### 3.3 Higher-order correlation functions

One of the most direct methods to evaluate the deviation from Gaussianity is to compute the
higher-order correlation functions. Suppose that now labels the position of the -th object (galaxy).
Then the two-point correlation function is defined also in terms of the
joint probability of the pair of objects located in the volume elements of and ,
where is the mean number density of the objects. This definition is generalized to three- and four-point
correlation functions, and , in a straightforward manner:
Apparently , , and are symmetric with respect to the change of the indices. Define the
following quantities with the same symmetry properties:
Then it is not unreasonable to suspect that the following relations hold:
where , , , and are constants. In fact, the analysis of the two-dimensional galaxy
catalogues [68] revealed
The generlization of those relations for -point correlation functions is suspected to hold generally,
and is called the hierarchical clustering ansatz. Cosmological -body simulations approximately support
the validity of the above ansatz, but also detect the finite deviation from it [82].