Asymptotic properties for partial sum processes of a Gaussian random field

Let {ξj;j∈Z+d} be a centered strictly stationary Gaussian random field, where Z+d is the d-dimensional lattice of all points in d  -dimensional Euclidean space RdRd having nonnegative integer coordinates. Put Sn=∑0⩽j⩽nξjSn=∑0⩽j⩽nξj for n∈Z+d and σ2(∥i-j∥)=E(Si-Sj)2σ2(∥i-j∥)=E(Si-Sj)2 for i≠ji≠j, where ∥·∥∥·∥ denotes the Euclidean norm and σ(·)σ(·) is a nondecreasing continuous regularly varying function. Under some additional conditions, we investigate asymptotic properties for increments of partial sum processes of {ξj;j∈Z+d}.