The Annals of Probability

Clump Counts in a Mosaic

Peter Hall

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A mosaic process is formed by centering independent and identically distributed random shapes at the points of a Poisson process in $k$-dimensional space. Clusters of overlapping shapes are called clumps. This paper provides approximations to the distribution of the number of clumps of a specified order within a large region. The approximations cover two different situations--"moderate-intensity" mosaics, in which the covered proportion of the region is neither very large nor very small; and "sparse" mosaics, in which the covered proportion is quite small. Both these mosaic types can be used to model observed phenomena, such as counts of bacterial colonies in a petri dish or dust particles on a membrane filter.

Article information

Ann. Probab., Volume 14, Number 2 (1986), 424-458.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G55: Point processes

Clump geometric probability mosaic normal approximation Poisson approximation random set


Hall, Peter. Clump Counts in a Mosaic. Ann. Probab. 14 (1986), no. 2, 424--458. doi:10.1214/aop/1176992525.

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