  We show that correlation functions have to satisfy contraint relations, owing
to the non-negativity of the power spectrum of the underlying random process.
Specifically, for any statistically homogeneous and (for more than one spatial
dimension) isotropic random field with correlation function $\xi(x)$, we derive
inequalities for the correlation coefficients $r_n\equiv \xi(n x)/\xi(0)$ (for
integer $n$) of the form $r_{n{\rm l}}\le r_n\le r_{n{\rm u}}$, where the lower
and upper bounds on $r_n$ depend on the $r_j$, with $j<n$. Explicit expressions
for the bounds are obtained for arbitrary $n$. These constraint equations very
significantly limit the set of possible correlation functions. For one
particular example of a fiducial cosmic shear survey, we show that the Gaussian
likelihood ellipsoid has a significant spill-over into the forbidden region of
correlation functions, rendering the resulting best-fitting model parameters
and their error region questionable, and indicating the need for a better
description of the likelihood function.
  We conduct some simple numerical experiments which explicitly demonstrate the
failure of a Gaussian description for the likelihood of $\xi$. Instead, the
shape of the likelihood function of the correlation coefficients appears to
follow approximately that of the shape of the bounds on the $r_n$, even if the
Gaussian ellipsoid lies well within the allowed region.
  For more than one spatial dimension of the random field, the explicit
expressions of the bounds on the $r_n$ are not optimal. We outline a
geometrical method how tighter bounds may be obtained in principle. We
illustrate this method for a few simple cases; a more general treatment awaits
future work.
