Convergence of sequences of ordered selections

TL;DR

This paper introduces a new convergence concept for ordered selections based on subpermutation densities and marginal distributions, extending permutation sequence convergence to a broader framework involving generalized permutons.

Contribution

It defines generalized permutons and proves their role as limits of ordered selection sequences, generalizing classical permutation convergence results.

Findings

Convergence notions for ordered selections are equivalent within generalized permutons.

Every convergent ordered selection sequence has a generalized permuton limit.

Any generalized permuton can be approximated by a sequence of ordered selections.

Abstract

In this paper, we introduce a convergence notion for ordered selections. Our convergence notion is based on subpermutation densities and convergences of the marginal distributions. A particular case of this convergence is the well-known convergence of permutation sequences. We also introduce a family of probability measures called generalized permutons. We show that in the family of generalized permutons several convergence notions are equivalent. We embed the set of ordered selections to the set of generalized permutons. We prove that any convergent sequence of ordered selections has a limit which is a generalized permuton. Moreover, any generalized permuton is the limit of a sequence of ordered selections. Our results are generalizations of well-known theorems on convergence of permutation sequences to permutons.