FOCUSING OF THE SCAN STATISTIC AND GEOMETRIC CLIQUE NUMBER
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