Constrained correlation functions

Measurements of correlation functions and their comparison with theoretical models are often employed in natural sciences, including astrophysics and cosmology, to determine best-fitting model parameters and their confidence regions. Due to a lack of better descriptions, the likelihood function of the correlation function is often assumed to be a multi-variate Gaussian. Using different methods, we show that correlation functions have to satisfy constraint 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) = xi(nx)/xi(0) (for integer n) of the form r(nl) <= r(n) <= r(nu), where the lower and upper bounds on r(n) depend on the r(j), with j < n, or more explicitly Xi(n-) {xi(0), xi(x), xi(2x), . . . , xi([n-1]x)} <= xi(nx) <= Xi(n+) {xi(0), xi(x), xi(2x), . . . , xi([n - 1]x)}. Explicit expressions for the bounds are obtained for arbitrary n. We show that 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 region of correlation functions forbidden by the aforementioned constraints, 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.. Instead, the shape of the likelihood function of the correlation coefficients appears to follow approximately the shape of the bounds on the rn, even if the Gaussian ellipsoid lies well within the allowed region. Therefore, we define a non-linear and coupled transformation of the rn, based on these bounds. Some numerical experiments then indicate that a Gaussian is a much better description of the likelihood in these transformed variables than of the original correlation coefficients - in particular, the full probability distribution then lies explicitly in the allowed region. For more than one spatial dimension of the random field, the explicit expressions of the bounds on the rn 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.