Asymptotic behavior of mean uniform norms for sequences of Gaussian processes and fields

We consider the mean uniform (mixed) norms for a sequence of Gaussian random functions. For a wide class of Gaussian processes and fields the (2 log n)1/2-asymptotic for mixed norms is found whenever the volume of the index set is of order n and tends to infinity, for example, n -length time interval for random processes. Some numerical examples demonstrate the rate of convergence for the obtained asymptotic. The developed technique can be applied to analysis of various linear approximation methods. As an application we consider the rate of approximation by trigonometrical polynomials in the mean uniform norm. © Springer Science + Business Media, LLC 2006.