  The Mat\'ern and the Generalized Cauchy families of covariance functions have
a prominent role in spatial statistics as well as in a wealth of statistical
applications. The Mat\'ern family is crucial to index mean-square
differentiability of the associated Gaussian random field; the Cauchy family is
a decoupler of the fractal dimension and Hurst effect for Gaussian random
fields that are not self-similar.
  Our effort is devoted to prove that a scale-dependent family of covariance
functions, obtained as a reparameterization of the Generalized Cauchy family,
converges to a particular case of the Mat\'ern family, providing a somewhat
surprising bridge between covariance models with light tails and covariance
models that allow for long memory effect.
