THE MAXIMUM OF A RATCHET SCANNING PROCESS OVER A POISSON RANDOM FIELD

Scan statistic is a popular method in searching non-random clusters over some random field. Motivated by a high energy particle detection problem, we are interested in a ratchet scan statistic whose scanning window is a grid set, instead of a rectangular box. We further generalize it to an r-dimensional problem for any r is an element of Z(+) and provide a tail probability approximation for a ratchet scan statistic over an r-dimensional homogeneous Poisson process. We show that the ratchet effect can be factored out as an overshoot function nu in each dimension.