Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2510.22066 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917041998397440 |
|---|---|
| author | Basrak, Bojan |
| author_facet | Basrak, Bojan |
| contents | Based on their earlier studies of the arcsine law, Pitman and Yor in \cite{PY97} constructed a widely adopted PD($α, θ)$ family of random mass-partitions with parameters $α\in [0,1),\ θ+α>0$. We propose an alternative model based on generalized perpetuities, which extends the PD family in a continuous manner, incorporating any $α\geq 0$. 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 $α$-stable subordinators for $α\in (0,1)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22066 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On generalized arcsine laws and residual allocation models Basrak, Bojan Probability 60G55 Based on their earlier studies of the arcsine law, Pitman and Yor in \cite{PY97} constructed a widely adopted PD($α, θ)$ family of random mass-partitions with parameters $α\in [0,1),\ θ+α>0$. We propose an alternative model based on generalized perpetuities, which extends the PD family in a continuous manner, incorporating any $α\geq 0$. 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 $α$-stable subordinators for $α\in (0,1)$. 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. |
| title | On generalized arcsine laws and residual allocation models |
| topic | Probability 60G55 |
| url | https://arxiv.org/abs/2510.22066 |