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Macroscopic properties of a linear mosaic

Published online by Cambridge University Press:  01 July 2016

Peter Hall
Peter Hall*
Affiliation:
The Australian National University
*
∗Postal address: Department of Statistics, The Faculties, The Australian National University, GPO Box 4, Canberra, ACf 2601, Australia.
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Abstract

A linear mosaic is a process of random line segments distributed according to a Poisson process. This paper presents a wide-ranging treatment of limit theory for the two major macroscopic properties of a linear mosaic: total vacancy, and total number of spacings (or clumps). These quantities do not admit an explicit, exact treatment, and are perhaps most informatively studied by means of limit theory. We permit segment length to have a general distribution, and study the implications of tail properties of this distribution. Necessary and sufficient conditions are given for vacancy on an interval [0, t] to admit a normal approximation as t →∞. An approximate formula is provided for the probability of complete coverage, in the case of general segment length distribution. In all cases, properties of vacancy and number of spacings are studied together, by means of joint limit theorems, rather than individually, as in some earlier work.

Keywords

CENTRAL LIMIT THEOREMCLUMPCOVERAGE PROBABILITYDOMAIN OF ATTRACTIONMOSAIC PROCESSPOISSON PROCESSSEGMENT LENGTHSPACING
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 2 , June 1985 , pp. 330 - 346
Copyright
Copyright © Applied Probability Trust 1985 

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