Let $\xi(t)=\xi(t_1,\dots,t_k)$ be a Gaussian random field. In this paper, some sufficient conditions for convergence of the sums $$ \sum_{\alpha_1,\dots,\alpha_k=1}^{2^n}F_n(\Delta_{2^{-n}}\xi(2^{-n}\alpha)), \quad \alpha=(\alpha_1,\dots,\alpha_k), $$ to a constant are obtained, where $\Delta_{2^{-n}}\xi(t)$ is the $k$th increment of the sample function $\xi(t)$ defined by (1) and $F_n$ are Borel functions. The results are analogues to those contained in [1]–[6] and can be considered as some generalizations of the theorem due to Berman in [5].