Place $n$ arcs, each of length $a_n$, uniformly at random on the circumference of a circle, choosing the arc length sequence $a_n$ so that the probability of completely covering the circle remains constant. We obtain the limiting distribution of the uncovered proportion of the circle. We show that this distribution has a natural interpretation as a noncentral chi-square distribution with zero degrees of freedom by expressing it as a Poisson mixture of mass at zero with central chi-square deviates having even degrees of freedom. We also treat the case of proportionately smaller arcs and obtain a limiting normal distribution. Potential applications include immunology, genetics, and time series analysis.
Publié le : 1979-08-14
Classification:
Asymptotic coverage distribution,
random arcs,
geometrical probability,
noncentral chi-square with zero degrees of freedom,
60D05,
60E05,
60F99,
62F99
@article{1176994988,
author = {Siegel, Andrew F.},
title = {Asymptotic Coverage Distributions on the Circle},
journal = {Ann. Probab.},
volume = {7},
number = {6},
year = {1979},
pages = { 651-661},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994988}
}
Siegel, Andrew F. Asymptotic Coverage Distributions on the Circle. Ann. Probab., Tome 7 (1979) no. 6, pp. 651-661. http://gdmltest.u-ga.fr/item/1176994988/