On complete convergence for weighted sums of martingale-difference random fields

Let {an,i,n∈Z+d,i≤n} be an array of real numbers, and let {Xi,i∈Z+d} be the martingale differences with respect to {Gn,n∈Z+d} satisfying E(E(X|Gk)|Gm)=E(X|Gk∧m) a.s., where k∧m denotes componentwise minimum, {Gk,k∈Z+d} is a family of σ-algebras such that ∀k≤n, Gk⊂Gn⊂G, and X is any integrable random variable defined on the initial probability space. The aim of this paper is to obtain some results concerning complete convergence of weighted sums ∑i≤nan,iXi.