Scan statistics of Lévy noises and marked empirical processes
Zakhar Kabluchko and Evgeny Spodarev
Source: Adv. in Appl. Probab. Volume 41, Number 1 (2009), 013-037.
Abstract
Let n points be chosen independently and uniformly in the unit cube [0,1]d, 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]d we associate its score 𝛘n(R) defined as the sum of marks of all points contained in R. The scan statistic is defined as the maximum of 𝛘n(R), taken over all cubes R contained in [0,1]d. 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 →∞ to the Gumbel distribution. We also prove a corresponding result for the scan statistic of a Lévy noise with negative mean. The more elementary cases of zero and positive mean are also considered.
Primary Subjects: 60G70
Secondary Subjects: 60F05
Keywords: Scan statistic with variable window size; marked empirical process; independently scattered Lévy random measure; extremes; Pickands' double sum method
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Permanent link to this document: http://projecteuclid.org/euclid.aap/1240319575
Digital Object Identifier: doi:10.1239/aap/1240319575
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