The two-parameter Poisson--Dirichlet point process

Details The two-parameter Poisson--Dirichlet point process The two-parameter Poisson--Dirichlet point process

2007-05-24

arxiv.org/abs/0705.3496v3

Bernoulli 2009, Vol. 15, No. 4, 1082-1116

Mathematics

Probability

Published in at http://dx.doi.org/10.3150/08-BEJ180 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/08-BEJ180

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 two-parameter Poisson--Dirichlet distribution.

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