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 Electronic Journal of Probability > Vol. 1 (1996) > Paper 4 open journal systems 


Random Discrete Distributions Derived from Self-Similar Random Sets

Jim Pitman, University of California, Berkeley
Marc Yor, Université Pierre et Marie Curie


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.


Full text: PDF

Pages: 1-28

Published on: February 20, 1996
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Electronic Journal of Probability. ISSN: 1083-6489