Asymptotic properties for partial sum processes of a Gaussian random field

Abstract

Let { ξ j ; 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 R d having nonnegative integer coordinates. Put S n = ∑ 0 ⩽ j ⩽ n ξ j for n ∈ Z + d and σ 2 ( ∥ i - j ∥ ) = E ( S i - S j ) 2 for i ≠ j , where ∥ · ∥ denotes the Euclidean norm and σ ( · ) is a nondecreasing continuous regularly varying function. Under some additional conditions, we investigate asymptotic properties for increments of partial sum processes of { ξ j ; j ∈ Z + d } .