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On the uniqueness of distance covariance


  • Székely, Gábor J.
  • Rizzo, Maria L.


Distance covariance and distance correlation are non-negative real numbers that characterize the independence of random vectors in arbitrary dimensions. In this work we prove that distance covariance is unique, starting from a definition of a covariance as a weighted L2 norm that measures the distance between the joint characteristic function of two random vectors and the product of their marginal characteristic functions. Rigid motion invariance and scale equivariance of these weighted L2 norms imply that the weight function of distance covariance is unique.

Suggested Citation

  • Székely, Gábor J. & Rizzo, Maria L., 2012. "On the uniqueness of distance covariance," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2278-2282.
  • Handle: RePEc:eee:stapro:v:82:y:2012:i:12:p:2278-2282
    DOI: 10.1016/j.spl.2012.08.007

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    Cited by:

    1. Mirosław Krzyśko & Tomasz Górecki & Waldemar Wołyński & Waldemar Ratajczak, 2016. "An Extension of the Classical Distance Correlation Coefficient for Multivariate Functional Data with Applications," Statistics in Transition new series, Główny Urząd Statystyczny (Polska), vol. 17(3), pages 449-466, September.
    2. repec:exl:29stat:v:17:y:2016:i:3:p:449-466 is not listed on IDEAS
    3. Fan, Yanan & de Micheaux, Pierre Lafaye & Penev, Spiridon & Salopek, Donna, 2017. "Multivariate nonparametric test of independence," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 189-210.
    4. Dueck, Johannes & Edelmann, Dominic & Richards, Donald, 2017. "Distance correlation coefficients for Lancaster distributions," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 19-39.


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