A packing dimension theorem for Gaussian random fields

Yimin Xiao

Statistics & Probability Letters, 2009, vol. 79, issue 1, 88-97

Abstract: Let be a Gaussian random field with values in defined by where X1,...,Xd are independent copies of a centered Gaussian random field X0. 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 [alpha]* and a lower index [alpha]* for X0 and showed that the Hausdorff dimensions of the range X([0,1]N) and graph are determined by the upper index [alpha]*. In this paper, we prove that the packing dimensions of X([0,1]N) and are determined by the lower index [alpha]* of X0. Namely, and 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].

Date: 2009
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