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

@misc{dolera2026variance,
  title = {{The variance of the Pitman--Yor process: Cifarelli--Regazzini-type identities and inversion formulas}},
  author = {Emanuele Dolera and Stefano Favaro},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.10215},
  eprint = {2608.10215},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.10215},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - The variance of the Pitman--Yor process: Cifarelli--Regazzini-type identities and inversion formulas
AU  - Emanuele Dolera
AU  - Stefano Favaro
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2608.10215
UR  - https://arxiv.org/abs/2608.10215
AB  - The Cifarelli--Regazzini identity lies at the foundation of the theory of the Dirichlet process and has played a central role in Bayesian nonparametric statistics. By providing the generalized Cauchy--Stieltjes transform of linear functionals of the Dirichlet process, together with an analytic inversion, it yields 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 less understood, with results largely confined to a few examples for 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. We then analytically invert this transform under general conditions on the base probability measure, thus obtaining explicit representations for both the distribution function and density. Corresponding formulas are also obtained for the Dirichlet variance, yielding more explicit representations than those currently available in the literature. We also establish a distributional representation of the Pitman--Yor variance in terms of a Dirichlet process whose base probability measure is the joint distribution of the mean and variance of a suitably specified Pitman--Yor process, mirroring the classical within--between decomposition of variance and providing a quadratic counterpart to a known distributional identity for Pitman--Yor means.
ER  -