Small ball probabilities for the Slepian Gaussian fields

The -dimensional Slepian Gaussian random field is a mean zero Gaussian process with covariance function for and . Small ball probabilities for are obtained under the -norm on , and under the sup-norm on which implies Talagrand's result for the Brownian sheet. The method of proof for the sup-norm case is purely probabilistic and analytic, and thus avoids ingenious combinatoric arguments of using decreasing mathematical induction. In particular, Riesz product techniques are new ingredients in our arguments.