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

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.