A Poisson limit theorem for a two-stage equiprobable scheme of allocating particles into cells

We consider a two-stage scheme of allocating particles into cells. At the first stage, $N_0$ initial particles are independently and equiprobably allocated into $N_1$ cells of the first layer. At the second stage, these $N_1$ cells are taken as particles which are independently and equiprobably allocated into $N_2$ cells of the second layer. We present conditions under which the distribution of the number of cells of the second layer containing exactly $r$ particles converges to the Poisson law.