A packing dimension theorem for Gaussian random fields

Abstract

Let X = { X ( t ) , t ∈ R N } be a Gaussian random field with values in R d defined by X ( t ) = ( X 1 ( t ) , … , X d ( t ) ) , ∀ t ∈ R N , where X 1 , … , 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, Qiman, Qian, Lianfen (Eds.), Asymptotic Theory in Probability and Statistics with Applications. Higher Education Press, Beijing, pp. 136–176] defined an upper index α ∗ and a lower index α ∗ 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 α ∗ . In this paper, we prove that the packing dimensions of X ( [ 0 , 1 ] N ) and Gr X ( [ 0 , 1 ] N ) are determined by the lower index α ∗ of X 0 . Namely, dim P X ( [ 0 , 1 ] N ) = min { d , N α ∗ } , a.s. and dim P Gr X ( [ 0 , 1 ] N ) = min { N α ∗ , 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].