SCAN STATISTICS OF LEVY NOISES AND MARKED EMPIRICAL PROCESSES

Abstract

Let n points be chosen independently and uniformly in the unit Cube 0, 1, and suppose that each point is supplied with a mark, the marks being independent and identically distributed random variables independent of the location of the points. To each cube R contained in 0, 1 we associate its score X-n(R) defined as the sum of marks of all points contained in R. The scan statistic is defined as the maximum of X-n(R), taken over all cubes R contained in 0, 1. We show that if the marks are nonlattice random variables with finite exponential moments, having negative mean and assuming positive values with nonzero probability, then the appropriately normalized distribution of the scan statistic converges as n -> infinity to the Gumbel distribution. We also prove a corresponding result for the scan statistic of a Levy noise with negative mean. The more elementary cases of zero and positive mean are also considered.