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1996 Random Discrete Distributions Derived from Self-Similar Random Sets
Jim Pitman, Marc Yor
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Electron. J. Probab. 1: 1-28 (1996). DOI: 10.1214/EJP.v1-4


A model is proposed for a decreasing sequence of random variables $(V_1, V_2, \cdots)$ with $\sum_n V_n = 1$, which generalizes the Poisson-Dirichlet distribution and the distribution of ranked lengths of excursions of a Brownian motion or recurrent Bessel process. Let $V_n$ be the length of the $n$th longest component interval of $[0,1]\backslash Z$, where $Z$ is an a.s. non-empty random closed of $(0,\infty)$ of Lebesgue measure $0$, and $Z$ is self-similar, i.e. $cZ$ has the same distribution as $Z$ for every $c > 0$. Then for $0 \le a < b \le 1$ the expected number of $n$'s such that $V_n \in (a,b)$ equals $\int_a^b v^{-1} F(dv)$ where the structural distribution $F$ is identical to the distribution of $1 - \sup ( Z \cap [0,1] )$. Then $F(dv) = f(v)dv$ where $(1-v) f(v)$ is a decreasing function of $v$, and every such probability distribution $F$ on $[0,1]$ can arise from this construction.


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Jim Pitman. Marc Yor. "Random Discrete Distributions Derived from Self-Similar Random Sets." Electron. J. Probab. 1 1 - 28, 1996.


Accepted: 20 February 1996; Published: 1996
First available in Project Euclid: 25 January 2016

zbMATH: 0891.60042
MathSciNet: MR1386296
Digital Object Identifier: 10.1214/EJP.v1-4

Primary: 60G18
Secondary: 60G57 , 60K05

Keywords: excursion lengths , interval partition , Regenerative set , structural distribution , Zero set

Vol.1 • 1996
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