Volume 13, January 2009
|Page(s)||409 - 416|
|Published online||22 September 2009|
Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays
Institut für Mathematische Stochastik,
Georg-August-Universität Göttingen, Maschmühlenweg 8-10, 37073 Göttingen,
Germany; firstname.lastname@example.org; email@example.com
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.
Mathematics Subject Classification: 60F15
Key words: Standardized increments / Lévy's continuity modulus / almost sure limit theorem / Erdös-Rényi law / multidimensional i.i.d. array / statistical multiscale parameter selection / scan statistics.
© EDP Sciences, SMAI, 2009
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