Limsup results and LIL for partial sum processes of a Gaussian random field

Abstract

Let {ξj; j ∈ ℤ+d be a centered stationary Gaussian random field, where ℤ+d is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝd, having nonnegative integer coordinates. For each j = (j1, ..., jd) in ℤ+d, we denote |j| = j1 ... jd and for m, n ∈ ℤ+d, define S(m, n] = Σm 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 < α < 1. Under some additional conditions, we study limsup results for increments of partial sum processes \( \{ S_n \} _{n \in \mathbb{Z}_ + ^d } \) and prove as well the law of the iterated logarithm for such partial sum processes.