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FOCUSING OF THE SCAN STATISTIC AND GEOMETRIC CLIQUE NUMBER

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By Mathew D. Penrose

Given sets C, R in d-dimensional space,
take a constant intensity Poisson point process on R;
the associated scan statistic S is the maximum number of
Poisson points in any translate of C.
As R becomes large with C fixed, bounded and open
but otherwise arbitrary, the distribution of S
becomes concentrated on at most two adjacent integers.
Moreover, a similar result holds when the underlying
Poisson process is replaced by a binomial point process,
and these results can be extended to a large class of
non-uniform distributions.