Hausdorff Measure of Space Anisotropic Gaussian Processes with Non-stationary Increments

Abstract

Let X = {X(t), t ∈ ℝ₊} be a centered space anisotropic Gaussian process values in ℝ^d with non-stationary increments, whose components are independent but may not be identically distributed. Under certain conditions, then almost surely c₁ ≤ ϕ − m(X([0, 1])) ≤ c₂, where ϕ denotes the exact Hausdorff measure associated with function \(\phi \left( s \right) = {s^{{1 \over {{\alpha _k}}} + \sum\limits_{i = 1}^k {\left( {1 - {{{\alpha _i}} \over {{\alpha _k}}}} \right)} }}\log \,\log\,{1 \over s}\) for some 1 ≤ k ≤ d, (α₁,⋯, α_d) ∈ (0, 1]^d. We also obtain the exact Hausdorff measure of the graph of X on [0, 1].