## 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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