@article{EJP4,
author = {Jim Pitman and Marc Yor},
title = {Random Discrete Distributions Derived from Self-Similar Random Sets},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {1},
year = {1996},
keywords = {interval partition, zero set, excursion lengths, regenerative set, structural distribution.},
abstract = {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.},
pages = {no. 4, 1-28},
issn = {1083-6489},
doi = {10.1214/EJP.v1-4},
url = {http://ejp.ejpecp.org/article/view/4}}