The variance of the Pitman--Yor process: a Cifarelli--Regazzini identity and inversion formula

Abstract

The celebrated Cifarelli--Regazzini identity for the Dirichlet process and its analytic inversion lie at the foundation of an elegant distributional theory for linear functionals of random probability measures, providing, in particular, an explicit formula for the density function of the Dirichlet mean. Nonlinear functionals of the Dirichlet process, by contrast, remain substantially less understood in the literature. In this paper, we develop a transform-and-inversion strategy beyond linear functionals by considering the variance of the Pitman--Yor process, which generalizes the Dirichlet process. We establish a Cifarelli--Regazzini identity for the Pitman--Yor variance, whose direct analytic inversion yields an explicit formula for its density function. The corresponding results for the Dirichlet variance are recovered as a limiting case.