The variance of the Pitman--Yor process: Cifarelli--Regazzini-type identities and inversion formulas

Abstract

The Cifarelli--Regazzini identity lies at the foundation of the distributional theory of the Dirichlet process and has played a central role in Bayesian nonparametrics. By providing the generalized Cauchy--Stieltjes transform of linear functionals of the Dirichlet process, together with an analytic inversion, it yields, in particular, explicit representations for the distribution function and density of the Dirichlet mean. This transform identity was subsequently extended to the Pitman--Yor process, a generalization of the Dirichlet process, leading to an analogous distributional theory for its linear functionals. Nonlinear functionals, by contrast, remain substantially less understood, with the available distributional literature limited to a small number of specific examples and confined to the Dirichlet process. In this paper, we move beyond linear functionals by considering the variance of the Pitman--Yor process, a genuinely quadratic functional. In particular, we establish a Cifarelli--Regazzini-type identity providing the generalized Cauchy--Stieltjes transform of the Pitman--Yor variance for general base probability measures under suitable integrability assumptions. For the Uniform base probability measure on $[0,1]$, analytic inversion yields explicit representations for both the distribution function and density. Corresponding formulas are also obtained for the Dirichlet variance, which provide more explicit representations than those currently available in the literature.