A Packing Dimension Theorem for Gaussian Random Fields

Let X = {X(t), t ∈ RN} be a Gaussian random field with values in R defined by X(t) = ( X1(t), . . . , Xd(t) ) , ∀ t ∈ R , where X1, . . . , Xd are independent copies of a centered Gaussian random field X0. Under certain general conditions, Xiao (2007a) defined an upper index α∗ and a lower index α∗ for X0 and showed that the Hausdorff dimensions of the range X ( [0, 1] ) and graph GrX ( [0, 1] ) are determined by the upper index α∗. In this paper, we prove that the packing dimensions of X ( [0, 1] ) and GrX([0, 1] ) are determined by the lower index α∗ of X0. Namely, dimPX ( [0, 1] ) = min { d, N α∗ } , a.s. and dimPGrX ( [0, 1] ) = min { N α∗ , N + (1− α∗)d } , a.s. This verifies a conjecture in Xiao (2007a). Our method is based on the potential-theoretic approach to packing dimension due to Falconer and Howroyd (1997). Running head: A packing dimension theorem for Gaussian random fields 2000 AMS Classification numbers: 60G15, 60G17; 28A80.