Demimartingale Approaches for ScanStatistics

Scan statistics are defined as random variables enumerating the moving windows in a sequence of binary outcome trials which contain a prescribed number of successes. The main objective of this contribution is to serve as a self-contained source of some recent results concerning both the simple and the multiple scan statistic. These results are innovative in the sense that they seem to be the first ones on scan statistics that were derived by means of demimartingale techniques. The demimartingale approach motivated also some classification questions for stochastic processes associated with scan statistics. These types of questions and some past results on scan statistics that can be regarded as relevant to the demimartingale approach are also discussed here. In order to illustrate how our results can be implemented in practice, our presentation is enriched with several numerical exhibitions.