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Random covering of the circle: the size of the connected components

Part of: Limit theorems Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Thierry Huillet
Thierry Huillet*
Affiliation:
Université de Cergy-Pontoise
*
∗ Postal address: Laboratoire de Physique Théorique et Modélisation, CNRS-UMR 8089 et Université de Cergy-Pontoise, 5 mail Gay-Lussac, 95031 Neuville sur Oise, France. Email address: [email protected]
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Abstract

Consider a circle of circumference 1. Throw n points at random onto this circle and append to each of these points a clockwise arc of length s. The resulting random set is a union of a random number of connected components, each with specific size. Using tools designed by Steutel, we compute the joint distribution of the lengths of the connected components. Asymptotic results are presented when n goes to ∞ and s to 0 jointly according to different regimes.

Keywords

Geometrical probabilitycoveringspacings

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60F05: Central limit and other weak theorems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 3 , September 2003 , pp. 563 - 582
Copyright
Copyright © Applied Probability Trust 2003 

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