A NEW PROOF OF THE MARKOV–KREIN IDENTITY FOR

DISTRIBUTIONS OF MEANS OF DIRICHLET PROCESSES

N. Tsilevich and A. Vershik and M. Yor

Abstract. We present a new simple proof of the Markov–Krein identity for distributions of means of Dirichlet processes. Our approach is based on a close relation of Dirichlet processes to gamma processes and basic properties of the gamma processes.

1. Introduction. The Dirichlet processes introduced in [F] play a key role in Bayesian nonparametric statistics. The classical definition of these processes is as follows. Denote by Δ_{n} = {x = (x_{0}, …, x_{n}) : x_{i} ≥ 0, ∑ x_{i} = 1} the ndimensional simplex. The Dirichlet distribution Dir(τ_{0}, …, τ_{n}) on Δ_{n} with parameters τ_{0}, …, τ_{n} > 0 is determined by the density \frac{Γ(τ_{0}+...+τ_{n})}{Γ(τ_{0})...Γ(τ_{n})}x_{0}^{τ_{0}−1} … x_{n}^{τ_{n}−1}.

Definition 1. Let (X, ν) be a standard Borel space with a non-atomic finite positive measure ν. The Dirichlet process on the space X with parameter measure ν is a random probability distribution P on X such that for every finite measurable partition X = A_{0} ∪ … ∪ A_{n}, the vector (P (A_{0}), …, P (A_{n})) has Dirichlet distribution Dir(ν(A_{0}), …, ν(A_{n})) on Δ_{n}.

Let θ = ν(X) be the total charge of the measure ν, and denote by ν̄ = ν/θ the normalized measure ν. An explicit construction of the Dirichlet process is given by the following formula:

(1)  P = ∑_{i=1}^{∞} Q_{i}δ_{Y_{i}},

where Y = (Y_{1}, Y_{2}, …) is a sequence of of i.i.d. variables with common distribution ν̄, and Q = (Q_{1}, Q_{2}, …) is a random point of the infinite-dimensional simplex Δ = {x = (x_{1}, x_{2}, …) : x_{1} ≥ x_{2} ≥ … ≥ 0, ∑ x_{i} = 1} which is independent of Y and has the Poisson–Dirichlet distribution P D(θ) with parameter θ.

Many papers are devoted to a particular problem of studying the distributions of random means of Dirichlet processes. Let a : X → ℝ be a measurable function.

Typeset by AMS-TEX

The part of this work was done when the second author stayed at MSRI. The first and the second authors are partially supported by RFBR grant 99–01–00098, and the second author — by CRDF grant RM1-2244

It defines a linear functional f_{a}(ξ) = ∫_{X} a(x)dξ(x) on the space of Borel measures on X. The problem is to describe the distribution μ_{a} of this functional with respect to the Dirichlet process P. The answer is as follows. Let ν_{a} be the distribution of the function a with respect to the normalized measure ν̄. Then the measures μ_{a} and ν_{a} are related by the following integral identity:

(∗)  ∫_{ℝ} \frac{1}{(1 + zu)^{θ}} dμ_{a}(u) = exp (− ∫_{X} log(1 + zu)^{θ}dν_{a}(u)) .

This formula was first obtained in [CR] by hard analytic arguments (see also simpler proofs in [DK, KT, RGN, HO]). In case of θ = 1, this identity means that the distribution μ_{a} is the Markov–Krein transform of the measure ν_{a}. This transform arises in many contexts, such as the Markov moment problem, continued fractions theory, exponential representations of functions of negative imaginary type, the spectral shift function of a self-adjoint operator, the Plancherel growth of Young diagrams, etc. (see [Ke] for a detailed survey). It appeared first in A. A. Markov’s paper [M] and was intensively studied by M. G. Krein and his school. In particular, this identity gives a link between the so-called Markov moment problem and the well-studied Hausdorff moment problem, see [KN].

The purpose of this paper is to present a new proof of the Markov–Krein identity for means of Dirichlet processes based on a relation of Dirichlet processes to gamma processes. Let γ(x) be the gamma process on (X, ν). Then the Dirichlet process on X with parameter measure ν is just the normalized gamma process γ̄(x) = γ(x)/γ(X). Thus the above problem about the distributions of means of Dirichlet processes may be formulated in terms of gamma processes as follows: given a linear functional f_{a}(ξ) = ∫_{X} a(x)dξ(x) defined by a function a : X → ℝ_{+}, describe the distribution μ_{a} of the functional f_{a} with respect to the normalized gamma process γ̄ in terms of the distribution ν_{a} of a. The key independence property of the gamma processes (the normalized gamma process γ̄ is independent on the total charge γ(X)) which leads to many distinguished properties of the gamma processes, e.g. the so-called multiplicative quasi-invariance, see [TV,TVY2], also allows one to obtain easily the Markov–Krein formula (∗) relating these two distributions. This formula may also be interpreted as a relation between the Laplace transform of the distribution of the functional f_{a} with respect to the gamma process and the Cauchy transform of the distribution of the same functional with respect to the normalized gamma process.

Our interpretation of the Markov–Krein identity leads to the following general problem: given an arbitrary Lévy process η on (X, ν) and a function a : X → ℝ_{+}, describe the relation between the distribution of the functional f_{a} with respect to η and the distribution of the same functional with respect to the normalized Lévy process η̄.

2. General Lévy processes. It is natural to consider Dirichlet processes in the context of general Lévy processes. We present this framework following [TV, TVY2].

Let (X, ν) be a standard Borel space with a non-atomic finite non-negative measure ν, and let ν(X) = θ be the total charge of ν. Denote by

D = {∑ z_{i}δ_{x_{i}}, x_{i} ∈ X, z_{i} ∈ ℝ, ∑ |z_{i}| < ∞}

the real linear space of all finite real discrete measures on X, and by D^{+} = {∑ z_{i}δ_{x_{i}} ∈ D : z_{i} > 0} ⊂ D the cone in D consisting of all positive measures.

Let Λ be a measure on ℝ_{+} satisfying the following conditions:

(2)  \begin{matrix}Λ(0, ∞) = ∞,  Λ(1, ∞) < ∞, \\ ∫_{0}^{1} sdΛ(s) < ∞,  Λ(\{0\}) = 0. \end{matrix}

Denote by ψ_{Λ} the Laplace transform of the infinitely divisible distribution F_{Λ} with Lévy measure Λ:

ψ_{Λ}(t) = exp (− ∫_{0}^{∞}(1 − e^{−ts})dΛ(s)) .

Each bounded Borel function a : X → ℝ defines a linear functional f_{a} on D, where f_{a}(η) = ∫_{X} a(x)dη(x) for η ∈ D.

Definition 2. A Lévy process on the space (X, ν) with Lévy measure Λ satisfying (2) is a generalized process on D whose law P_{Λ} has Laplace transform

(3)  𝔼 [exp (− ∫_{X} a(x)dη(x))] = exp (∫_{X} log ψ_{Λ}(a(x))dν(x)) ,

where a is an arbitrary non-negative bounded Borel function on X.

Consider the cone C = {z = (z_{1}, z_{2}, …) : z_{1} ≥ z_{2} ≥ … ≥ 0, ∑ z_{i} < ∞} ⊂ l^{1}, and define a map T : D^{+} → C × X^{∞} by

T η = ((Q_{1}, Q_{2}, …), (X_{1}, X_{2}, …)), if η = ∑ Q_{i}δ_{X_{i}}.

Definition 3. Let P be a distribution on the space D^{+}, and let η be a random process obeying P. The random sequence of charges (Q_{1}, Q_{2}, …) is called the conic part of the process η, and its distribution on the cone C is called the conic part of the law P.

It is not difficult to show that the conic part of the Lévy process with Lévy measure Λ is the ordered sequence of points of the Poisson process on ℝ_{+} with mean measure |ν|Λ. Thus the conic part depends only on Λ and on the full charge of the parameter measure ν. In fact, the following theorem shows that studying a Lévy process may be essentially reduced to studying its conic part, since the construction of the process involves the parameter measure in a trivial way. This fundamental property of Lévy processes is a particular case of the representation theorem first proved in [FK]. A simpler proof of this fact is presented in [TVY1].

Theorem 1 ([FK, TVY1]). Let η = ∑ Q_{i}δ_{X_{i}} be a Lévy process on the space (X, ν) with Lévy measure Λ. Then T P_{Λ} = κ_{|ν|Λ} × ν̄^{∞}, i.e. X_{1}, X_{2}, … is a sequence of i.i.d. random variables with common distribution ν̄ = ν/|ν|, and this sequence is independent of the conic part (Q_{1}, Q_{2}, …).

Denote by D_{1}^{+} ⊂ D^{+} the simplex of all normalized atomic measures. Then D^{+} = D_{1}^{+} × [0, ∞), i.e. each η ∈ D^{+} can be represented as

η = (η/η(X), η(X)).

The second coordinate in this decomposition is the total charge of the measure η. It follows from the definition of the Lévy process that η(X) obeys the infinite divisible law F_{Λ} corresponding to the Lévy measure Λ. The first coordinate is called the normalization of the measure η. In general, the law of a Lévy process is not a product measure in this decomposition. It will be a product measure only for gamma processes, see below.

Using this decomposition, consider a map T′ : D^{+} → ℝ_{+} × Σ × X^{∞}, where Σ = {y = (y_{1}, y_{2}, …) : y_{1} ≥ y_{2} ≥ … ≥ 0, y_{1} + y_{2} + … = 1} is the infinitedimensional simplex, and

T′η = (η(X), (Q_{1}/η(X), Q_{2}/η(X), …), (X_{1}, X_{2}, …)), if η = ∑ Q_{i}δ_{X_{i}}.

Definition 4. The normalized sequence of charges (Q_{1}/η(X), Q_{2}/η(X), …) is called the simplicial part of the process and its distribution on Σ is called the simplicial part of the law P_{Λ}.

See [TVY1] for a detailed treatment of general Lévy prosesses, characterization of conic parts, etc.

3. The gamma process. In this section we summarize basic properties of the gamma processes which we need to prove the Markov–Krein identity (∗).

Definition 5. The standard gamma process on the space (X, ν) is a Lévy process γ on (X, ν) with Lévy measure dΛ(z) = z^{−1}dz, z > 0. Thus the law G of the gamma process has Laplace transform

(4)  𝔼_{G} [exp (− ∫_{X} a(x)dγ(x))] = exp (− ∫_{X} log (1 + a(x)) dν(x)) .

It is easy to show that formula (4) holds for any measurable function a ∈ M = {a : X → ℝ_{+} : ∫_{X} log(a(x) + 1)dν(x) < ∞}.

Lemma 1 presents the key independence property of the gamma process. It follows immediately from the corresponding property of gamma variables: if Y and Z are independent gamma variables with the same scale parameter, then the variables Y + Z and \frac{Y}{Y +Z} are independent.

Lemma 1. The total charge γ(X) of the gamma process and the normalized gamma process γ̄ = γ/γ(X) are independent. The distribution of the total charge is the gamma distribution with shape parameter θ = |ν|.

As follows from a remarkable result of Lukacs [L] for gamma variables, this property is characteristic of the gamma processes. That is, if η is a Lévy process such that η̄ and η(X) are independent, then η is a gamma process (maybe with some scale parameter).

Lemma 2 ([Ki]). The simplicial part of the gamma process with |ν| = θ is the Poisson–Dirichlet measure P D(θ).

It is easy to see from (3) that for every measurable subset A ⊂ X, the random variable γ(A) is governed by the gamma distribution with shape parameter ν(A), and for every measurable partition A_{0}, …, A_{n} of X, the variables γ(A_{0}), … γ(A_{n}) are independent. Then it follows from the known property of the Dirichlet distributions that the vector (γ(A_{0})/γ(X), …, γ(A_{n})/γ(X)) has Dirichlet distribution on the simplex with parameters (ν(A_{0}), …, ν(A_{n})). Thus the Dirichlet process on the space X with parameter measure ν is just the normalized gamma process on D(X, ν). One can also establish this relation in terms of the explicit construction (1). Indeed, in view of Theorem 1 and Lemmas 1 and 2, the right-hand side of (1) presents the normalized gamma process on (X, ν).

4. The Markov–Krein identity for means of Dirichlet processes. In this section we prove the Markov–Krein identity for means of Dirichlet processes which may be interpreted as a formula relating the distribution of a linear functional with respect to the gamma process and the distribution of the same functional with respect to the normalized gamma process. This interpretation allows one to prove it immediately, using only a formula for the Laplace transform of the gamma process and the independence property of the gamma processes.

Given a function a ∈ M, denote by μ_{a} the distribution of the linear functional η ↦ f_{a}(η) = ∫_{X} a(x)dη(x) on D with respect to the law D of the Dirichlet process on (X, ν) (i.e., the law of the normalized gamma process), and let ν_{a} be the distribution of the function a with respect to the normalized parameter measure ν̄.

Theorem 2. The measures μ_{a} and ν_{a} are related by the following integral identity,

(5)  ∫_{ℝ} \frac{1}{(1 + zu)^{θ}} dμ_{a}(u) = exp (− ∫_{X} log(1 + zu)^{θ}dν_{a}(u)) .

Note that the left-hand side of (5) is the generalized Cauchy–Stieltjes transform of the distribution μ_{a}. It is natural to call the right-hand side the multiplicative version of the generalized Cauchy–Stieltjes transform of the distribution ν_{a}. In view of (4), it is equal to the Laplace transform of the gamma process calculated on the function a. Hence one may regard formula (5) as relating an integral transform (Cauchy–Stieltjes) of the distribution μ_{a} of the functional f_{a} with respect to the normalized gamma process and an integral transform (Laplace) of its distribution with respect to the non-normalized gamma process.

Proof. Using (4), Lemma 1 and the Fubini theorem we obtain that the right-hand side of (5) equals

exp (− ∫_{X} log(1 + za(x))dν(x)) = 𝔼_{G} [exp (−z ∫_{X} a(x)dγ(x))]
=𝔼_{G} [exp (−zγ(X) ∫_{X} a(x)dγ̄(x))]
=𝔼_{D} [\frac{1}{Γ(θ)} ∫_{0}^{∞} t^{θ−1} exp (−t − zt ∫_{X} a(x)dγ̄(x))]
=𝔼_{D} [\frac{1}{(1 + z ∫_{X} a(x)dγ̄(x))^{θ}} ] ,

and Theorem follows.

According to a personal communication of P. Diaconis to S. Kerov, the idea of proving formula (6) using the Laplace transform formula for the gamma process was used by F. Huffer in case of discrete parameter measure ν (in this case the gamma process is just a sum of independent gamma variables, and the normalized gamma process is a random point of a finite-dimensional simplex obeying a Dirichlet distribution). But the fact that this argument works as well for continuous parameter measures seems to have been overlooked. But after the preliminary version [TVY1] of this paper had been published, other independent close proofs of the identity (5) appeared, see [RGN, HO].

A multivariate version of the Markov–Krein identity for the common distributions of several linear functionals of the Dirichlet process was first obtained in [KT]. It is easy to extend our proof of Theorem 2 to obtain this result.

Theorem 3. Let a_{1}, …, a_{n} ∈ M. Denote by μ_{a} the common distribution of the linear functionals (f_{a_{1}}, …, f_{a_{n}}) on D with respect to the law D of the Dirichlet process. Let ν_{a} be the common distribution of the functions a_{1}, …, a_{n} with respect to the normalized parameter measure ν̄. Then the measures μ_{a} and ν_{a} are related by the multivariate Markov–Krein identity

(6) ∫_{ℝ^{n}} \frac{1}{(1 + z_{1}u_{1} + … + z_{n}u_{n})^{θ}} dμ_{a}(u)

= exp (− ∫_{ℝ^{n}} log(1 + z_{1}u_{1} + … + z_{n}u_{n})^{θ}dν_{a}(u)) .

Proof. Goes exactly as the proof of Theorem 2 with function za(x) replaced by z_{1}a_{1}(x) + … + z_{n}a_{n}(x).

5. Two-parameter generalization of the Markov–Krein identity. S. Kerov [Ke] and J. Pitman [P] suggested independently the same class of generalizations of the Dirichlet processes. These generalizations are indexed by an arbitrary distribution σ on the simplex Δ, and they are defined by (1), where the sequence Q obeys σ instead of P D(θ). An important particular case of these generalized Dirichlet processes is obtained when σ is the so-called two-parameter Poisson–Dirichlet distribution P D(α, θ) [PY]. The range of admissible parameters is the union of the sets {(α, θ) : α ∈ (0, 1), θ > −α} and {(α, −mα) : α < 0, m ∈ ℕ}. When α = 0, the measure P D(0, θ) coincides with the ordinary Poisson–Dirichlet distribution P D(θ). Denote by D(α, θ) the law of the generalized Dirichlet process associated with the two-parameter Poisson–Dirichlet distribution P D(α, θ).

An analogue of the Markov–Krein identity for the distribution of a linear functional with respect to D(α, θ) is obtained in [T]. We present here a new proof of this identity based on relation of the two-parameter Poisson–Dirichlet distributions to the stable processes.

Definition 6. Let α ∈ (0, 1). The standard α-stable process on the space (X, ν) is a Lévy process with Lévy measure dΛ_{α} = \frac{α}{Γ(1−α)}s^{−α−1}ds, s > 0. Thus the law P_{α} of the α-stable process has Laplace transform

(7)  𝔼_{P_{α}} [exp (− ∫_{X} a(x)dη(x))] = exp (− ∫_{X} a(x)^{α}dν(x)) ,

where a : X → ℝ_{+} is an arbitrary measurable function with ∫_{X} a(x)^{α}dν(x) < ∞.

Lemma 3 [PY]. The simplicial part of the α-stable process is the Poisson–Dirichlet distribution P D(α, 0).

The Poisson–Dirichlet distribution P D(α, θ) with α, θ ≠ 0 cannot be represented as the simplicial part of any Lévy process. However, one may obtain it as the simplicial part of the process that has density with respect to a stable process. Namely, let θ > −α and consider the law P_{α,θ} on D which has density

(8)  \frac{dP_{α,θ}}{dP_{α}} (η) = \frac{c_{α,θ}}{η(X)^{θ}}

with respect to the α-stable law P_{α}. Here c_{α,θ} = \frac{Γ(θ+1)}{Γ(θ/α+1)} is a normalizing constant.

Lemma 4 [PY]. The simplicial part of the measure P_{α,θ} is the Poisson–Dirichlet distribution P D(α, θ).

It follows from Lemmas 3 and 4 that the generalized Dirichlet process with parameters (α, 0) is just the normalized α-stable process, and the generalized Dirichlet process with parameters (α, θ) is the normalized processed governed by the law P_{α,θ}.

Given an arbitrary measurable function a : X → ℝ_{+} with ∫_{X} a(x)^{α}dν(x) < ∞, let μ_{a} be the distribution of the functional f_{a} with respect to D(α, θ), and let ν_{a} be the distribution of a with respect to the normalized parameter measure ν̄.

Theorem 4. The measures μ_{a} and ν_{a} are related by the following integral identity:

1) if θ ≠ 0,

(9)  (∫_{ℝ}(1 + zu)^{−θ}dμ_{a}(u))^{− \frac{1}{θ}} = (∫_{ℝ}(1 + zu)^{α}dν_{a}(u))^{\frac{1}{α}} ;

2) if θ = 0,

(10)  exp (∫_{ℝ} log(1 + zu)^{α}dμ_{a}(u)) = ∫_{ℝ}(1 + zu)^{α}dν_{a}(u).

Proof. 1) Denote the left-hand side of the desired identity by A^{−1/θ} and the righthand side by B^{1/α}. Using the identity

\frac{1}{r^{θ}} = \frac{1}{Γ(θ)} ∫_{0}^{∞} t^{θ−1}e^{−rt}dt,

we obtain

A =c_{α,θ}𝔼^{α} [(η(X) + z ∫_{X} a(x)dη(x))^{−θ}]

=\frac{c_{α,θ}}{Γ(θ)}𝔼^{α} [∫_{0}^{∞} t^{θ−1} exp (−t (η(X) + z ∫_{X} a(x)dη(x))) dt]

=\frac{c_{α,θ}}{Γ(θ)} ∫_{0}^{∞} t^{θ−1}𝔼^{α} [exp (− ∫_{X} t(1 + za(x))dη(x))] dt.

By the Laplace transform formula (7), the expectation equals precisely e^{−t^{α}B}, thus

A = \frac{c_{α,θ}}{Γ(θ)} ∫_{0}^{∞} t^{θ−1}e^{−t^{α}B}dt,

and (9) follows by changing variables.

2) Follows from (9) by letting θ → 0.

References

[CR] D. M. Cifarelli, E. Regazzini, Some remarks on the distribution functions of means of a Dirichlet process, Ann. Statist. 18 (1990), 429–442.

[DK] P. Diaconis, J. Kemperman, Some New Tools for Dirichlet Priors, Bayesian Statistics 5 (J. M. Bernardino, J. O. Berger, A. P. Dawid and A. F. M. Smith eds.), Oxford University Press, 1996, pp. 97–106.

[F] T. S. Ferguson, A Bayesian analysis of some nonparametric problems, Ann. Statist. 1 (1973), 209–230.

[FK] T. S. Ferguson, M. J. Klass,, A representation of independent increment processes without Gaussian components, Ann. Math. Statist. 43 (1972), 1634–1643.

[HO] N. L. Hjort, A. Ongaro, On the distribution of random Dirichlet means, Statistical Research Report, University of Oslo (2001).

[Ke] S. V. Kerov, Interlacing Measures, Amer. Math. Soc. Transl. (2) 181 (1998), 35–83.

[KT] S. V. Kerov, N. V. Tsilevich, The Markov–Krein correspondence in several dimensions, PDMI Preprint 1 (1998).

[Ki] J. F. C. Kingman, Random discrete distributions, J. Roy. Statist. Soc. B 37 (1975), 1–22.

[KN] M. G. Krein, A. A. Nudelman, The Markov moment problem and extremal problems, Nauka, Moscow, 1973; English translation: Translations of Mathematical Monographs, vol. 50, AMS, Providence, RI, 1978.

[L] E. Lukacs, A characterization of the gamma distribution, Annals of Math. Stat. 26 (1965), 319–324.

[M] A. A. Markov, Nouvelles applications des fractions continues, Math. Ann. 47 (1896), 579–597.

[P] J. Pitman, Some developments of the Blackwell–MacQueen urn scheme, Statistics, Probability and Game theory (T. S. Ferguson, L. S. Shapley, J. B. MacQueen eds.), IMS Lecture Notes – Monograph series, vol. 30, 1996, pp. 245–267.

[PY] J. Pitman, M. Yor, The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator, Ann. Prob. 25 (1997), 855–900.

[RGN] E. Regazzini, A. Guglielmi, G. di Nunno, Theory and numerical analysis for exact distributions of functionals of a Dirichlet process, IAMI preprint (2000), no. 00.12.

[T] N. V. Tsilevich, Distributions of mean values for some random measures, Zapiski nauchn. seminarov POMI 240 (1997), 268–279; English translation in J. Math. Sciences 96 (1999), no. 5, 3616–3623.

[TV] N. V. Tsilevich, A. M. Vershik, Quasi-invariance of the gamma process and multiplicative properties of the Poisson–Dirichlet measures, C. R. Acad. Sci. Paris, Série I 329 (1999), no. 2, 163–168.

[TVY1] N. V. Tsilevich, A. M. Vershik, M.Yor, Distinguished properties of the gamma process and related topics, Prépublications du Laboratoire de Probabilités et Modèles aléatoires (2000), no. 575.

[TVY2] N. V. Tsilevich, A. M. Vershik, M.Yor, An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process, to appear in J. Funct. Anal.

N. Tsilevich, Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia

E-mail address: natalia@pdmi.ras.ru

A. Vershik, Steklov Mathematical Institute at St. Petersburg, 27 Fontanka, 191011 St. Petersburg, Russia

E-mail address: vershik@pdmi.ras.ru

M. Yor, Laboratoire de Probabilités et Modèles Aléatoires, Tour 56, 4 Place Jussieu, 75252 Paris Cedex 05, France