A simple procedure calculating the generalized Stieltjes transform of the mean of a Dirichlet process
Let \(P_\alpha\) denote a Dirichlet process with parameter \(\alpha\) on the real line; i.e., \(P_\alpha\) is a random measure on \(\mathbb{R}\) such that, for any partition \(A_1,\dots, A_n\) of \(\mathbb{R}\), the vector \(P_\alpha(A_1),\dots, P_\alpha(A_n)\) has a Dirichlet distribution with parameters \(\alpha(A_1),\dots,\alpha(A_n)\). The distribution of the mean \(\Gamma_\alpha=\int x P_\alpha(dx)\) of \(P_\alpha\) was obtained by \textit{D. M. Cifarelli} and \textit{E. Regazzini} [Ann. Stat. 18, No. 1, 429-442 (1990; Zbl 0706.62012)] through calculating the generalized Stieltjes transform of \(\Gamma_\alpha\). The short note under review now suggests a simplified calculation based on an integral equation for the Laplace transform of \(\Gamma_\alpha\).
scientific article; zbMATH DE number 1202803