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Supports of quasi-copulas

Juan Fernández-Sánchez, José Juan Quesada-Molina, Manuel Úbeda-Flores

Abstract

It is known that for every $s\in]1,2[$ there is a copula whose support is a self-similar fractal set with Hausdorff -- and box-counting -- dimension $s$. In this paper we provide similar results for (proper) quasi-copulas, in both the bivariate and multivariate cases.

Supports of quasi-copulas

Abstract

It is known that for every there is a copula whose support is a self-similar fractal set with Hausdorff -- and box-counting -- dimension . In this paper we provide similar results for (proper) quasi-copulas, in both the bivariate and multivariate cases.
Paper Structure (8 sections, 12 theorems, 31 equations, 2 figures)

This paper contains 8 sections, 12 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasi-transformation matrices
  4. Quasi-copulas and fractals
  5. Self-similarity, Hausdorff and box-counting dimensions in $]1,2[$
  6. Hausdorff dimension less than 1?
  7. The multidimensional case
  8. Conclusions

Key Result

Theorem 1

Let $(X,d)$ be a complete metric space. If $\{X;(w_{n})\}$ is a hyperbolic IFS, then there exists a unique nonempty compact subset $K$ of $X$ for which $K=W(K)$. Moreover, if each $w_n$ is a similarity and Moran's open set condition is satisfied, then $\dim_{\mathcal{H}}(K)=\dim_{\mathcal{B}}(K)=s$

Figures (2)

  • Figure 1: The support of $T_{0}(\Pi )$.
  • Figure 2: The support of $T_{1/2}(\Pi )$ from Theorem \ref{['frac1']}.

Theorems & Definitions (28)

  • Theorem 1
  • Definition 1: FerRodUb2011
  • Remark 1
  • Theorem 2
  • Remark 2
  • Lemma 3
  • proof
  • Definition 2
  • Remark 3
  • Theorem 4
  • ...and 18 more