Summary
A mosaic is formed by centring independent and identically distributed random sets at points of a Poisson process in Euclidean space. We derive high-intensity approximations to the distributions of size, structure and number of uncovered regions in a mosaic. A limit theorem is proved for vacancy, and leads to a general approximation to the probability that a given region is completely covered by random shapes.
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Fisher, R.A.: On the similarity of the distributions found for the test of significance in harmonic analysis, and in Stevens's problem in geometrical probability. Ann. Eugenics10, 14–17 (1940)
Domb, C.: The problem of random intervals on a line. Proc. Cambridge Philos. Soc.43, 329–341 (1947)
Gilbert, E.N.: The probability of covering a sphere withN circular caps. Biometrika52, 323–330 (1965)
Glaz, J., Naus, J.: Multiple coverage of the line. Ann. Probab.7, 900–906 (1979)
Hall, P.: On the coverage ofk-dimensional space byk-dimensional spheres. Ann. Probab. (submitted)
Hall, P.: Macroscopic properties of a linear mosaic. J. Appl. Probab. (submitted)
Holst, L.: On multiple covering of a circle with random arcs. J. Appl. Probab.17, 284–290 (1980)
Holst, L.: On the lengths of the pieces of a stick broken at random. J. Appl. Probab.17, 623–634 (1980)
Holst, L.: On convergence of the coverage by random arcs on a circle and the largest spacing. Ann. Probab.9, 648–655 (1981)
Holst, L.: A note on random arcs on the circle. In: Probability and Mathematical Statistics: Essays in Honour of C.-G. Esseen. Gut, A., Holst, L., Eds. Uppsala University, Uppsala pp. 40–46.
Hüsler, J.: Random coverage of the circle and asymptotic distributions. J. Appl. Probability19, 578–587 (1982)
Janson, S.: Random coverings of the circle with arcs of random lengths. In: Probability and Mathematical Statistics: Essays in Honour of C.-G. Esseen. Gut, A., Holst, L. Eds. Uppsala University, Uppsala pp. 62–73.
Kendall, M.G., Moran, P.A.P.: Geometrical Probability, London: Griffin, 1963
Matheron, G.: Random Sets and Integral Geometry, New York: Wiley 1975
Miles, R.E.: The asymptotic values of certain coverage probabilities. Biometrika56, 661–680 (1969)
Miles, R.E.: A synopsis of “Poisson flats in Euclidean spaces”. Izv. Akad. Nauk. Armen. SSR.5, 263–285 (1970)
Miles, R.E.: On the homogeneous planar Poisson point process. Math. Biosci.6, 85–127 (1970)
Miles, R.E.: Poisson flats in Euclidean spaces. Part II: Homogeneous Poisson flats and the Complementary Theorem. Adv. Appl. Prob.3, 1–43 (1971)
Miles, R.E.: The random division of space. Suppl. Adv. Appl. Probab.2, 243–266 (1972)
Moran, P.A.P.: The random volume of interpenetrating spheres in space. J. Appl. Probab.10, 483–490 (1973)
Moran, P.A.P., Fazekas de St. Groth, S.: Random circles on a sphere. Biometrika49, 389–396 (1962)
Siegel, A.F.: Random space filling and moments of coverage in geometrical probability. J. Appl. Probab.15, 340–355 (1978)
Siegel, A.F.: Asymptotic covergae distributions on the circle. Ann. Probab.7, 651–661 (1979)
Siegel, A.F., Holst, L.: Covering the circle with random arcs of random sizes. J. Appl. Probab.19, 373–381 (1982)
Silverman, B. and Brown, T.: Short distances, flat triangles and Poisson limits. J Appl. Probab.15, 815–825 (1978)
Solomon, H.: Geometrical Probability. SIAM, Philadelphia, 1978.
Stevens, W.L.: Solution to a geometrical problem in probability. Ann. Eugenics9, 315–320 (1939)
Yadin, M., Sacks, S.: Random coverage of a circle with application to a shadowing problem. J. Appl. Probab.19, 562–577 (1982)
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Hall, P. Distribution of size, structure and number of vacant regions in a high-intensity mosaic. Z. Wahrscheinlichkeitstheorie verw Gebiete 70, 237–261 (1985). https://doi.org/10.1007/BF02451430
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DOI: https://doi.org/10.1007/BF02451430