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Convergence of sequences of ordered selections

B. Fazekas, I. Fazekas

TL;DR

The paper addresses the limits of ordered selections with $\frac{m}{n}\to \lambda$ by extending permutation limit theory to generalized permutons. It develops a convergence framework based on subpermutation densities and marginal distributions, introducing $\lambda$-permutons and generalized permutons as limit objects. It proves that convergent $(n,m)$-permutation sequences have generalized permuton limits and that every generalized permuton is realizable as a limit by some $(n,m)$-permutation sequence, unifying several convergence notions via $d_\infty$, $d_\square$, and weak convergence. This generalizes Hoppen's results on permutation limits to ordered selections and provides a robust toolset for analyzing asymptotics of these structures.

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.

Convergence of sequences of ordered selections

TL;DR

The paper addresses the limits of ordered selections with by extending permutation limit theory to generalized permutons. It develops a convergence framework based on subpermutation densities and marginal distributions, introducing -permutons and generalized permutons as limit objects. It proves that convergent -permutation sequences have generalized permuton limits and that every generalized permuton is realizable as a limit by some -permutation sequence, unifying several convergence notions via , , and weak convergence. This generalizes Hoppen's results on permutation limits to ordered selections and provides a robust toolset for analyzing asymptotics of these structures.

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.

Paper Structure

This paper contains 4 sections, 8 theorems, 66 equations, 4 figures.

Table of Contents

  1. Introduction
  2. $(n,m)$-permutations and $\lambda$-permutons
  3. Distances of generalized permutons
  4. Convergence of generalized permutons

Key Result

Theorem 1.1

Let $\nu_{n}=\nu_{n,m}$ be a convergent sequence of $(n,m)$-permutations: $\nu_{n} \xrightarrow[]{t}$. Let $\mu_{n}$ be the generalized permuton corresponding to $\nu_{n}$, $n=1,2, \dots$. Then the sequence $\mu_n$ is convergent, i.e. there exists a generalized permuton $\mu$ such that $\mu_{n} \xri

Figures (4)

  • Figure 1: The measure $\mu$ in Example \ref{['ex1']}.
  • Figure 2: The marginal densities and distribution functions of the measure $\mu$ in Example \ref{['ex1']}.
  • Figure 3: The measure $\mu_\sigma$ in Example \ref{['ex2']}.
  • Figure 4: Four steps in the proof of Theorem \ref{['invMain']} for the particular case given in Example \ref{['exStripes']}.

Theorems & Definitions (34)

  • Theorem 1.1: Main theorem
  • Theorem 1.2
  • Example 2.1
  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.2
  • Definition 2.4
  • Example 2.2
  • ...and 24 more