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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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References

Chung, K. L. (1974) A Course in Probability Theory. Academic Press, New York.Google Scholar
Domb, C. (1947) The problem of random intervals on a line. Proc. Camb. Phil. Soc. 43, 329–341.Google Scholar
Fisher, R. A. (1940) On the similarity of the distributions found for the test of significance in harmonic analysis, and in Stevens’s problem in geometrical probability. Ann. Eugenics 10, 14–17.Google Scholar
Flatto, L. and Konheim, A. G. (1962) The random division of an interval and the random covering of a circle. SIAM Rev. 4, 211–222.Google Scholar
Gnedenko, B. V. and Kolmogorov, A. N. (1968) Limit Distributions for Sums of Random Variables. Addison-Wesley, Reading, Mass.Google Scholar
Hall, P. (1983a) On the vacancy in a mosaic process.Google Scholar
Hall, P. (1983b) Heavy traffic approximations for busy period in an M/G/? queue. Stoch. Proc. Appl. To appear.Google Scholar
Hall, P. (1986) Mosaics: An Introduction to Coverage and Clumping Processes.Google Scholar
Holst, L. (1980a) On multiple covering of a circle with random arcs. J. Appl. Prob. 17, 284–290.Google Scholar
Holst, L. (1980b) On the lengths of the pieces of a stick broken at random. J. Appl. Prob. 17, 623–634.Google Scholar
Holst, L. (1981) On convergence of the coverage by random arcs on a circle and the largest spacing. Ann. Prob. 9, 648–655.Google Scholar
Holst, L. (1983) A note on random arcs on the circle. In Probability and Mathematical Statistics: Essays in Honour of C.-G. Esseen , Ed. Gut, A. and Holst, L.. Uppsala University, Uppsala, 40–46.Google Scholar
Hüsler, J. (1982) Random coverage of the circle and asymptotic distributions. J. Appl. Prob. 19, 578–587.Google Scholar
Ibragimov, I. A. and Linnik, Yu. V. (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.Google Scholar
Janson, S. (1983) Random coverings of the circle with arcs of random lengths. In Probability and Mathematical Statistics: Essays in Honour of C.-G. Esseen , Ed. Gut, A.. and Holst, L., Uppsala University, Uppsala, 62–73.Google Scholar
Seneta, E. (1976) Regularly Varying Functions. Springer-Verlag, Berlin.Google Scholar
Siegel, A. F. (1978) Random arcs on the circle. J. Appl. Prob. 15, 774–789.Google Scholar
Siegel, A. F. (1979) Asymptotic coverage distributions on the circle. Ann. Prob. 7, 651–661.Google Scholar
Siegel, A. F. and Holst, L. (1982) Covering the circle with random arcs of random sizes. J. Appl. Prob. 19, 373–381.Google Scholar
Solomon, H. (1978) Geometric Probability. SIAM, Philadelphia, PA.Google Scholar
Stevens, W. L. (1939) Solution to a geometrical problem in probability. Ann. Eugenics 9, 315–320.Google Scholar
Takács, E. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
Votaw, D. F. (1946) The probability distribution of the measure of a random linear set. Ann. Math. Statist. 17, 240–244.Google Scholar
Yadin, M. and Zachs, S. (1982) Random coverage of a circle with application to a shadowing problem. J. Appl. Prob. 19, 562–577.Google Scholar