On generalized arcsine laws and residual allocation models

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.