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.
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