Limit distributions of the number of sets of $H$-equivalent segments in an equiprobable polynomial scheme of arrays

In this paper we study random variables which characterise collections of segments in an equiprobable polynomial scheme related by the $H$-equivalence. We give an upper bound for the variation distance between the distribution of the random variable $\xi_k(H)$ equal to the number of collections of $H$-equivalent segments and the Poisson distribution. We present sufficient conditions for the convergence of the distribution functions of the number of $H$-equivalent segments in the triangular array scheme of equiprobable polynomial trials to the normal law, the Poisson law, and the compound Poisson law.