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.
Mots-clés : standardized increments, Lévy's continuity modulus, almost sure limit theorem, Erdös-Rényi law, multidimensional i.i.d. array, statistical multiscale parameter selection, scan statistics
@article{PS_2009__13__409_0,
author = {Kabluchko, Zakhar and Munk, Axel},
title = {Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays},
journal = {ESAIM: Probability and Statistics},
pages = {409--416},
publisher = {EDP-Sciences},
volume = {13},
year = {2009},
doi = {10.1051/ps:2008020},
mrnumber = {2554963},
zbl = {1188.60014},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2008020/}
}
TY - JOUR AU - Kabluchko, Zakhar AU - Munk, Axel TI - Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays JO - ESAIM: Probability and Statistics PY - 2009 SP - 409 EP - 416 VL - 13 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2008020/ DO - 10.1051/ps:2008020 LA - en ID - PS_2009__13__409_0 ER -
%0 Journal Article %A Kabluchko, Zakhar %A Munk, Axel %T Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays %J ESAIM: Probability and Statistics %D 2009 %P 409-416 %V 13 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2008020/ %R 10.1051/ps:2008020 %G en %F PS_2009__13__409_0
Kabluchko, Zakhar; Munk, Axel. Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays. ESAIM: Probability and Statistics, Tome 13 (2009), pp. 409-416. doi : 10.1051/ps:2008020. http://www.numdam.org/articles/10.1051/ps:2008020/
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