A packing dimension theorem for Gaussian random fields

Let X={X(t),t∈RN}X={X(t),t∈RN} be a Gaussian random field with values in RdRd defined by X(t)=(X1(t),…,Xd(t)),∀t∈RN, where X1,…,XdX1,…,Xd are independent copies of a centered Gaussian random field X0X0. Under certain general conditions, Xiao [Xiao, Y., 2007. Strong local nondeterminism and the sample path properties of Gaussian random fields. In: Lai, Tze Leung, Shao, Qiman, Qian, Lianfen (Eds.), Asymptotic Theory in Probability and Statistics with Applications. Higher Education Press, Beijing, pp. 136–176] defined an upper index α∗α∗ and a lower index α∗α∗ for X0X0 and showed that the Hausdorff dimensions of the range X([0,1]N)X([0,1]N) and graph GrX([0,1]N) are determined by the upper index α∗α∗. In this paper, we prove that the packing dimensions of X([0,1]N)X([0,1]N) and GrX([0,1]N) are determined by the lower index α∗α∗ of X0X0. Namely, dimPX([0,1]N)=min{d,Nα∗},a.s. and dimPGrX([0,1]N)=min{Nα∗,N+(1−α∗)d},a.s. This verifies a conjecture of Xiao in the above-cited reference. Our method is based on the potential-theoretic approach to packing dimension due to Falconer and Howroyd [Falconer, K.J., Howroyd, J.D., 1997. Packing dimensions for projections and dimension profiles. Math. Proc. Cambridge Philos. Soc. 121, 269–286].