Small ball probabilities for the Slepian Gaussian fields

The $d$-dimensional Slepian Gaussian random field $\{S({\mathbf{t}}), {\mathbf{t}} \in \mathbb{R}_+^d\}$ is a mean zero Gaussian process with covariance function $\mathbb{E} S({\mathbf{s}})S({\mathbf{t}})= \prod_{i=1}^d \max (0, a_i-\left\vert s_i-t_i\right\vert )$ for $a_i>0$ and ${\mathbf{t}}=(t_1, \cdots, t_d) \in \mathbb{R}_+^d$. Small ball probabilities for $S({\mathbf{t}})$ are obtained under the $L_2$-norm on $[0,1]^d$, and under the sup-norm on $[0,1]^2$ 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.