## Random Discrete Distributions Derived From Self-Similar Random Sets (1996)

Venue: | Electronic J. Probability |

Citations: | 14 - 10 self |

### BibTeX

@ARTICLE{Pitman96randomdiscrete,

author = {Jim Pitman and Marc Yor},

title = {Random Discrete Distributions Derived From Self-Similar Random Sets},

journal = {Electronic J. Probability},

year = {1996},

volume = {1},

pages = {1--28}

}

### OpenURL

### Abstract

: A model is proposed for a decreasing sequence of random variables (V 1 ; V 2 ; \Delta \Delta \Delta) with P 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 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 \Gamma1 F (dv) where the structural distribution F is identical to the distribution of 1 \Gamma sup(Z " [0; 1]). Then F (dv) = f(v)dv where (1 \Gamma v)f(v) is a decreasing function of v, and every such probability distribution F on [0; 1] can arise from this construction. Keywords: interval partition, zero set, excursion lengths, regenerative set, structural distribution. AMS subject classificat...