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Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays

Published online by Cambridge University Press:  22 September 2009

Zakhar Kabluchko  and
Axel Munk
Zakhar Kabluchko
Affiliation:
Institut für Mathematische Stochastik, Georg-August-Universität Göttingen, Maschmühlenweg 8-10, 37073 Göttingen, Germany; kabluch@math.uni-goettingen.de; munk@math.uni-goettingen.de
Axel Munk
Affiliation:
Institut für Mathematische Stochastik, Georg-August-Universität Göttingen, Maschmühlenweg 8-10, 37073 Göttingen, Germany; kabluch@math.uni-goettingen.de; munk@math.uni-goettingen.de
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Abstract

We generalize a theorem of Shao [Proc. Amer. Math. Soc.123 (1995) 575–582] on the almost-sure limiting behavior of the maximum of standardized random walk increments to multidimensional arrays of i.i.d. random variables. The main difficulty is the absence of an appropriate strong approximation result in the multidimensional setting. The multiscale statistic under consideration was used recently for the selection of the regularization parameter in a number of statistical algorithms as well as for the multiscale signal detection.

Keywords

Standardized incrementsLévy's continuity modulusalmost sure limit theoremErdös-Rényi lawmultidimensional i.i.d. arraystatistical multiscale parameter selectionscan statistics.

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 13 , July 2009 , pp. 409 - 416
Copyright
© EDP Sciences, SMAI, 2009

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