An approach to complete convergence theorems for dependent random fields via application of Fuk–Nagaev inequality

Let {Xn,n∈Nd}{Xn,n∈Nd} be a random field i.e. a family of random variables indexed by NdNd, d≥2d≥2. Complete convergence, convergence rates for non-identically distributed, negatively dependent and martingale random fields are studied by application of Fuk–Nagaev inequality. The results are proved in asymmetric convergence case i.e. for the norming sequence equal n1α1⋅n2α2⋅…⋅ndαd, where (n1,n2,…,nd)=n∈Nd(n1,n2,…,nd)=n∈Nd and min1≤i≤d⁡αi≥12.