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Poisson flats in Euclidean spaces Part II: Homogeneous Poisson flats and the complementary theorem

Published online by Cambridge University Press:  01 July 2016

R. E. Miles
R. E. Miles*
Affiliation:
Australian National University
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Extract

Part I [21] treated the case of a finite number of independent random uniform s-flats in an ‘admissible’ subset of Ed (s = 0, · · ·, d − 1). In this second part, the natural and fruitful ‘Poisson extension’ to a ‘countable number of independent random uniform s-flats in Ed itself” is considered. It is worth mentioning at the outset that to have read Part I is not a prerequisite for reading the present paper. Although results of that part are often applied here, they serve only in an auxiliary capacity, thereby allowing the main thread of the theory to be developed without interruption.

Type
Research Article
Information
Advances in Applied Probability , Volume 3 , Issue 1 , Spring 1971 , pp. 1 - 43
Copyright
Copyright © Applied Probability Trust 1971 

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References

[1] Bartlett, M. S. (1966) An Introduction to Stochastic Processes with Special Reference to Methods and Applications. 2nd Edition. Cambridge University Press.Google Scholar
[2] Bartlett, M. S. (1967) The spectral analysis of line processes. Proc. Fifth Berkeley Symp. Math. Statist. and Prob. Vol. III, 135–153. University of California Press.Google Scholar
[3] Crain, I. K. and Miles, R. E. Monte Carlo estimates of the distributions of the random polygons determined by random lines in the plane. (In preparation).Google Scholar
[4] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[5] Feller, W. (1957) An Introduction to Probability Theory and its Applications. Vol. I. 2nd Edition. Wiley, New York.Google Scholar
[6] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. Wiley, New York.Google Scholar
[7] Giger, H. and Hadwiger, H. (1968) Über Treffzahlwahrscheinlichkeiten im Eikörperfeld. Z. Wahrscheinlichkeitsth. 10, 329–334.CrossRefGoogle Scholar
[8] Goldman, J. R. (1967) Stochastic point processes: limit theorems. Ann. Math. Statist. 38, 771–779.Google Scholar
[9] Goldman, J. R. (1967) Infinitely divisible point processes in R n . J. Math. Anal. Appl. 17, 133–146.Google Scholar
[10] Goudsmit, S. (1945) Random distribution of lines in a plane. Rev. Mod. Phys. 17, 321–322.CrossRefGoogle Scholar
[11] Hadwiger, H. (1957) Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag.Google Scholar
[12] Haight, F. A. (1967) Handbook of the Poisson Distribution. Wiley, New York. (Publications in Operations Research, Number 11).Google Scholar
[13] Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.Google Scholar
[14] James, A. T. (1954) Normal multivariate analysis and the orthogonal group. Ann. Math. Statist. 25, 40–75.CrossRefGoogle Scholar
[15] Kingman, J. F. C. (1967) Completely random measures. Pacific J. Math. 21, 59–78.Google Scholar
[16] Kuznecov, P. I. and Stratonovič, R. L. (1968) On the mathematical theory of correlated random points. IMS/AMS Selected Translations Math. Statist. Prob. 7, 1–16.Google Scholar
[17] Loève, M. (1963) Probability Theory. 3rd Edition. Van Nostrand.Google Scholar
[18] Miles, R. E. (1961) Random Polytopes: The Generalisation to n Dimensions of the Intervals of a Poisson Process. , Cambridge University.Google Scholar
[19] Miles, R. E. (1964) Random polygons determined by random lines in a plane. Proc. Nat. Acad. Sci. U.S.A. 52, 901–907, II 1157–1160.CrossRefGoogle ScholarPubMed
[20] Miles, R. E. (1969) Probability distribution of a network of triangles (a solution to problem 67–15). SIAM Rev. 11, 399–402.CrossRefGoogle Scholar
[21] Miles, R. E. (1969) Poisson flats in Euclidean spaces. Part I: A finite number of random uniform flats. Adv. Appl. Prob. 1, 211–237.CrossRefGoogle Scholar
[22] Miles, R. E. (1969) The asymptotic values of certain coverage probabilities. Biometrika 56, 661–680.CrossRefGoogle Scholar
[23] Miles, R. E. (1970) On the homogeneous planar Poisson point process. Mathematical Biosciences 6, 85–127.CrossRefGoogle Scholar
[24] Moyal, J. E. (1962) The general theory of stochastic population processes. Acta Math. 108, 1–31.Google Scholar
[25] Nachbin, L. (1965) The Haar Integral. Van Nostrand, Princeton.Google Scholar
[26] Neyman, J. and Scott, E. L. (1958) Statistical approach to problems of cosmology. J. R. Statist. Soc. B 20, 1–43.Google Scholar
[27] Ogston, A. G. (1959) The spaces in a uniform random suspension of fibres. Trans. Faraday Soc. 54, 1754–1757.Google Scholar
[28] Rényi, A. (1967) Remarks on the Poisson process. Stud. Sci. Math. Hung. 2, 119–123.Google Scholar
[29] Ryll-Nardzewski, C. (1961) Remarks on processes of calls. Proc. Fourth Berkeley Symp. Math. Statist. and Prob. Vol. II, 455–465. University of California Press.Google Scholar
[30] Santaló, L. A. (1955) Sur la mesure des espaces linéaires qui coupent un corps convexe et problèmes qui s'y rattachent. Colloque sur les questions de réalité en géométrie, Liège, 177–190. Georges Thone, Liège; Masson et Cie, Paris.Google Scholar
[31] Streit, F. (1970) On multiple integral geometric integrals and their applications to probability theory. Canadian J. Math. 22, 151–163.Google Scholar
[32] Takács, L. (1957) On secondary stochastic processes generated by a multidimensional Poisson process. Publ. Math. Inst. Hung. Acad. Sci. 2, 71–79.Google Scholar
[33] Takács, L. (1958) On the probability distribution of the measure of the union of random sets placed in a Euclidean space. Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 1, 89–95.Google Scholar
[34] Varga, O. (1935) Integralgeometrie 3. Croftons Formeln für den Raum. Math. Z. 40, 387–405.Google Scholar
[35] Wiener, N. (1939) The ergodic theorem. Duke Math. J. 5, 1–18.Google Scholar