Electronic Journal of Probability

Random Discrete Distributions Derived from Self-Similar Random Sets

Jim Pitman and Marc Yor

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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.

Article information

Electron. J. Probab., Volume 1 (1996), paper no. 4, 28 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G18: Self-similar processes
Secondary: 60G57: Random measures 60K05: Renewal theory

interval partition zero set excursion lengths regenerative set structural distribution

This work is licensed under aCreative Commons Attribution 3.0 License.


Pitman, Jim; Yor, Marc. Random Discrete Distributions Derived from Self-Similar Random Sets. Electron. J. Probab. 1 (1996), paper no. 4, 28 pp. doi:10.1214/EJP.v1-4. https://projecteuclid.org/euclid.ejp/1453756467

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