Random Discrete Distributions Derived from Self-similar Random Sets

A model is proposed for a decreasing sequence of random variables (V 1 ; V 2 ;) with P n V n = 1, which generalizes the Poisson-Dirichlet distribution and the distribution of ranked lengths of excursions of a Brow-nian motion or recurrent Bessel process. Let V n be the length of the nth longest component interval of 0; 1]nZ, where Z is an a.s. non-empty random closed of (0; 1) 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 a < b 1 the expected number of n's such that V n 2 (a; b) equals R b a v ?1 F(dv) where the structural distribution F is identical to the distribution of 1 ? sup(Z \ 0; 1]). Then 1 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.