On generalized arcsine laws and residual allocation models

TL;DR

This paper introduces a new model based on generalized perpetuities that extends the PD family of random partitions, providing novel proofs and insights into generalized arcsine laws and their applications to stable subordinators and Bessel processes.

Contribution

It proposes an alternative, continuous extension of the PD family using generalized perpetuities, offering new proofs and broader applicability of arcsine laws.

Findings

New continuous extension of PD distributions via generalized perpetuities

Concise proof of stick-breaking representations of PD distributions

General arcsine laws for excursions of Bessel processes and Brownian motion

Abstract

Based on their earlier studies of the arcsine law, Pitman and Yor in \cite{PY97} constructed a widely adopted PD($\alpha, \theta)$ family of random mass-partitions with parameters $\alpha \in [0,1),\ \theta+\alpha>0$. We propose an alternative model based on generalized perpetuities, which extends the PD family in a continuous manner, incorporating any $\alpha\geq 0$. This perspective yields a new, concise proof for the stick-breaking (or residual allocation) representations of PD distributions, recovering the classical results of McCloskey and Perman in particular. We apply this framework to provide a constructive and intuitive proof of Pitman and Yor's generalized arcsine law concerning the partitions arising from $\alpha$-stable subordinators for $\alpha \in (0,1)$. The result shows that the random partitions generated by stable subordinators have identical distributions when observed over temporal or spatial intervals. This theorem has a number of significant implications for excursion theory. As a corollary, using purely probabilistic arguments, we obtain general arcsine laws for excursions of $d$-dimensional Bessel process for $0<d<2$, and Brownian motion in particular.