The two-parameter Poisson-Dirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (that is, the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Using this, we apply the theory of point processes to reveal the mathematical structure of the two-parameter Poisson-Dirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we are able to extend several results previously known for the one-parameter case. The Markov-Krein identity for the generalized Dirichlet process is discussed from the point of view of functional analysis based on the twoparameter Poisson-Dirichlet distribution. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2009, Vol. 15, No. 4, 1082-1116. This reprint differs from the original in pagination and typographic detail.
1350-7265 c 2009 ISI/BSThe two-parameter Poisson-Dirichlet point processbe the decreasing order statistics of ( V i ), namely V 1 ≥ V 2 ≥ · · · are the ranked values of ( V i ), and define PD(α, θ) to be the law of (V i ) 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 (V i ) governed by PD(α, θ) is studied through the random point process ξ := δ Vi , 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 i ). For example, we have ξ([t, ∞)) = 0 precisely when V 1 < t. More generally, for each n = 2, 3, . . . , quantitative information on V 1 , . . . , V n can be derived from ξ by the principle of inclusion-exclusion. Among the many ways of characterizing a point process, we choose the one prescribed in terms of correlation functions. For each positive integer n, the nth correlation function of a point process is informally defined as the mean density of tuples of n distinct points in the process. As far as the one-parameter family ...