Means of a Dirichlet process and multiple hypergeometric functions
Antonio Lijoi and Eugenio Regazzini
Source: Ann. Probab. Volume 32, Number 2 (2004), 1469-1495.
Abstract
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.
Primary Subjects: 60E05
Secondary Subjects: 62E10, 33C65
Keywords: Functional Dirichlet probability distribution; distribution of means of a random probability measure; generalized gamma convolutions; Lauricella functions; Markov–Krein identity
Full-text: Access granted (open access)
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1084884858
Digital Object Identifier: doi:10.1214/009117904000000270
Mathematical Reviews number (MathSciNet):
MR2060305
Zentralblatt MATH identifier:
02100699
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