Mathematics – Probability
Scientific paper
2004-10-06
Annals of Probability 2004, Vol. 32, No. 2, 1469-1495
Mathematics
Probability
Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imsta
Scientific paper
10.1214/009117904000000270
The Lauricella theory of multiple hypergeometric functions is used to shed some light on certain distributional properties of the mean of a Dirichlet process. This approach leads to several results, which are illustrated here. Among these are a new and more direct procedure for determining the exact form of the distribution of the mean, a correspondence between the distribution of the mean and the parameter of a Dirichlet process, a characterization of the family of Cauchy distributions as the set of the fixed points of this correspondence, and an extension of the Markov-Krein identity. Moreover, an expression of the characteristic function of the mean of a Dirichlet process is obtained by resorting to an integral representation of a confluent form of the fourth Lauricella function. This expression is then employed to prove that the distribution of the mean of a Dirichlet process is symmetric if and only if the parameter of the process is symmetric, and to provide a new expression of the moment generating function of the variance of a Dirichlet process.
Lijoi Antonio
Regazzini Eugenio
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