The Minkowski functionals, including the Euler characteristic statistics, are
standard tools for morphological analysis in cosmology. Motivated by
cosmological research, we examine the Minkowski functional of the excursion set
for an isotropic central limit random field, the k-point correlation
functions (kth order cumulants) of which have the same structure as that
assumed in cosmic research. We derive the asymptotic expansions of the expected
Euler characteristic density incorporating skewness and kurtosis, which is a
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 precisely approximates the true Euler
characteristic density.Comment: 28 pages, 3 figures, 1 tabl