The Minkowski functionals, including the Euler characteristic statistics, are standard tools for morphological analysis in cosmology. Motivated by cosmic research, we examine the Minkowski functional of the excursion set for an isotropic central limit random field, the $k$-point correlation functions ($k$th order cumulants) of which have the same structure as that assumed in cosmic research. Using 3- and 4-point correlation functions, we derive the asymptotic expansions of the Euler characteristic density, which is the building block of the Minkowski functional. The resulting formula reveals the types of non-Gaussianity that cannot be captured by the Minkowski functionals. As an example, we consider an isotropic chi-square random field and confirm that the asymptotic expansion accurately approximates the true Euler characteristic density.

翻译：Minkowski 函数,包括 Euler 特性统计,是宇宙学形态分析的标准工具。根据宇宙研究,我们检查了用于异向中央限制随机场的出游功能Minkowski, 美元-点相关函数(k$th coulants)与宇宙研究中假定的结构相同。我们利用3-和4点相关函数,得出了Euler 特性密度的无症状扩展,这是 Minkowski 功能的构件。由此产生的公式揭示了非Gausianity 的种类, Minkowski 函数无法捕捉到。举例来说,我们考虑的是一个异向正方形奇形随机字段,并证实无症状扩展准确接近了真正的 Euler 特性密度。