the central moments for the multivariate case

Here we assume , and define the second central moment , the fourth central moment , etc. The central moments of odd degrees are zero. Considering the symmetry of the normal distribution, these central moments of even degrees are expressed by the following partition indicators.

Then,  , , , etc.

Check if this rule works for one-dimension case; are given by

They should be equal to .

In the below, we rather use superscript to indicate the components of . For example, we use , instead of x=. denotes a quadratic form with symmetric matrix

The expectation is

Similarly, for the symmetric 4-form

Similarly, for the symmetric 6-form