Sufficient conditions for asymptotic normality of sums of values of discrete random fields, dependent in strips

Sufficient conditions are given for the asymptotic normality of a sum of values of a field on a square of the integer lattice in $\mathbf{R}^2$ with the side $n\to\infty$ for normalization by the quantity $(2n+1)^{3/2}$. Moreover, the values of the field are dependent on strips of fixed width. The method of moments is used for proof.