Statistical properties and probability densities of correlation functions

在最近的一篇论文(Schneider&Hartlap，2009)中，我们发现统计过程的相关函数必须服从一组不等式，因此，测量的相关函数的概率密度不能是多变量高斯的，正如在比较观测和模型预测以获得参数估计时几乎总是假设的那样。正如我们可以证明的那样，这种影响是严重的，并影响到至少一些最近的宇宙学调查的分析结果--在那些情况下，高斯甚至不是一个合理的近似值。这个项目的目标是在不违反严格的不等式的情况下，获得关联函数的概率密度的近似描述，这不仅是精密宇宙学所需的。基于上述论文和两篇正在进行的文凭论文，将采用解析和数值相结合的方法，推导出服从所有必要的不等式的概率密度的近似，并将该概率推广到随机过程(如宇宙密度场)的其他二阶统计测量。然后，所得到的密度将被用于使用最大似然法和贝叶斯方法对几个模拟的和可能的真实的宇宙学调查进行分析。

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.