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A Topological Proof of the Archimedean Axiom for Archimedean Copulas

Victory Idowu

TL;DR

This work establishes a topological proof of the Archimedean axiom for Archimedean copulas in the presence of non-continuous distributions. By employing C-volume $V_C$ and recursive volumes $V_C^n(u)$, the authors show the sequence $f_n(u)=C(u,f_{n-1}(u))$ decreases to 0 for any $u\in(0,1)$, ensuring the Archimedean axiom holds without assuming continuity. The key contribution is a rigorous demonstration that the Archimedean property extends to discrete and mixed distributions via a partitioned, limit-based argument, with convergence governed by topological properties of $V_C$. This broadens the applicability of Archimedean copulas and clarifies the role of topological volumes in dependence modeling for noncontinuous settings.

Abstract

Archimedean copulas are a popular type of copulas in which a variant of the Archimedean axiom apply. We provide a topological proof of the Archimedean Axiom which is applicable for non-continuous distribution functions.

A Topological Proof of the Archimedean Axiom for Archimedean Copulas

TL;DR

This work establishes a topological proof of the Archimedean axiom for Archimedean copulas in the presence of non-continuous distributions. By employing C-volume and recursive volumes , the authors show the sequence decreases to 0 for any , ensuring the Archimedean axiom holds without assuming continuity. The key contribution is a rigorous demonstration that the Archimedean property extends to discrete and mixed distributions via a partitioned, limit-based argument, with convergence governed by topological properties of . This broadens the applicability of Archimedean copulas and clarifies the role of topological volumes in dependence modeling for noncontinuous settings.

Abstract

Archimedean copulas are a popular type of copulas in which a variant of the Archimedean axiom apply. We provide a topological proof of the Archimedean Axiom which is applicable for non-continuous distribution functions.
Paper Structure (4 sections, 3 theorems, 13 equations)

This paper contains 4 sections, 3 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Copulas and the Archimedean axiom
  3. Proof of Archimedean axiom
  4. Proof of Theorem \ref{['c1c-thm-ArchimedeanAxiom']}

Key Result

Theorem 2.1

The Archimedean axiom for $([0,1],C)$ is, for any two $u,v \in (0,1)$, there exists $n \in \mathbb{Z}^+$ such that $f_n(u) < v$.

Theorems & Definitions (11)

  • Definition 1: H-volume, $V_H$ nelsen2006introductioncherubini2009computing
  • Definition 2: Cumulative distribution function $F$durante2013topological
  • Definition 3
  • Definition 4: Archimedean copulas joe2014dependence
  • Definition 5: $C$-power nelsen2006introduction
  • Theorem 2.1
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 1 more