Statistical properties and probability densities of correlation functions

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In a recent publication (Schneider & Hartlap 2009), we discovered that correlation functions of statistical processes have to obey a set of inequalities and that, as a consequence, the probability density of measured correlation functions cannot be a multi-variate Gaussian, as almost always assumed when comparing observations with model predictions to obtain parameter estimates. As we could demonstrate, this effect is severe and affects the results of the analysis of at least some recent cosmological surveys – in those cases, the Gaussian is not even a reasonable approximation. The goal of this project is to obtain an approximate description of the probability density of correlation functions without violation of the strict inequalities, needed not only for precision cosmology. Based on the aforementioned paper and two ongoing diploma theses, a combination of analytical and numerical methods will be employed to derive approximations for this probability density which obey all the necessary inequalities and to generalize this probability to other second-order statistical measures of a random process (such as the cosmological density field). The resulting densities will then be applied to the analysis of several mock and possibly real cosmological surveys, using Maximum Likelihood and Bayesian methods.