On the Asymptotic Distribution of the Scan Statistic for Point Clouds

We derive the large-sample distribution of several variants of the scan statistic applied to a point process on an interval, which can be applied to detect the presence of an anomalous interval with any length. The main ingredients in the proof are Kolmogorov's theorem, a Poisson approximation, and recent technical results by Kabluchko et al (2014).