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The Asymptotic Distribution of Random Molecules

Wulf Rehder
Advances in Applied Probability
Vol. 12, No. 3 (Sep., 1980), pp. 640-654
DOI: 10.2307/1426424
Stable URL: http://www.jstor.org/stable/1426424
Page Count: 15
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The Asymptotic Distribution of Random Molecules
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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 nV(Kn)=log(an/(log n)(d-1)(s-1)), $a>0$, Ln(s) is asymptotically Poisson. This result can be generalized to obtain a theorem about the convergence of a sequence of stochastic processes towards a Poisson point process.

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