Structural equivalence of s-tuples in random discrete sequences

Abstract

Recently there appeared a signi cant number of papers investigating structural properties of random discrete sequences. They provide a large group of results on structurallyequivalent intervals in such sequences. The goal of the present paper is to give a survey of the problems and the known results in this interesting area of discrete probability theory.

The Poisson approximation for the number of matches of values of a discrete function on segments of a sequence of random variables

“…Under the hypotheses of Corollary 2, for s 2 the asymptotic behaviour of the random variable 0 n and the random variable n , which is equal to the number of pairs of s-tuples with identical frequencies of symbols in a segment of the sequence X of length n C s 1, is different. It can be shown that the asymptotic behaviour of n is determined by neighbouring overlapping tuples, and under the hypotheses of Corollary 2 E n grows without bound (see [11]). The asymptotic behaviour of s in different domains of variation of the parameters s and N is studied in [14].…”

“…The match of frequencies of symbols in s-tuples defines on the set A s the permutation equivalence relation (see [11]), which is a special case of the G-equivalence (see [13]). In other words, the match of frequencies of symbols in two tuples is equivalent to the match of these tuples up to permutation of symbols.…”

“…As in Section 3.3, we assume that the sequence X is obtained by the equiprobable polynomial scheme with outcomes in the set A D f1; : : : ; N g. We say that two tuples Y i 1 .s/ and Y j 1 .s/ are of identical structure (are structurally equivalent, see [11]) if…”

“…The equivalence relation such that a class is constituted by the tuples of the same structure is called (see [11]) the structural equivalence relation. It is not difficult to see that the structural equivalence of a pair of tuples is the same as their match up to coordinate-wise replacement of symbols in accordance to some permutation in the symmetric permutation group defined on the set A, so the structural equivalence relation is a special case of the H-equivalence (see [13]).…”

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

“…Along with the problem about exact matches of tuples, we point out the direction related to the asymptotic behaviour of the number of pairs of structurally or permutationally equivalent tuples, which is under active development over the last decade (see [2,17,18] and the survey [12]). …”