Small ball probabilities for the Slepian Gaussian fields

Abstract:

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 $\mathbb {E} S({\mathbf {s}})S({\mathbf {t}})= \prod _{i=1}^d \max (0, a_i-\left | s_i-t_i\right | )$ 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.