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

Let $\{X_{\mathbf{n}},\mathbf{n}\in {\mathbb{N}}^d\}$ be a random field i.e. a family of random variables indexed by ${\mathbb{N}}^d$, $d\ge 2$. 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 $n_1^{\alpha_1}\cdot n_2^{\alpha_2}\cdot\ldots\cdot n_d^{\alpha_d}$,where$(n_1,n_2,\ldots, n_d)=$\mathbf{n}$ \in $\mathbb{N}$^d$and$\min\limits_{1\leq i \leq d}\alpha_i \geq \frac{1}{2}.$