Limit theorem for the size of an image of subset under compositions of random mappings

Let $\mathcal{X_N}$ be a set consisting of $N$ elements and $F_1,F_2,\ldots$ be a sequence of random independent equiprobable mappings $\mathcal{X_N}\to\mathcal{X_N}$. For a subset $S_0\subset \mathcal{X_N}$, $|S_0|=n$, we consider a sequence of its images $S_t=F_t(\ldots F_2(F_1(S_0))\ldots)$, $t=1,2\ldots$ The conditions on $n$, $t$, $N\to\infty$ under which the distributions of image sizes $|S_t|$ are asymptotically connected with the standard normal distribution are presented.