Asymptotic properties for partial sum processes of a Gaussian random field

Let be a centered strictly stationary Gaussian random field, where is the d-dimensional lattice of all points in d-dimensional Euclidean space having nonnegative integer coordinates. Put Sn=[summation operator]0[less-than-or-equals, slant]j[less-than-or-equals, slant]n[xi]j for and [sigma]2([short parallel]i-j[short parallel])=E(Si-Sj)2 for i[not equal to]j, where [short parallel]·[short parallel] denotes the Euclidean norm and [sigma](·) is a nondecreasing continuous regularly varying function. Under some additional conditions, we investigate asymptotic properties for increments of partial sum processes of .