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On generalized arcsine laws and residual allocation models

Bojan Basrak

TL;DR

The paper develops a unifying beta/gamma perpetuity framework that extends the Poisson-Dirichlet family to all $\alpha\ge0$ and yields stick-breaking representations for a broad class of residual allocation models (RAM). It shows that all PD distributions arise as beta/gamma perpetuities and provides constructive proofs of Pitman-Yor’s generalized arcsine law for partitions generated by $\alpha$-stable subordinators, with important consequences for excursion theory and Brownian/Bessel processes. The authors introduce RAM($\alpha, a_1, c$) as a three-parameter extension that naturally integrates $\alpha$ into the RAM framework, and they develop thinning and size-bias techniques to derive new representations and corollaries, including connections to Dickman’s subordinator. The results yield new arcsine-type laws for excursions and offer versatile probabilistic tools for studying mass-partitions, subordinators, and related stochastic processes, with potential impact on excursion theory and Bayesian nonparametrics.

Abstract

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.

On generalized arcsine laws and residual allocation models

TL;DR

The paper develops a unifying beta/gamma perpetuity framework that extends the Poisson-Dirichlet family to all and yields stick-breaking representations for a broad class of residual allocation models (RAM). It shows that all PD distributions arise as beta/gamma perpetuities and provides constructive proofs of Pitman-Yor’s generalized arcsine law for partitions generated by -stable subordinators, with important consequences for excursion theory and Brownian/Bessel processes. The authors introduce RAM() as a three-parameter extension that naturally integrates into the RAM framework, and they develop thinning and size-bias techniques to derive new representations and corollaries, including connections to Dickman’s subordinator. The results yield new arcsine-type laws for excursions and offer versatile probabilistic tools for studying mass-partitions, subordinators, and related stochastic processes, with potential impact on excursion theory and Bayesian nonparametrics.

Abstract

Based on their earlier studies of the arcsine law, Pitman and Yor in \cite{PY97} constructed a widely adopted PD( family of random mass-partitions with parameters . We propose an alternative model based on generalized perpetuities, which extends the PD family in a continuous manner, incorporating any . 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 -stable subordinators for . 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 -dimensional Bessel process for , and Brownian motion in particular.
Paper Structure (24 sections, 17 theorems, 105 equations, 1 figure)

This paper contains 24 sections, 17 theorems, 105 equations, 1 figure.

Table of Contents

  1. Keywords
  2. Introduction
  3. Self normalized point processes and subordinators.
  4. Stick-breaking distributions.
  5. Generalized beta/gamma perpetuities.
  6. Outline and main results.
  7. Stick-breaking theorem
  8. Size-biasing
  9. Standard Poisson-Dirichlet distribution.
  10. Standard Pitman-Yor distribution.
  11. Poisson-Dirichlet distributions
  12. Residual allocation model
  13. Arcsine laws
  14. Mass-partitions generated by stable subordinators
  15. Thinning
  16. ...and 9 more sections

Key Result

Proposition 2.2

Each PD($\alpha,\theta$) distribution belongs to the beta/gamma family of distributions with independent random variables $G_n \sim \textit{gamma}(1-\alpha,1)$ and $U_n \sim \textit{beta}(a_n,1)$, where $a_n=\theta+ n\alpha$, $n \geq 1$.

Figures (1)

  • Figure 1: An illustration of the points of $N^\vee$ and $N^\wedge$ in the proof of \ref{['thm:1']} (in blue). By other circles (in white and red) we denote corresponding exponential marks for the points above a certain threshold. Note that for points of $N^\wedge$, their exponential mark in red falls below their respective blue point. Random variable $\Gamma$ in \ref{['eq:Gamma']} corresponds to the total length of vertical blue lines.

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Corollary 2.5: McCloskey, Perman, Pitman, Yor
  • Definition 2.6
  • Lemma 3.1
  • ...and 26 more