Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays
ESAIM: Probability and Statistics, Tome 13 (2009), pp. 409-416.

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.

DOI : 10.1051/ps:2008020
Classification : 60F15
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 
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     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},
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     year = {2009},
     doi = {10.1051/ps:2008020},
     mrnumber = {2554963},
     zbl = {1188.60014},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps:2008020/}
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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://archive.numdam.org/articles/10.1051/ps:2008020/

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