Small ball probabilities for the Slepian Gaussian fields

Source abstract

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