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 non-uniform distributions.