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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Tuesday, August 7, 2001}
\centerline{Sequoia Hall Rm. 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Eugenio Regazzini}
\centerline{\sl Universit\`a degli Studi di Pavia, Italia}
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\centerline{\bf Functionals of Dirichlet Processes and Multiple 
Hypergeometric Functions}

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Given a Dirichlet process $D$ -- with parameter $\alpha$ -- some
issues, related to the probability distribution (p.d.) $\mu_\alpha$ of
the random integral
$\widetilde d = \int_R x D (dx),$

\noindent are discussed. In particular, the Lauricella theory of
multiple hypergeometric functions (1893) is taken as a starting point
to establish -- in a very simple way -- a relation between a transform
of the Stieltjes type of $\mu_\alpha$ and the characteristic function
of $\widetilde g = \int_{\IR} x G (dx)$, $G$ being a gamma process
with parameter $\alpha$. Such a relation represents a substantial
extension of an identity, given by Cifarelli and Regassini more than
twenty years ago, recently referred to as Markov-Krein identity. Here,
this relation is used to state a one-to-one correspondence between
$\alpha$ and $\mu_\alpha$. Subsequently it is shown that it allows a
plain determination of explicit formulas for $\mu_\alpha$. Moreover,
after observing that the p.d. of $\widetilde g$ is infinitely
divisible, the corresponding L\'evy measure is used to verify absolute
continuity of $\mu_\alpha$. Finally, an explicit formula for the
characteristic function of $\widetilde d$ is provided, connected with
a confluent form of the Lauricella fourth function. This formula is
applied to show that $\mu_\alpha$ is symmetric if and only if $\alpha$
is symmetric.

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