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

Zakhar Kabluchko (1) , Axel Munk (1)

1 Institut für Mathematische Stochastik

We generalize a theorem of Shao [ (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: Mathematics

Zakhar Kabluchko, Axel Munk. Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays. ESAIM: Probability and Statistics, 2009, 13, pp.409-416. ⟨10.1051/ps:2008020⟩. ⟨hal-00520011⟩