Distribution of the main value for certain random measures (Q1966357)

This paper studies a generalized Dirichlet point process \({\mathcal D}(\alpha,\theta,\tau)\) on \([0,1]\), which can be constructed as a random probability measure \(F=\sum_{i=1}^\infty P_i\delta_{X_i}\), where \(X_i\) are i.i.d. sequences on \([0,1]\) with common distribution \(\tau\), and the random weights \((P_1,P_2,\dots)\) have a two-parameter Poisson-Dirichlet distribution \(PD(\alpha,\theta)\). The main result of the paper is the following identity for the distribution \(\mu\) of the random mean \(\int x F(dx)\): If \(\alpha,\theta\neq 0\), then \[ \bigg(\int(z-u)^{-\theta} \mu(du)\bigg)^{-1/\theta}= \bigg(\int(z-u)^{\alpha} \mu(du)\bigg)^{1/\alpha} ,\qquad z\in{\mathbb C}\backslash{\mathbb R} . \] Corresponding identities are also given for the limiting cases \(\alpha=0\) or \(\theta=0\). In the case \(\alpha=0\) and \(\theta=1\), the result was obtained by \textit{P. Diaconis} and \textit{J. Kemperman} [in: Bayesian Statistics 5, 97-106 (1996)].