Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T04:54:54.245Z Has data issue: false hasContentIssue false
  • English
  • Français

The asymptotic distribution of random molecules

Published online by Cambridge University Press:  01 July 2016

Wulf Rehder
Wulf Rehder*
Affiliation:
Technische Universitat Berlin
Get access Rights & Permissions [Opens in a new window]

Abstract

If n solid spheres Kn of some volume V(Kn) are scattered randomly in the unit cube of euclidean d-space, some of them will overlap to form Ln(s) molecules with exactly s atoms Kn. The random variable Ln(s) has a limit distribution if V(Kn) tends to zero but nV(Kn) tends to infinity at a certain rate: it is shown that for

Ln(s) is asymptotically Poisson.

Keywords

POISSON PROCESSLIMIT THEOREMRANDOM CONVEX SETSFACTORIAL MOMENTS
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 3 , September 1980 , pp. 640 - 654
Copyright
Copyright © Applied Probability Trust 1980 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Bonnesen, T. and Fenchel, W. (1934) Theorie der konvexen Körper. Springer-Verlag, Berlin. (Chelsea, New York (1948).) Google Scholar
2. Bouligand, G. (1928) Ensembles impropres et nombre dimensionnel. Bull. Sci. Math. 52, 320–344.Google Scholar
3. Eberl, W. and Hafner, R. (1971) Die asymptotische Verteilung von Koinzidenzen. Z. Wahrscheinlichkeitsth. 18, 322–332.Google Scholar
4. Hafner, R. (1972) Die asymptotische Verteilung von mehrfachen Koinzidenzen. Z. Wahrscheinlichkeitsth. 21, 96–108.Google Scholar
5. Hafner, R. (1972) The asymptotic distribution of random clumps. Computing 10, 335–351.Google Scholar
6. Loève, M. (1977) Probability Theory I. Springer-Verlag, Berlin.Google Scholar
7. Rehder, W. (1980) On the volume of unions of translates of a convex set. Amer. Math. Monthly. Google Scholar
8. Rehder, W. (1978) Die Konvergenz von Koinzidenzen konvexer Körper gegen einen Poisson-Prozess. Ph. D. Dissertation, TU Berlin.Google Scholar