Means of a Dirichlet Process and Multiple Hypergeometric Functions

• Published in: The Annals of probability 2004-04, Vol.32 (2), p.1469-1495
• Main Authors: Lijoi, Antonio, Regazzini, Eugenio
• Format: Article
• Language: English
• Subjects: 33C65; 60E05; 62E10; Dirichlet problem; distribution of means of a random probability measure; Distribution theory; Eigenfunctions; Exact sciences and technology; Functional Dirichlet probability distribution; generalized gamma convolutions; Generating function; Hypergeometric functions; Lauricella functions; Lebesgue measures; Markov analysis; Markov–Krein identity; Mathematical analysis; Mathematical independent variables; Mathematics; Probabilities; Probability; Probability and statistics; Probability theory and stochastic processes; Random variables; Sciences and techniques of general use; Special functions; Statistical variance; Statistics; Studies
• ISSN: 0091-1798; 2168-894X
• DOI: 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.