A packing dimension theorem for Gaussian random fields

Let X = {X(t), t is an element of R-N} be a Gaussian random field with values in R-d defined by X (t) = (X-1 (t), ..., X-d (t)), for all(t) is an element of R-N, where X-l,..., X-d are independent copies of a centered Gaussian random field X-0. 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, Qman, Qian, Lianfen (Eds.), Asymptotic Theory in Probability and Statistics with Applications. Higher Education Press, Beijing, pp. 136-176] defined an upper index alpha* and a lower index a.. for X-0 and showed that the Hausdorff dimensions of the range X ([0, 1](N)) and graph Gr X ([0, 1](N)) are determined by the upper index a*. In this paper, we prove that the packing dimensions of X ([0. 1](N)) and GrX([0, 1](N)) are determined by the lower index alpha* of X-0. Namely, dim(P) X ([0, 1](N)) = min {d, N/alpha(*)}, a.s. and dim(P) Gr X ([0, 1](N)) = min {N/alpha(*,) N + (1-alpha(*))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]. (c) 2008 Elsevier B.V. All rights reserved.