A quasi-Gaussian approximation for the probability distribution of
  correlation functions
  Context. Whenever correlation functions are used for inference about
cosmological parameters in the context of a Bayesian analysis, the likelihood
function of correlation functions needs to be known. Usually, it is
approximated as a multivariate Gaussian, though this is not necessarily a good
approximation.
  Aims. We show how to calculate a better approximation for the probability
distribution of correlation functions, which we call "quasi-Gaussian".
  Methods. Using the exact univariate PDF as well as constraints on correlation
functions previously derived, we transform the correlation functions to an
unconstrained variable for which the Gaussian approximation is well justified.
From this Gaussian in the transformed space, we obtain the quasi-Gaussian PDF.
The two approximations for the probability distributions are compared to the
"true" distribution as obtained from simulations. Additionally, we test how the
new approximation performs when used as likelihood in a toy-model Bayesian
analysis.
  Results. The quasi-Gaussian PDF agrees very well with the PDF obtained from
simulations; in particular, it provides a significantly better description than
a straightforward copula approach. In a simple toy-model likelihood analysis,
it yields noticeably different results than the Gaussian likelihood, indicating
its possible impact on cosmological parameter estimation.
