发表机构
University of Pavia; University of Torino; Collegio Carlo Alberto（帕维亚大学; 都灵大学; 卡洛·阿尔贝托学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对推广了Dirichlet过程的Pitman--Yor过程的方差，建立了对应的Cifarelli--Regazzini恒等式并推导其密度显式公式，还可从该结果导出Dirichlet方差的对应极限结果。
AI 中文摘要
Dirichlet过程的著名Cifarelli--Regazzini恒等式及其解析反演是随机概率测度线性泛函的优美分布理论的基础，尤其为Dirichlet均值的密度函数提供了显式公式。相比之下，Dirichlet过程的非线性泛函在文献中仍鲜为人知。本文针对推广了Dirichlet过程的Pitman--Yor过程的方差，开发了一种超越线性泛函的变换-反演策略。我们建立了Pitman--Yor方差的Cifarelli--Regazzini恒等式，其直接解析反演得到了该方差密度函数的显式公式，而Dirichlet方差的对应结果可作为极限情形被推导出来。
英文摘要
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.
CommentsThe generalized Cauchy--Stieltjes transform identities have been simplified, the inversion formula has been corrected, and numerical experiments have been added
