On the Joint Typicality of Permutations of Sequences of Random Variables

TL;DR

This paper analyzes the statistical behavior of permuted correlated sequences of random variables, establishing how permutation structure influences joint typicality and providing bounds for various permutation classes.

Contribution

It introduces the concept of standard and Bell permutation vectors, extending typicality analysis to complex permutation structures in correlated sequences.

Findings

Probability of joint typicality depends on permutation cycle structure.

Upper bounds on typicality are derived for standard permutations.

Extended analysis to Bell permutation vectors for arbitrary sequences.

Abstract

Permutations of correlated sequences of random variables appear naturally in a variety of applications such as graph matching and asynchronous communications. In this paper, the asymptotic statistical behavior of such permuted sequences is studied. It is assumed that a collection of random vectors is produced based on an arbitrary joint distribution, and the vectors undergo a permutation operation. The joint typicality of the resulting permuted vectors with respect to the original distribution is investigated. As an initial step, permutations of pairs of correlated random vectors are considered. It is shown that the probability of joint typicality of the permuted vectors depends only on the number and length of the disjoint cycles of the permutation. Consequently, it suffices to study typicality for a class of permutations called 'standard permutations', for which, upper-bounds on the probability of joint typicality are derived. The notion of standard permutations is extended to a class of permutation vectors called 'Bell permutation vectors'. By investigating Bell permutation vectors, upper-bounds on the probability of joint typicality of permutations of arbitrary collections of random sequences are derived.