Limit distributions of the number of collections of H -equivalent segments in the triangular array scheme of equiprobable polynomial trials

Abstract

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 ξ

Structural equivalence of s-tuples in random discrete sequences

“…Note that for r D 2 this result was obtained by A. M. Shoitov (see [37] (8) is not directly related to the type of the limit distribution in Theorem 5. This equality is due to the fact that we deal with H -connected tuples of lengths suf cient to include all N alphabet letters.…”

“…It was done by A. M. Shoitov in [37], where he studied the case of an arbitrary subgroup of the symmetric group S N . The main result of [37] can be formulated as follows. …”

“…Inequalities (40) help to study the asymptotics of PfE.s; N /g under condition (37). Theorem 27 below is based on (40), the Poisson limit theorem, and two other results.…”

“…Let X 1 ; : : : ; X s have an arbitrary polynomial distribution, while Y 1 ; : : : ; Y s be distributed according to (37).…”

The compound Poisson distribution of the number of matches of values of a discrete function of s-tuples in segments of a sequence of random variables

“…In Section 4.2, we give a well-known result on the limit distribution of the number of pairs of H-equivalent (see [13]) tuples in segments of sequences in a polynomial scheme (see [16,18]). In Section 4.3, we study conditions for convergence to a compound Poisson law of the distribution of the number of pairs of non-overlapping G-equivalent (see [13]) tuples in segments of sequences in a polynomial scheme; as corollaries to the results given in Section 3 we obtain a new compound Poisson and normal limit theorems for tuples in segments of sequences with identical vectors of frequencies of occurrences of symbols (that is, permutationally equivalent tuples).…”