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

Let 𝓧𝓝 be a set consisting of N elements and F 1,F 2, … be a sequence of random independent equiprobable mappings 𝓧𝓝 → 𝓧𝓝. For a subset S 0 ⊂ 𝓧𝓝, |S 0| = n, we consider a sequence of its images St = Ft (… F 2(F 1(S 0))…), t = 1, 2 … The conditions on n, t, N → ∞ under which the distributions of image sizes |St | are asymptotically connected with the standard normal distribution are presented.