On the asymptotic normality in a scheme of allocation of non-equiprobable particle sets

A scheme of allocation of non-equiprobable $m$-dependent particle sets into cells is considered. The number of cells and the number of sets of particles are supposed to tend to $\infty$ and to have the same order of growth. Asymptotic formulas for mathematical expectation and variance of the number of empty cells are obtained along with the sufficient conditions for their asymptotic normality.