Constrained correlation functions

Abstract

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.