ρ (t) 1)=0 { − − } α,θ − 0 because dg(t) on (0, ) equals δ (dt). Integration by parts yields (4.9). − ∞ 1 Remarks. (i) (4.9) with α = 0 or its variant can be found in [19, 21, 22, 54]; see also (4.25) in [1]. (ii) One can easily see that (4.9) with θ = 0 is derived from a functional equation for H (t)=1 ρ (t) obtained in [35] ((3.5), page 730). α − α,0 In the next section, we calculate not only the marginal distributions Pα,θ(Vm ) for m =2, 3,..., but also multidimensional distributions of PD(α, θ), by developing∈ point process calculus based on Theorem 2.1. 5. Distributional results for PD(α, θ) In this section, we apply the theory of point processes to deduce from (2.3) some distribu- tional information on PD(α, θ). An essential idea underlying the subsequent argument is the principle of inclusion-exclusion, which was already being used in [19]. By the same rea- soning as in the proof of Theorem 2 there, namely, by a version of Fr´echet’s formula (see, for example, Section IV.5 in [15]), the following relationship between one-dimensional distributions and correlation functions holds with great generality. Lemma 5.1. Let a k P (Zm dz) 1(a,b)(z) ∞ ( 1) ∈ = − dy1 dym+k 1 qm+k(z,y1,...,ym+k 1). dz (m 1)! k! [z,b)m+k−1 − − − k =0 The two-parameter Poisson–Dirichlet point process 1097 Example 5.1. Consider a Poisson point process Z1 >Z2 > on (a,b) with mean measure Λ(dz)= h(z) dz such that Λ((a,b)) = and Λ([c,b)) < for all c (a,b). Lemma 5.1 then gives ∞ ∞ ∈ b m 1 b P (Z dz) h(z) − m ∈ = h(y) dy exp h(y) dy 1 (z) (5.2) dz (m 1)! − (a,b) − z z because the nth correlation function is h(y1) h(yn); see, for example, Example 2.5 in [27]. Setting ρm,α,θ(s)= Pα,θ(sVm < 1) (m =1, 2,...), we obtain the following proposition containing a two-parameter generalization of the aforementioned result in [19]. Proposition 5.2. Let m be a positive integer. Then, for all s> 0, 1 ( 1)k ρ (s)=1 − c I (s) (5.3) m,α,θ − (m 1)! (m + k)k! m+k,α,θ m+k,α,θ 0 k s m − ≤ ≤ − and for all λ> 0, λz θ ∞ θ 1 λs α ∞ e− λ dss − e− (ρ (s) 1)= R λ− dz , (5.4) m,α,θ − m,α,θ − zα+1 0 1 where m 1 u ( 1) − cm 1,α,θ m 1 Rm,α,θ(u) := − − x − Rα,θ+α(m 1)′(x) dx (m 1)! 0 − − (5.5) m 1 u Γ(θ + 1)( θ) − m 1 θx − x − e dx, α =0, (m 1)! 0 = − m 1 u m 1 Γ(θ + 1)Γ(θ/α + m)( α) − x − − dx, 0 <α< 1. (m 1)!Γ(θ/α + 1)Γ(1 α)m (1 C x)θ/α+m − − 0 − α Also, for v (0, 1), ∈ Pα,θ(Vm dv) Γ(θ + 1) (α+1) θ+α 1 ∈ = v− (1 v) − dv Γ(θ + α)Γ(1 α) − − (5.6) 1 ( 1)k 1 v − cm+k 1,α,θ+αIm+k 1,α,θ+α − . × (m 1)! k! − − v 0 k 1/v m − ≤ ≤ − Remarks. (i) Since ρ1,α,θ = ρα,θ, it is worth noting that the right-hand sides of (5.3), (5.5) and (5.6) with m = 1 equal the right-hand sides of (3.5), (4.1) and (4.4), respectively. (ii) In [34], Theorem 3 on page 47 gives the expression (5.3) with (α, θ)=(0, 1) to the limit distribution of the mth maximal cycle length in a random permutation. 1098 K. Handa (iii) At the end of this section, another expression of the density (5.6) will be given in terms of the generalized Dickman function. Proof of Proposition 5.2. First, (5.1) and Theorem 2.1 together imply that 1 ∞ ( 1)k P (sV 1)= − c I (s). α,θ m ≥ (m 1)! (m + k)k! m+k,α,θ m+k,α,θ − k =0 By removing the terms which actually vanish, (5.3) follows. Next, showing (5.4) reduces to verifying that for u sufficiently close to 0, m 1 ( 1) − ∞ Γ(θ + α(m + k)) R (u)= − c um+k. (5.7) m,α,θ (m 1)! (m + k)k! m+k,α,θ − k =0 For, if (5.7) is true, then (5.3) and Lemma 3.1(ii) with R = Rm,α,θ and G = 1[1, ) show − ∞ that (5.4) is valid, at least for λ sufficiently large, and the expression (5.5) allows us to extend (5.4) to all λ> 0 by analytic continuation. Turning to (5.7), one can easily verify it by using (term-by-term differentiation of) (3.6) or, alternatively, by substituting the θx (θ/α+m) expansion of e or (1 C x)− into (5.5). This proves (5.4). − α Lastly, since (5.3) is at hand, the proof of (5.6) is similar to that of (4.4) based on (4.5). The details are left to the reader. To obtain multidimensional results for PD(α, θ), one needs more developed arguments still based essentially on inclusion-exclusion. In the theory of point processes, one such calculus is formulated as a connection between correlation measures and Jonassy mea- sures; see Section 5 of [7]. Its significance is that the local probabilistic structure of points in the process is revealed in terms of correlation measures. Lemma 5.3. Let a ∞ M (c) n (1 + ε)n < for some ε = ε(c) > 0. (5.8) n! ∞ n=1 Let Z >Z > be the decreasing sequence of points in ξ. Then, for each m =1, 2,..., 1 2 the joint probability density fm of (Z1,...,Zm) is given by ∞ ( 1)k fm(z)= − dy1 dyk qm+k(z,y1,...,yk), (5.9) k k! [zm,b) k=0 where z = (z ,...,z ) is such that b>z > >z >a. 1 m 1 m The two-parameter Poisson–Dirichlet point process 1099 Proof. Let b =: z0 >z1 > >zm >a be given arbitrarily. Consider the localized process (m) (m) ξ (dx) := ξ(dx [zm,b)), the nth correlation function of which is given by qn (xn) := q (x )1 (x )∩ 1 (x ), where x = (x ,...,x ). By applying Theorem 5.4.II in n n [zm,b) 1 [zm,b) n n 1 n [7], then, we have P (Z1 >z1 Z2 >z2 >zm 2 Zm 1 >zm 1,zm >Zm) ≥ ≥ − ≥ − − (m) (m) (m) = P (ξ ([zm,z0]) = m 1, ξ ((z1,z0]) = = ξ ((zm 1,zm 2]) = 1) − − − k m 1 ∞ ( 1) − (m) = − 1(zi,zi−1](xi)qm 1+k(xm 1+k) dx1 dxm 1+k k! m−1+k − − − (a,b) i=1 k=0 zm−2 z1 b = dxm 1 dx2 dx1 − zm−1 z2 z1 ∞ ( 1)k − dxm dxm+1 dxm 1+k qm 1+k(xm 1+k) (5.10) k − − − × k! [zm,b) k=0 zm−2 z1 b = dxm 1 dx2 dx1 − zm−1 z2 z1 b b b ∞ k ( 1) dxm dxm+1 dxm 1+k qm 1+k(xm 1+k), (5.11) − − − × − zm xm xm−2+k k=0 where the last equality is due to symmetry in xm,...,xm 1+k and the terms for − k = 0 in (5.10) and (5.11) are understood as qm 1(xm 1) with the convention that − − q 1. This shows (5.9) by the symmetry of the integrand in (5.11) in k 1 variables 0 ≡ − xm+1,...,xm 1+k. − Example 5.2. If ξ is a Poisson point process with mean measure density h as in Example 5.1, the joint density of the first m largest points in ξ is given by m k m b ∞ ( 1)k − dy1 dyk h(zi) h(yj )= h(zi)exp h(z) dz , (5.12) k! k − [zm,b) i=1 j=1 i=1 zm k =0 provided that b>z > >z >a. 1 m By applying (5.9), we deduce the next result for joint densities, which is essentially contained in [45], although an explicit formula is not given. (Indeed, the authors obtained in their Corollary 41 a corresponding result for the variables y1,y2,... in (2.6) instead of v1, v2,.... Alternatively, the formula below can be retrieved from their Proposition 47 after some additional calculations. See also Lemma 3.1 in [16].) This simultaneously generalizes the formula due to Watterson [54] for α = 0 and the one-dimensional result (4.4). Also, our expression (5.13) below will be quite useful in Section 6.1. 1100 K. Handa Theorem 5.4. Let (Vi) be governed by PD(α, θ). Then, for each m =1, 2,..., the joint probability density of (V ,...,V ) at v = (v ,...,v ) is 1 m 1 m ∈ ∇m θ+αm 1 m m − m (α+1) 1 j=1 vj f (v) := c v− 1 v ρ − . (5.13) m,α,θ m,α,θ i − j α,θ+αm v i=1 j=1 m Proof. First note that the PD(α, θ) process ξ = δVi is a point process on (0, 1) which satisfies all of the assumptions in Lemma 5.3 since, for each 0 du1 duk qm+k,α,θ(v,u1,...,uk) k [vm,1) m (α+1) = cm,α,θ vi− ck,α,θ+αm (5.14) i=1 θ+α(m+k) 1 k m k − 1 uj vm { ≥ } α+1 1 vi uj du1 duk × m u − − ∆k(1 Pi=1 vi) j=1 j i=1 j=1 − with notation (3.4). Because the last integral in (5.14) is equal to θ+αm 1 m − 1 m v 1 v I − i=1 i , (5.15) − i k,α,θ+αm v i=1 m the desired expression (5.13) is derived from (5.14), (5.15) and (4.1). Since fm,α,θ is too complicated to compute quantities concerning the joint distribution directly from this, it is worth providing a moment formula. For the one-parameter case where α = 0 and θ> 0, Griffiths [18] showed that E [V a1 V am ] equals 0,θ 1 m m m z θ Γ(θ) ai 1 zi ∞ e− dz (z − e− )exp θ dz , (5.16) Γ(θ + a + + a ) i − z 1 m z1> >zm>0 i=1 zm provided that θ +a1 + +am > 0, where dz = dz1 dzm. The proof reduces to calculus of a gamma process by virtue of the well-known independence property of it; see [50] and [51] for extensive discussions and related topics. In this regard, we remark only that the integrand in (5.16) contains the density function, computed from (5.12), of the first m largest points in a gamma process with parameter θ, that is, a Poisson point process on The two-parameter Poisson–Dirichlet point process 1101 (0, ) with mean measure θ dz/(zez). Although the case where 0 <α< 1 and θ> α ∞ − can be handled by means of representing a PD(α, θ)-distributed random element by an α-stable subordinator ([45, 50] combined with (5.12)), we prefer to exploit the previous results in order to make the proof self-contained. Corollary 5.5. Let 0 <α< 1 and θ> α. If a + + a > θ, then − 1 m − E [V a1 V am ] α,θ 1 m Γ(θ + 1)Γ(θ/α + m)αm 1 = − (5.17) Γ(θ + a + + a )Γ(θ/α + 1)Γ(1 α)m 1 m − m (θ/α+m) z − ai (α+1) zi ∞ e− dz (z − e− ) 1+ C dz . × i α zα+1 z1> >zm>0 i=1 zm Proof. By (5.13), E [V a1 V am ]/c α,θ 1 m m,α,θ θ+αm 1 m m − 1 m v ai (α+1) j=1 j = dv dv v − 1 v ρ − . 1 m i − j α,θ+αm v m i=1 j=1 m ∇ θ+A 1 y Multiplying both sides by Γ(θ + A)= ∞ y − e− dy with A = a + + a , we intro- 0 1 m duce an additional integration with respect to a new variable y on the right-hand side. For the resulting (m + 1)-dimensional integral, perform the change of variables m 1 s := v− 1 v , z := v y (i =1,...,m), m − j i i j=1 or, equivalently, y = (z + + z )+ z s, v = z /y (i =1,...,m), to get 1 m m i i E [V a1 V am ]Γ(θ + A)/c α,θ 1 m m,α,θ m ai (α+1) θ+αm 1 ∞ m y θ+αm 1 = dz z − z − dsy e− J s − ρ (s) i m | | α,θ+αm z1> >zm>0 i=1 0 m with Jacobian J = ∂(v1,...,vm,y)/∂(z1,...,zm,s)= y− zm. Thanks to (4.8), with θ + αm > 0 in place of θ, we arrive at (5.17). Remarks. (i) It can be seen that the formula for E [V p ](p> θ) in Proposition 17 of α,θ m − [45] is recovered by putting a1 = = am 1 = 0 and am = p in (5.17). − (ii) Let 0 6. Asymptotics of PD(α, θ) for large θ In this section, we study certain asymptotic behaviors of PD(α, θ) as θ , gener- alizing results of Griffiths [18] and of Joyce, Krone and Kurtz [28], who all→ worked ∞ on PD(0,θ)’s with motivation coming from the study of population genetics. Although there is a context [24] in which such an extension could be applicable, this section is mainly intended to demonstrate that the results we have thus far obtained provide efficient meth- ods for the study of two-parameter Poisson–Dirichlet distributions. It turns out that the presence of the parameter α does not affect the validity of the assertions analogous to those for PD(0,θ)’s, except for some minor changes of sub-leading terms or multiplicative constants in rescaling. 6.1. Convergence to a Gumbel point process Given 0 α< 1 and θ> 1, put ≤ β = log θ (α + 1) log log θ log Γ(1 α). (6.1) α,θ − − − (θ) Under the assumption that (Vi )i∞=1 is distributed according to PD(0,θ), it was shown (θ) in [18] that as θ , (θV β )∞ converges in law to a Poisson point process with → ∞ i − 0,θ i=1 mean measure density exp( z), (α,θ) Theorem 6.1. Fix an α [0, 1) arbitrarily. For each θ> 1, let (Vi )i∞=1 have PD(α, θ) distribution and define ∈ Z(α,θ) = θV (α,θ) β (i =1, 2,...). (6.2) i i − α,θ Then, as θ , (Z(α,θ),Z(α,θ),...) converges in joint distribution to the decreasing order → ∞ 1 2 statistics (Z1∗,Z2∗,...) of a Poisson point process on the whole real axis with mean measure exp( z) dz. − The two-parameter Poisson–Dirichlet point process 1103 The proof of this theorem reduces to showing pointwise convergence of joint densities, thanks to Scheff´e’s theorem (cf. [2]). It follows from (5.12) that m zm P (Z∗ dz ,...,Z∗ dz )=exp z e− dz dz , (6.3) 1 ∈ 1 m ∈ m − i − 1 m i=1 (α,θ) (α,θ) while the density of (Z1 ,...,Zm ) derived from (5.13) involves the generalized Dick- man function as a key factor. It is therefore essential to study the asymptotic behavior of ρα,θ with a properly rescaled variable for large θ. This is the content of the following lemma, in which is derived the Gumbel distribution, the law of Z1∗. Lemma 6.2. Fix 0 α< 1. Let θ > 1 and b :[θ , ) R be bounded. If V (α,θ) and ≤ 0 0 ∞ → 1 Z(α,θ) are as in Theorem 6.1, then for each x R, 1 ∈ − (α,θ) (α,θ) θ e x P (Z < x (x + β )b(θ)V )= ρ + b(θ) e− (6.4) 1 − α,θ 1 α,θ x + β → α,θ (α,θ) as θ . In particular, Z Z∗ weakly and ρ (s) 1 for all s R. → ∞ 1 → 1 α,θ → ∈ Proof. First, the equality in (6.4) is clear from (6.2). Rather than using (4.1), which is not very informative for the estimation of the value of ρα,θ, we employ (4.8), that is, z ∞ e Γ(θ)exp θ dz − , α = 0, ∞ − z θ θ 1 λs λ λ dss − e− ρα,θ(s)= z θ/α (6.5) 0 ∞ e− − Γ(θ) 1+ Cα dz , 0 <α< 1, zα+1 λ where λ> 0 is arbitrary. We will give only a proof for the case where 0 <α< 1 because the case α = 0 can be handled in much the same way. By Fubini’s theorem, the left-hand side of (6.5) is written as −1 −1 V1 λV1 θ θ 1 λs θ 1 t 1 λ Eα,θ dss − e− = Eα,θ dtt − e− = Eα,θ[fλ,θ(V1− )], 0 0 λs θ 1 t where fλ,θ(s)= 0 dtt − e− . For all λ> 0 and s> 0, we get 1 z θ/α E [f (V − )] Γ(θ) ∞ e − ρ (s) α,θ λ,θ 1 = 1+ C dz − (6.6) α,θ ≤ f (s) f (s) α zα+1 λ,θ λ,θ λ 1/4 as a Chebyshev-type bound. Set ε(θ)= θ− , so that ε(θ) log θ 0, ε(θ)√θ (θ ). (6.7) → → ∞ → ∞ 1104 K. Handa Given arbitrary x R, choose ∈ θ s = s(x, θ) := + b(θ) and λ = λ(x, θ) := (1 + ε(θ))(x + βα,θ), x + βα,θ both of which are positive for sufficiently large θ. Then ε(x, θ), defined implicitly by λ(x, θ)s(x, θ)=(1+ ε(x, θ))θ, has the same properties as (6.7). The latter property, com- bined with the standard argument in the proof of Stirling’s formula (that is, the Laplace method [9]), shows that ∞ θ 1 t ∞ θ 1 θu Γ(θ) 0 dtt − e− 0 duu − e− = (1+ε(x,θ))θ = 1+ε(x,θ) 1 fλ(x,θ),θ(s(x, θ)) e θ 1 t e θ 1 θu → 0 dtt − e− 0 duu − e− as θ . Consequently, by (6.6), with the above choice of s and λ, we have → ∞ z θ/α θ ∞ e− − lim sup ρα,θ + b(θ) lim sup 1+ Cα dz α+1 . (6.8) θ x + βα,θ ≤ θ λ(x,θ) z →∞ →∞ For two functions a1(θ) and a2(θ), which may have parameters α, x, etcetera, the notation a (θ) a (θ) will mean that a (θ)/a (θ) 1 as θ . By (6.7) and (6.1), 1 ∼ 2 1 2 → → ∞ z λ(x,θ) (x+βα,θ) ∞ e− e− e− Γ(1 α) x dz − e− . zα+1 ∼ λ(x, θ)α+1 ∼ (x + β )α+1 ∼ θ λ(x,θ) α,θ Combining this with (6.8) yields θ x lim sup ρα,θ + b(θ) exp( e− ). (6.9) θ x + βα,θ ≤ − →∞ The converse estimate can be shown in an analogous way. Indeed, considering this time 1 1 ρ (s)= P (V s− ), we again have, by Fubini’s theorem, − α,θ α,θ 1 ≥ θ ∞ θ 1 λs λ dss − e− (1 ρ (s)) = E [ϕ (V )], (6.10) − α,θ α,θ λ,θ 1 0 ∞ θ 1 t where ϕλ,θ(s)= λ/s dtt − e− . Therefore, a Chebyshev-type bound we get is z θ/α Eα,θ[ϕλ,θ(V1)] Γ(θ) ∞ e− − 1 ρα,θ(s) = 1 1+ Cα dz , − ≤ ϕ (s 1) ∞ dttθ 1e t − zα+1 λ,θ − λs − − λ where the last equality follows from (6.10) and (6.5) together. By setting s = s(x, θ) and λ = (1 ε(θ))(x + β ), one can easily modify the previous argument to obtain − α,θ θ x lim sup 1 ρα,θ + b(θ) 1 exp( e− ). θ − x + βα,θ ≤ − − →∞ The two-parameter Poisson–Dirichlet point process 1105 This, together with (6.9), proves the convergence in (6.4). The other assertions can be seen simply by taking b( ) 0 and noting that ρ ( ) is non-increasing. ≡ α,θ Proof of Theorem 6.1. Fix a positive integer m and z1 > >zm arbitrarily. By (α,θ) (α,θ) (5.13), the joint probability density of (Z1 ,...,Zm ) at (z1,...,zm) is m z1 + βα,θ zm + βα,θ θ− f ,..., m,α,θ θ θ θ+αm 1 m m − (α+1) zj + βα,θ = c (z + β )− 1 (6.11) m,α,θ i α,θ − θ i=1 j=1 m αm θ j=1(zj + βα,θ) z1 + βα,θ zm + βα,θ θ ρα,θ+αm − 1 m ,..., . × z + β ∇ θ θ m α,θ It is easy to see that the factors in (6.11) behave as m (α+1) (1 α)m m (1+α)m c (z + β )− θ − Γ(1 α)− (log θ)− m,α,θ i α,θ ∼ − i=1 and θ+αm 1 m − m zj + βα,θ m (1+α)m m 1 exp z θ− (log θ) Γ(1 α) , − θ ∼ − j − j=1 j=1 respectively. In view of (6.3), it remains to show that the factor involving ρ ( ) α,θ+αm converges to exp( exp( zm)). But this follows from (6.4) with b( ) defined appropriately. The proof of Theorem− 6.1− is thus complete. (α,θ) The weak convergence of Zm to Zm∗ implies the following. Corollary 6.3. For each m =1, 2,... and x R, as θ , ∈ → ∞ m 1 − kx θ x e− ρm,α,θ exp( e− ) . (6.12) x + βα,θ → − k! k =0 (α,θ) Proof. As a consequence of Theorem 6.1, the left-hand side of (6.12), that is, P (Zm ≤ x), tends to P (Z∗ x), which is calculated by (5.2): m ≤ x dy y 1 ∞ m 1 t P (Zm∗ x)= exp( my e− )= t − e− dt. ≤ (m 1)! − − (m 1)! e−x −∞ − − This changes into the right-hand side of (6.12) after repeated integration by parts. 1106 K. Handa Theorem 6.1 immediately implies that (V (α,θ), V (α,θ),...) (0, 0,...) as θ . Feng 1 2 → → ∞ [16] has recently obtained associated large deviation estimates with rate function (v ) i → log(1 v ). The proof is based on logarithmic asymptotics of the finite-dimensional − − i densities. We note that, in view of (5.13), one can grasp the validity of such a result because the factor involving ρα,θ+αm stays away from 0 by (6.4). 6.2. Asymptotic normality of the population moments As the final topic on PD(α, θ) for large θ, we discuss asymptotic normality of the popu- lation moment, which is defined for each p> 0 by the sum H(α,θ) = (V (α,θ))p + (V (α,θ))p + , p 1 2 (α,θ) (α,θ) where (V1 , V2 ,...) continues to be PD(α, θ)-distributed. We can verify that (α,θ) p Hp < , a.s. for all p > α. Indeed, setting g (s) = exp( s ), one observes that λ (g ) ∞ p − α p in Theorem 3.2 is finite for p > α and the claim follows from (3.7) with g = gp. Also, (2.12) with φ(v)= gp(sv) 1 makes it possible to derive the Laplace transform of the (α,θ) − (α,θ) p (α,θ) law of Hp because i∞=1 gp(sVi ) = exp( s Hp ). As far as the PD(0,θ) case (0,θ) − is concerned, the statistics H2 , called the population homozygosity, is known to play (0,θ) a special role in some population genetics contexts (see [19, 20, 29]), where Vi are (0,θ) regarded as the ranked frequencies of alleles in a population. More generally, Hp is re- (0,θ) ferred to as the pth population moment in [28], where the asymptotic normality of Hp (α,0) for large θ has been established. On the other hand, covariances of Hp ’s are calculated in [10] ((12), page 188), motivated by certain physical problems. In this subsection, we will provide an extension of these results to the two-parameter setting. In the case α = 0, it was shown in [28] that for each p =2, 3,..., as θ , the limit (θ) p 1 (0,θ) → ∞ distribution of Wp := √θ(θ − Hp /Γ(p) 1) is the normal distribution with mean − 0 and variance Γ(2p)/Γ(p)2 p2. In fact, in [28], much stronger results were obtained, − (θ) showing, for example, asymptotic normality of random vectors (Wm )m∞=2. To seek an appropriate rescaling for the two-parameter case, we give some auxiliary results by means of the first and second correlation functions, namely, q1,α,θ and q2,α,θ. Lemma 6.4. Let p > α and p′ > α. Then Γ(θ + 1)Γ(p α) E[H(α,θ)]= − =: h(α,θ) (6.13) p Γ(θ + p)Γ(1 α) p − (α,θ) (α,θ) (α,θ) (α,θ) and the covariance Cov(Hp ,Hp′ ) of Hp and Hp′ is given by 2 (α,θ) Γ(p α)Γ(p′ α) Γ(θ + 1)(θ + α) Γ(θ + 1) h ′ + − − . (6.14) p+p Γ(1 α)2 Γ(θ + p + p ) − Γ(θ + p)Γ(θ + p ) − ′ ′ The two-parameter Poisson–Dirichlet point process 1107 Moreover, as θ , → ∞ p 1 (α,θ) Γ(p α) θ − E[H ] − (6.15) p → Γ(1 α) − and ′ p+p 1 (α,θ) (α,θ) Γ(p + p′ α) Γ(p α)Γ(p′ α) θ − Cov(H ,H ′ ) − + − − (α pp′). (6.16) p p → Γ(1 α) Γ(1 α)2 − − − 1 p Proof. (6.13) and (6.14) are immediate by calculating 0 v q1,α,θ(v) dv and 1 ′ ′ (α,θ) (α,θ) p+p p p E[Hp Hp′ ]= v q1,α,θ(v) dv + v1 v2 q2,α,θ(v1, v2) dv1 dv2, 0 ∆2 1 p respectively. By virtue of the well-known formula Γ(θ + 1)/Γ(θ + p) θ − implied ∼ by Stirling’s formula, (6.15) follows from (6.13). Showing (6.16), however, requires a more accurate version of Stirling’s formula: with the standard o-notation, Γ(θ +1)= θ 1 1 θ √2πθ exp( θ + Cθ− + o(θ− )) as θ for some universal constant C; see, for ex- − → ∞ ample, [9]. This makes it possible to show that Γ(θ + 1) 1 p 1 1 = θ − exp( p(p 1)(2θ)− + o(θ− )). (6.17) Γ(θ + p) − − The proof of (6.16) is a matter of straightforward calculation, combining (6.17) with (6.14). The details are omitted. Lemma 6.4 suggests a rescaling of the form (α,θ) Γ(1 α) p 1 (α,θ) W := √θ − θ − H 1 . (6.18) p Γ(p α) p − − We can now state a desired result for 0 <α< 1. Theorem 6.5. Fix an α (0, 1) arbitrarily. Suppose that p > α and p =1. For each (α,θ) ∈ (α,θ) θ> 0, let Wp be defined as above. Then, as θ , Wp converges in law to a → ∞ normal random variable with mean 0 and variance Γ(1 α)Γ(2p α) σ 2 := − − + α p2 > 0. (6.19) α,p Γ(p α)2 − − Proof. Fixing an arbitrary x R, we will prove that as θ , ∈ → ∞ σ 2 ψ(α,θ)(x) := E[exp(√ 1xW (α,θ))] exp α,p x2 . (6.20) p − p → − 2 1108 K. Handa The proof is based on (3.8), a consequence of Theorem 3.2 which can be written as z θ/α 1 ∞ θ 1 s ∞ ∞ e− − dss − e− E g(sV ) = 1+ C dz (1 g(z)) , (6.21) Γ(θ) α,θ i α zα+1 − 0 i=1 0 p provided that 1 > λα∗ (g). Let g be of the form g(z) = exp(c1z + d1z) and observe that ∞ p (α,θ) Eα,θ g(sVi) = E[exp(c1s Hp + d1s)]. (6.22) i=1 Note that (6.21) is valid as long as the integral on the right-hand side is small enough in modulus. With the choices x Γ(1 α) x c1 = √ 1 − and d1 = √ 1 , (6.23) − √θ Γ(p α) − − √θ − this requirement for the validity of (6.21) is fulfilled for θ large enough since as θ , → ∞ z z ∞ e− x ∞ e− Γ(1 α) dz (1 g(z)) = √ 1 dz − zp z zα+1 − − − √θ zα+1 Γ(p α) − 0 0 − 2 z 2 x ∞ e− Γ(1 α) p 1 + dz − z z + o(θ− ) 2θ zα+1 Γ(p α) − 0 − 2 x 2 2 1 = Γ(1 α)(σ + (p 1) )+o(θ− ), 2θ − α,p − where the last equality is achieved by direct calculations. This also implies that the right-hand side of (6.21) tends to exp( σ 2x2/2 (p 1)2x2/2). On the other hand, − α,p − − the left-hand side of (6.21) with (6.23) becomes, by (6.22), after change of variable s = θt, ∞ θ 1 θt p (α,θ) p θ 1 dtt − e− E[exp(c1θ Hp t + d1θt)](Γ(θ)θ− )− 0 ∞ 1 θh(t) (α,θ) p p θ θ 1 = dtt− e− ψ (xt ) exp(√ 1x√θ(t t))(Γ(θ)θ− e )− , p − − 0 where h(t)= t log t 1. Thus, by Stirling’s formula, (6.20) is equivalent to − − 1 θh(t) (α,θ) p p ∞ dtt− e− ψp (xt ) exp(√ 1x√θ(t t)) 2 2 0 − − ψ(α,θ)(x)e (p 1) x /2 0, π p − − 2 /θ − → the proof of which will be divided into proofs of ∞ 1 θh(t) (α,θ) p (α,θ) p √θ dtt− e− (ψ (xt ) ψ (x)) exp(√ 1x√θ(t t)) 0 (6.24) p − p − − → 0 The two-parameter Poisson–Dirichlet point process 1109 and θ ∞ 1 θh(t) p (p 1)2x2/2 dtt− e− exp(√ 1x√θ(t t)) e− − 0. (6.25) 2π − − − → 0 Note that the integral in (6.24) can be bounded in modulus by ∞ 1 θh(t) (α,θ) p (α,θ) (α,θ) ∞ 1 θh(t) p dtt− e− ψ (xt ) ψ (x) x E[ W ] dtt− e− t 1 | p − p | ≤ | | | p | | − | 0 0 (α,θ) and that sup E[ Wp ] < since, by Lemma 6.4 and (6.17), θ>1 | | ∞ (E[ W (α,θ) ])2 Var(W (α,θ)) + (E[W (α,θ)])2 σ2 . | p | ≤ p p → α,p √ ∞ 1 θh(t) p Therefore, our task is reduced to establishing that θ 0 dtt− e− t 1 0 and (6.25). Both are purely analytic problems which can be solved without| difficu− lty| → by the Laplace method (or by the change of variable t =1+ u/ √θ). 2 It remains to show that σα,p > 0. A stronger result which can be shown is 2 Γ(1 α) p 1 α 2 σ = (1 α)Var − Y − + (p 1) , (6.26) α,p − Γ(p α) α 1 α − − − 1 α z where Yα has the density z − e− /Γ(2 α), 0 Remark. The above proof also works in the case α = 0 if (6.21) is replaced by the corresponding equality for α = 0. This gives a proof which is completely different from that in [28] and which is not based on the independence property underlying PD(0,θ). One may consider an extension of Theorem 1 in [28] in the two-parameter setting, (α,θ) that is, convergence of Wp : p > α to a centered Gaussian system as θ . The convergence of every finite-dimensional{ } law is shown by a suitable modification→ ∞ of the proof of Theorem 6.5. It means, for instance, that g in (6.21) is taken to be √ n 1 Γ(1 α) pk g(s)=exp − xk − s s , √θ Γ(pk α) − k=1 − where x R, p > >p > α and n =1, 2,... are arbitrary. The limit process is k ∈ 1 n described by its covariance Γ(1 α)Γ(p + p′ α)/(Γ(p α)Γ(p′ α))+ α pp′ =: C(p,p′) − − − − − and the associated quadratic form C(pk,pl)xkxl has two different expressions: 2 2 z n n 1 ∞ e− Γ(1 α) pk dz α+1 xk − z z xk(pk 1) Γ(1 α) 0 z Γ(pk α) − − − − k=1 − k=1 n n 2 Γ(1 α) pk 1 α = (1 α)Var xk − Yα − + xk(pk 1) , − Γ(pk α) 1 α − k=1 − − k =1 1110 K. Handa where Yα is as in (6.26). We do not produce the whole proof, a routine matter which is left to the reader. 7. The generalized Dirichlet process This section discusses the two-parameter generalization of Dirichlet processes from the point of view of our previous results. There are a number of motivations to study the original Dirichlet process, for example, as a prior distribution in Bayesian nonparametric statistics [17] and as a stationary state of a certain diffusion process arising in population genetics. The well-known relationship between the Dirichlet process and the Poisson– Dirichlet distribution is described as follows. Let X ∞ be a sequence of i.i.d. random { i}i=1 variables and (V )∞ be governed by PD(0,θ). Assuming independence of X ∞ and i i=1 { i}i=1 (Vi)i∞=1, we know that a random distribution η := ViδXi defines a Dirichlet process with underlying parameter measure θν, where ν is the common law of the Xi’s. In this sense, PD(0,θ) is the simplicial part of a Dirichlet process. By replacing PD(0,θ) by PD(α, θ), a two-parameter generalization of a Dirichlet process was introduced in [49]. In the case of Dirichlet processes, there are many articles, including [5, 11, 23, 36, 46], in which exact forms (the distribution function, for example) of the law of the (random) mean XiVi of η are obtained in terms of ν. A key tool is an integral identity due to Cifarelli and Regazzini [5] which connects these two laws. As explained in [31, 50, 53], it is called the Markov–Krein identity because in the case θ = 1, an integral transform is involved, analogous to the one studied by Markov and Krein in the context of moment problems; see [30] for background and various applications. Its extension was proven by Tsilevich [49] for the generalized Dirichlet process; see also [31] for a further extension and [50, 53] for a simple proof under some restriction on the support of ν. This identity gives a one-to-one correspondence between ν and the law of XiVi with (Vi) having PD(α, θ) distribution. However, such a correspondence is implicit and seems very subtle in general. One therefore needs some procedure of inversion in order to obtain explicit information. Recently, James, Lijoi, and Pr¨unster [25] obtained some distributional results for the generalized Dirichlet process by means of a Perron–Stieltjes- type inversion formula. Since PD(0,θ) and PD(α, 0) are related to gamma processes and stable subordinators, respectively [45], the corresponding problem is naturally considered for a more general class of random distributions derived from subordinators, as posed in [53]. For results of this kind, we refer the reader to works by Regazzini, Lijoi and Pr¨unster [47] and by Nieto-Barajas, Pr¨unster and Walker [38], both of which make essential use of the Gurland inversion formula. Our focus now will be on what is implied by Theorem 3.2 in the aforementioned context. It will be pointed out that the basic identity (3.8) involving the probability generating functional exhibits a mathematical structure underlying these kinds of identities. This should be compared with the proofs in [11, 31, 49], where the Markov–Krein identity is viewed as a relation between moment sequences through the Ewens(–Pitman) sampling formula [13, 42]. Let us introduce some notation used to describe the domain of a map defined via the Markov–Krein identity. Denoting by the totality of Borel probability P The two-parameter Poisson–Dirichlet point process 1111 measures on R, let 0 = ν : ν(dx) log(1 + x ) < P ∈P R | | ∞ and set = ν : ν(dx) x α < for 0 <α< 1. The Markov–Krein identity al- Pα { ∈P R | | ∞} ready mentioned is the following. Taking a ν , let X ∞ be i.i.d. random variables ∈Pα { i}i=1 with each law of X being ν and let (V )∞ be PD(α, θ)-distributed. If X ∞ and i i i=1 { i}i=1 (V )∞ are mutually independent, then X V < a.s. (as was shown in [14] for α =0 i i=1 | i| i ∞ and in Proposition 1 of [47] for 0 <α< 1) and the law, denoted α,θν, of M := XiVi is characterized by one of the following equalities: M (i) for α = 0 and θ> 0, θ 0,θν(dx)(z x)− = exp θ ν(dx) log(z x) (z C R); (7.1) R M − − R − ∈ \ (ii) for α (0, 1) and θ> 0, ∈ θ/α θ α − α,θν(dx)(z x)− = ν(dx)(z x) (z C R); (7.2) R M − R − ∈ \ (iii) for α (0, 1) and θ ( α, 0), α,θν θ and (7.2) holds true; (iv) for α ∈ (0, 1) and θ ∈= 0,− νM and∈P− ∈ Mα,0 ∈P0 1 α α,0ν(dx) log(z x)= α− log ν(dx)(z x) (z C R). (7.3) R M − R − ∈ \ Thus the correspondence ν α,θν defines a map from α to , for which we also write , by a slight abuse of → notation. M Since we are not awareP ofP any reference in which Mα,θ integrability of the transformed measure α,θν claimed in (iii) or (iv) has been shown, the proof shall be given below by applyingM our result. As will be seen later (Proposition 7.2(i)), such a property is also needed for further discussion. In the subsequent argument, we often use the following three equalities: ∞ θ 1 s us θ dss − e− e− = Γ(θ)(1 + u)− (θ> 0), (7.4) 0 ∞ α 1 s us α C dss− − e− (e− 1)=1 (1 + u) (0 <α< 1), (7.5) α − − 0 ∞ 1 s us dss− e− (e− 1) = log(1 + u), (7.6) − − 0 where u C is such that Re u> 1. ∈ − Lemma 7.1. Let α (0, 1) and θ ( α, 0]. Then α,θν θ for any ν α. ∈ ∈ − M ∈P− ∈P 1112 K. Handa Proof. Define M ∗ := X V . By the assumed independence, | i| i ∗ ∞ sM sVi x E[e− ]= Eα,θ ν(dx)e− | | (s 0). (7.7) R ≥ i=1 We will apply Theorem 3.2 by taking g(s)= R ν(dx)exp( s x ). For this purpose, with the help of Fubini’s theorem, observe from (7.5) that for 0−<α<| | 1, λz Cα ∞ e− 1 α α dz α+1 g(z) 1 = ν(dx)(1 + λ− x ) 1 < , (7.8) λ z | − | R | | − ∞ 0 which implies that λα∗ (g) =0. Therefore, (3.8) holds for all λ> 0 and reads θ ∗ λ ∞ θ 1 λs sM dss − e− (E[e− ] 1) Γ(θ + 1) 0 − (7.9) θ/α 1 1 α − 1 ν(dx)(1 + λ− x ) , 0 <α< 1, θ = 0, θ R | | − θ = 1 1 α log ν(dx)(1 + λ− x ) , 0 <α< 1, θ = 0. −α R | | The desired integrability can be seen as follows. For θ = 0, the left-hand side of (7.9) with λ = 1 becomes E[log(1 + M ∗)] by (7.6), while the right-hand side is finite by − ν α. This proves that α,0ν 0 whenever ν α. Similarly, for θ ( α, 0), by using∈P (7.5) with θ in placeM of α∈P, we see that the∈P left-hand side of (7.9∈) with− λ =1 − θ equals (E[(1 + M ∗)− ] 1)/θ. Therefore, α,θν θ is implied by ν α. − M ∈P− ∈P Remarks. (i) In view of (7.9), the reader will be able to see that the dichotomy result of Feigin–Tweedie type [14] (originally shown for PD(0,θ)’s) holds true for PD(α, θ) with θ > 0. More precisely, if ν and M ∗ is as above, then M ∗ = a.s. ∈P\Pα ∞ Thus the maximal domain of α,θ with θ > 0 is identified with α. In the Dirich- let process case, Cifarelli andM Regazzini [6] gave a proof of this factP by using (7.1). Their cutoff argument also applies for 0 <α< 1, as seen in the following. Considering Mn∗ := Xi 1 Xi n Vi for each n =1, 2,..., which corresponds to the image measure | | {| |≤ } νn of ν under x x 1[ n,n](x), we have (7.9) with Mn∗ and νn in place of M ∗ and ν, re- → | | − θ spectively. Under the assumption that ν / , letting n yields E[(1 + M ∗/λ)− ]=0 ∈Pα → ∞ and hence M ∗ = a.s., as long as θ> 0. (ii) The above∞ proof makes it almost obvious that the Markov–Krein identity itself can be recovered from Theorem 3.2 with g(s)= ψν ( s), where ψν is the characteristic function of ν. These choices are allowed for any ν ± because for all x R, ∈Pα ∈ z ∞ e− √ 1xz A0 log(1 + x )+ B0 min x , 1 , α = 0, dz e − 1 α | | {| | } zα+1 | − |≤ Aα x + Bα min x , 1 , 0 <α< 1, 0 | | {| | } where Aα and Bα are constants depending only on α. (For the proof of this inequality, first observe that the left-hand side is bounded above by x Γ(1 α) and then show it | | − The two-parameter Poisson–Dirichlet point process 1113 for x> 1 by, for example, dividing the domain of integration into (0, 1/x) and [1/x, ).) ∞ In addition, (7.5) and (7.6) show that λ∗ (ψ ( )) = 0. Therefore, analogously to (7.7), α ν ± ∞ ψ α,θν ( s)= E[exp( √ 1sM)] = Eα,θ ψν ( sVi) , (7.10) M ± ± − ± i=1 and we obtain, by (3.8), λz θ ∞ θ 1 λs α ∞ e− λ dss − e− (ψ α,θ ν ( s) 1)= Rα,θ λ− dz (ψν ( z) 1) (7.11) M ± − zα+1 ± − 0 0 for all λ> 0. In the same way as before, (7.11) can be converted to (7.1)–(7.3) with z = √ 1λ by making use of (7.4)–(7.6). ± − The rest of the paper is devoted to describing various consequences of the above cal- culations and the Markov–Krein identity. Proposition 7.2. (i) Let 0 <β<α< 1 and θ> β. Then, as a map defined on , − Pα β,θ α, β = α,θ. (7.12) M ◦M − M (ii) Let 0 α< 1 and θ> α. Suppose that ν . Then, for a.e. t R, ≤ − ∈Pα ∈ θ+αn 1 n n − ∞ cn,α,θ ψν (tvi) 1 ψ α,θ ν (t)=1+ − 1 vj dv1 dvn. (7.13) M n! α+1 − n=1 ∆n i=1 vi j=1 (iii) Let 0 <α< 1 and θ> α. Then defined on is injective. − Mα,θ Pα Proof. (i) First, note that by Lemma 7.1, the composition β,θ α, β is well defined M ◦M − on α under the conditions assumed. (7.12) with θ = 0 becomes β,0 α, β = α,0, P M ◦M − M the proof of which requires careful handling of branches. Taking an arbitrary ν and ∈Pα putting ν′ = ( β,0 α, β)ν, we observe from (7.2) and (7.3) that M ◦M − 1 β ν′(dx) log(z x)= log α, βν(dx)(z x) R − β R M − − 1 β/α 1 β = log ν(dx)(z x)α = log exp log ν(dx)(z x)α . β R − β α R − It should here be noted that since 0 <β/α< 1, β β argexp √ 1 arg ν(dx)(z x)α = arg ν(dx)(z x)α. − α R − α R − 1114 K. Handa Combining these equalities yields 1 β α ν′(dx) log(z x)= log ν(dx)(z x) = α,0ν(dx) log(z x), R − β α R − R M − where the last equality follows from (7.3). Since z C R is arbitrary, this implies ∈ \ ν′ = α,0ν, as required. It only remains to deal with the case where 0 <β<α< 1 and θ = 0M is such that θ> β. Calculations are quite similar to those in the previous case and so are left to the reader.− The proof of (i) has already been done. (ii) By virtue of (7.11), (7.13) is a direct consequence of Lemma 3.1(i). (iii) First, suppose that α,0ν = α,0ν′ for some ν,ν′ α. Then, by (7.3), 1 α M M 1 α ∈ P ν(dx)(1 + √ 1λ− x) = ν′(dx)(1 + √ 1λ− x) for all λ R 0 . This, together R − R − ∈ \{ } with (7.5), implies that ν = ν′. Next, let θ = 0 be such that θ> α and assume that − α,θν = α,θν′ for some ν,ν′ α. Taking care with branches, we see by (7.2) that for Meach λ RM 0 , there exists an∈P integer n(λ) such that ∈ \{ } 1 α 1 α 1 log ν(dx)(1+ √ 1λ− x) log ν′(dx)(1 + √ 1λ− x) =2πn(λ)αθ− . (7.14) R − − R − However, since the left-hand side is continuous in λ and tends to 0 as λ , we have | | → ∞ n(λ) 0. This proves ν = ν′ and therefore is injective. ≡ Mα,θ Let us make some comments on Proposition 7.2. The assertion (i) extends Theorem 2.1 of [25], where (7.12) with β = 0 is shown and applied for a sampling procedure. We can also understand (7.12) via (7.11) combined with Rβ,θ Rα, β = Rα,θ. In general, a ◦ − random variable M having ( α,β γ,δ)ν distribution (if such a law is well defined) is M (◦Mγ,δ) (α,β) constructed by M = j ( i Xij Vij )Vj , where Xij is a family of i.i.d. random (α,β) { } (γ,δ) variables with common law ν, (Vj )j∞=1 has PD(α, β) distribution and (Vi1 )i∞=1, (γ,δ) (Vi2 )i∞=1,... are PD(γ,δ)-distributed. Moreover, these random elements are required to be mutually independent. (ii) and (iii) are motivated by some results of Lijoi and Regazzini [36] concerned with 0,θ. Applying to (7.11) a general complex inversion for- mula for Laplace transforms (theM Bromwich integral), the authors provided an expression for the characteristic function of 0,θν. It is not clear how to verify equivalence between such a formula and (7.13) directly.M They also proved a prototype of (iii), that is, that 0,θ : 0 is injective for each θ> 0. MAs observedP →P [26] (Remark 4.2) in a much more general scheme called a species sampling model, Cauchy distributions are fixed points of every α,θ. Alternatively, this also follows from (7.10). It should be emphasized that any propertyM of PD(α, θ) as a law on is irrelevant to the validity of this fact, except the one which ensures that X V∇<∞ | i| i ∞ a.s. The study of the inverse problem would require a deeper understanding of α,θ itself or PD(α, θ). In this respect, Yamato [55] gave a partial result concerning theM case α = 0. It was shown in [36] that for any ν 0, 0,1ν = ν occurs only when ν is Cauchy or degenerate. It seems reasonable to expect∈P thatM such an equivalence would hold true for any and we could approach this problem by the use of results presented in Section Mα,θ The two-parameter Poisson–Dirichlet point process 1115 3. But here we shall show only a minor extension by mimicking the proof of Theorem 5 in [36], that is, by reducing to an ordinary differential equation via the Markov–Krein identity. Proposition 7.3. Let 0 <α< 1 and ν α. Then α,1 αν = ν if and only if ν is Cauchy or degenerate. ∈P M − Proof. Supposing that α,1 αν = ν, we only have to show that ν is Cauchy or degen- M − erate. By combining this equality with (7.2), we easily see that f (t) := R ν(dx)(t ± α 1 1 1/α 1/α ± √ 1x) solves the equation α− f ′(t)= f (t) − (t> 0) and therefore f (t) = − 1/α ± ± ± t+f (0) (t 0). Here, since f+(0) and f (0) are complex conjugate with arg f (0) πα/2,± we have≥ arg(f (0)1/α) π/2 so that− f (0)1/α = σ2 √ 1m for| some±σ2 |≤0 and m R. Consequently,| ± f |≤(t)1/α = t + σ2 ±√ 1m(t 0).± Furthermore,− by noting≥ the continuity∈ of arg f (t), we± get f (t) = (t±+ σ−2 √ ≥1m)α for every t 0. With the help of (7.5), we± conclude that ±ν is a Cauchy± or− degenerate distribution≥ with ψ (t) = exp(√ 1mt σ2 t ). ν − − | | Acknowledgements This work was supported in part by the exchange program between the Japan Society for the Promotion of Science and Centre National de la Recherche Scientifique. It was initiated during a stay at Laboratoire de Probabilit´eet Mod`eles Al´eatoires, Paris. The author would like to express his sincere gratitude to Professor Marc Yor for stimulat- ing discussions and gratefully acknowledges the warm hospitality of the laboratory. He wishes to thank the anonymous referee for providing comments and help in improving the presentation of this paper, and is also grateful to S. Feng, who informed him of the recent article [16]. 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MR0723745 Received December 2007 and revised August 2008