The Annals of Probability

Means of a Dirichlet process and multiple hypergeometric functions

Antonio Lijoi and Eugenio Regazzini

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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.

Article information

Source
Ann. Probab., Volume 32, Number 2 (2004), 1469-1495.

Dates
First available in Project Euclid: 18 May 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1084884858

Digital Object Identifier
doi:10.1214/009117904000000270

Mathematical Reviews number (MathSciNet)
MR2060305

Zentralblatt MATH identifier
1061.60078

Subjects
Primary: 60E05: Distributions: general theory
Secondary: 62E10: Characterization and structure theory 33C65: Appell, Horn and Lauricella functions

Keywords
Functional Dirichlet probability distribution distribution of means of a random probability measure generalized gamma convolutions Lauricella functions Markov–Krein identity

Citation

Lijoi, Antonio; Regazzini, Eugenio. Means of a Dirichlet process and multiple hypergeometric functions. Ann. Probab. 32 (2004), no. 2, 1469--1495. doi:10.1214/009117904000000270. https://projecteuclid.org/euclid.aop/1084884858


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