约束关联函数
Constrained correlation functions
摘要：
我们表明，关联函数必须满足约束关系，由于非负的功率谱的潜在的随机过程。具体地说，对于任何统计均匀和（对于一个以上的空间维度）各向同性的随机场与关联函数$ξ(x)$，我们推导出不等式的相关系数$r_n\equiv ξ(n x)/ξ(0)$（整数$n$）的形式$r_{n{\rm l}}\le r_n\le r_{n{\rm u}}$，其中上下限和$r_n$取决于$r_j$，与$j<n$。对于任意的$n$，得到了该边界的显式表达式.这些约束方程非常显著地限制了可能的关联函数的集合。对于一个特定的例子的基准宇宙剪切调查，我们表明，高斯似然椭球有一个显着的溢出到禁止区域的关联函数，渲染得到的最佳拟合模型参数和它们的误差区域值得怀疑，并表示需要一个更好的描述的似然函数。我们进行了一些简单的数值实验，明确表明失败的高斯描述的可能性$ξ$。相反，相关系数的似然函数的形状似乎近似地遵循$r_n$上的边界的形状，即使高斯椭圆正好位于允许的区域内。对于多个空间维的随机场，$r_n$上的边界的显式表达式不是最优的。我们概述了一个几何方法如何更严格的界限可能获得的原则。我们举例说明这种方法的几个简单的情况下，更一般的治疗等待未来的工作。
英文摘要：
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 $ξ(x)$, we derive inequalities for the correlation coefficients $r_n\equiv ξ(n x)/ξ(0)$ (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 $ξ$. 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.
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