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Clump counts in a mosaic. (English) Zbl 0606.60017

Asymptotic expressions are derived for the mean and variance of the number of isolated grains of a Boolean model within a bounded sampling region as the grain volume tends to zero while the proportion of volume covered by the process tends to a positive constant. When the volume fraction tends to zero, the number of clumps of any specified order n within the sampling region is shown to be asymptotically Poisson distributed. For \(n\geq 2\), an integral expression is given for the mean number of clumps of order n, which involves the probability that n independent realisations of a grain placed at n specified positions form a connected set.
Reviewer: P.J.Davy

MSC:

60D05 Geometric probability and stochastic geometry
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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