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Expected measure of the union of random rectangles

Part of: Geometric probability and stochastic geometry Approximations and expansions

Published online by Cambridge University Press:  14 July 2016

Chern-Ching Chao ,
Hsien-Kuei Hwang  and
Wen-Qi Liang
Chern-Ching Chao*
Affiliation:
Academia Sinica
Hsien-Kuei Hwang*
Affiliation:
Academia Sinica
Wen-Qi Liang*
Affiliation:
Academia Sinica
*
∗Postal address: Institute of Statistical Science, Academia Sinica, Taipei, 115, Taiwan.
∗Postal address: Institute of Statistical Science, Academia Sinica, Taipei, 115, Taiwan.
∗Postal address: Institute of Statistical Science, Academia Sinica, Taipei, 115, Taiwan.
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Abstract

An asymptotic expression for the expected area of the union of n random rectangles is derived by Mellin transforms, where their two diagonal corners are independently and uniformly distributed over (0,1)2. The general result for d-dimensional hyper-rectangles is also stated.

Keywords

Computational geometryconvex hullgeometric probabilitygeometry of rectanglesMellin transformcalculus of finite difference

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Type
Short Communications
Information
Journal of Applied Probability , Volume 35 , Issue 2 , June 1998 , pp. 495 - 500
Copyright
Copyright © Applied Probability Trust 1998 

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References

Buchta, C. (1989). On the average number of maxima in a set of vectors. Inform. Process. Lett. 33, 63–65.CrossRefGoogle Scholar
Buchta, C., Müller, J., and Tichy, R.F. (1985). Stochastic approximation of convex bodies. Math. Ann. 271, 225–235.CrossRefGoogle Scholar
Efron, B. (1965). The convex hull of a random set of points. Biometrika 52, 331–343.CrossRefGoogle Scholar
Erdélyi, A. (1985). Higher Transcendental Functions, Volume I. Robert E. Krieger Publishing Company, Malabar, Florida. Original edition, 1953.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications, Volume II. John Wiley & Sons, New York.Google Scholar
Flajolet, P., and Sedgewick, R. (1995). Mellin transforms and asymptotics: finite differences and Rice's integrals. Theoret. Comput. Sci. 144, 101–124.CrossRefGoogle Scholar
Groeneboom, P. (1988). Limit theorems for convex hulls. Prob. Theory Rel. Fields 79, 327–368.CrossRefGoogle Scholar
Hsing, T. (1994). On the asymptotic distribution of the area outside a random convex hull in a disk. Ann. Appl. Prob. 4, 478–493.CrossRefGoogle Scholar
Klee, V. (1977). Can the measure of ∪[ai,bi ] be computed in less than O(n log n) steps. Amer. Math. Monthly 84, 284–285.Google Scholar
Mead, C., and Conway, L. (1980). Introduction to VLSI-systems. Addison-Wesley, Reading, Mass.Google Scholar
Mehlhorn, K. (1984). Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry. Springer-Verlag, Berlin, New York.CrossRefGoogle Scholar
Preparata, F.P., and Shamos, M.I. (1985). Computational Geometry: An Introduction. Springer-Verlag, New York.CrossRefGoogle Scholar
Raynaud, H. (1970). Sur l'enveloppe convexe des nuages de point aléatoires dans R n . I. J. Appl. Prob. 7, 35–48.Google Scholar
Rényi, A., and Sulanke, R. (1963). Über die konvexe Hulle von n zufallig gewahlten Punkten. I. Z. Wahrscheinlichkeitsth. 2, 75–84.CrossRefGoogle Scholar
Rényi, A., and Sulanke, R. (1964). Über die konvexe Hulle von n zufallig gewahlten Punkten. II. Z. Wahrscheinlichkeitsth. 3, 138–148.CrossRefGoogle Scholar