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Scan statistics of Lévy noises and marked empirical processes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Zakhar Kabluchko  and
Evgeny Spodarev
Zakhar Kabluchko*
Affiliation:
Georg-August-Universität Göttingen
Evgeny Spodarev*
Affiliation:
Universität Ulm
*
∗ Postal address: Institut für Mathematische Stochastik, Georg-August-Universität Göttingen, Goldschmidtstr. 7, D-37077 Göttingen, Germany. Email address: kabluch@math.uni-goettingen.de
∗∗ Postal address: Institut für Stochastik, Universität Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany.
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Abstract

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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 defined as the sum of marks of all points contained in R. The scan statistic is defined as the maximum of 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.

Keywords

Scan statistic with variable window sizemarked empirical processindependently scattered Lévy random measureextremesPickands' double sum method

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F05: Central limit and other weak theorems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 41 , Issue 1 , March 2009 , pp. 13 - 37
Copyright
Copyright © Applied Probability Trust 2009 

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