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Let $\{ X_{\bf n}, {\bf n}\in \mathbb{N}^d \}$ be a random field i.e. afamily of random variables indexed by $\mathbb{N}^d $, $d\ge 2$. Completeconvergence, convergence rates for non identically distributed, negativelydependent and martingale random fields are studied by application of Fuk-Nagaevinequality. The results are proved in asymmetric convergence case i.e. for thenorming sequence equal $n_1^{\alpha_1}\cdot n_2^{\alpha_2}\cdot\ldots\cdotn_d^{\alpha_d}$, where $(n_1,n_2,\ldots, n_d)=\mathbf{n} \in \mathbb{N}^d$ and$\min\limits_{1\leq i \leq d}\alpha_i \geq \frac{1}{2}.$

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