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April 2004 Means of a Dirichlet process and multiple hypergeometric functions
Antonio Lijoi, Eugenio Regazzini
Ann. Probab. 32(2): 1469-1495 (April 2004). DOI: 10.1214/009117904000000270

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.

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Antonio Lijoi. Eugenio Regazzini. "Means of a Dirichlet process and multiple hypergeometric functions." Ann. Probab. 32 (2) 1469 - 1495, April 2004. https://doi.org/10.1214/009117904000000270

Information

Published: April 2004
First available in Project Euclid: 18 May 2004

zbMATH: 1061.60078
MathSciNet: MR2060305
Digital Object Identifier: 10.1214/009117904000000270

Subjects:
Primary: 60E05
Secondary: 33C65 , 62E10

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

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 2 • April 2004
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