The two - parameter Poisson - Dirichlet point process

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 (i.e., the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Relying on this, we apply the theory of point processes to reveal mathematical structure of the two-parameter Poisson-Dirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we will be able to extend several results previously known for the one-parameter case, and the Markov-Krein identity for the generalized Dirichlet process is discussed from a point of view of functional analysis based on the twoparameter Poisson-Dirichlet distribution.