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Measure-Preserving Functions and the Independence Copula

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Abstract

We solve a problem recently proposed by Kolesárová et al. Specifically, we prove that a necessary and sufficient condition for a given copula to be the independence or product copula is for the pair of measure-preserving transformations representing the copula to be independent as random variables.

We provide examples of such pairs for the well-known Cantor, Peano, and Hilbert curves. Moreover, a general constructive method is given for the representation of copulae in terms of measure-preserving transformations. In particular, we apply numbers representation systems to the study of self-similar copulae properties.

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Authors and Affiliations

  1. Departamento de Álgebra y Análisis Matemático, Universidad de Almería, La Cañada de San Urbano, 04120, Almería, Spain

    Enrique de Amo & Juan Fernández-Sánchez

  2. Departamento de Análisis Matemático, Universidad de Granada, Campus de Fuente Nueva s/n, 18071, Granada, Spain

    Manuel Díaz Carrillo

Authors
  1. Enrique de Amo
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  2. Manuel Díaz Carrillo
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  3. Juan Fernández-Sánchez
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Correspondence to Enrique de Amo.

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de Amo, E., Díaz Carrillo, M. & Fernández-Sánchez, J. Measure-Preserving Functions and the Independence Copula. Mediterr. J. Math. 8, 431–450 (2011). https://doi.org/10.1007/s00009-010-0073-9

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  • DOI: https://doi.org/10.1007/s00009-010-0073-9

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