Hausdorff-type Measures of the Sample Path of Gaussian Random Fields

Abstract

Let ϕ be a Hausdorff measure function and Λ be an infinite increasing sequence of positive integers. The Hausdorff-type measure ϕ—m Λ associated to ϕ and Λ is studied. Let X(t)(t ∈R N ) be certain Gaussian random fields in R d . We give the exact Hausdorff measure of the graph set G r X([0, 1] N ), and evaluate the exact ϕ—m Λ measure of the image and graph set of X(t). A necessary and sufficient condition on the sequence Λ is given so that the usual Hausdorff measure function for X([0, 1] N ) and G r X([0, 1] N ) are still the correct measure functions. If the sequence Λ increases faster, then some smaller measure functions will give positive and finite (ϕ, Λ)-Hausdorff measure for X([0, 1] N ) and G r X([0, 1] N ).