Higher order statistics of the galaxy distribution using generating functions
István Szapudi, Alexander S. Szalay · The Astrophysical Journal · 1993
In this paper we lay the groundwork and develop a formalism for subsequent analyses of the higher order statistics of the galaxy distribution in various catalogs, applying the method of generating functions. The generating function of the distribution is connected to the continuum limit characteristic function. The continuum moments are replaced by the factorial moments of the discrete distribution during the transition to the discrete. The generating function of the distribution is related to the generating function of other relevant quantities like the Q_N_'s describing the hierarchy of correlation functions, the moments, and factorial moments, and the combinants, a new and natural statistic to characterize the distribution. We calculate the probability distribution of projected cell counts and show that it uniquely translates into the projection of the N-point correlation functions via Limber's equation. Most of our results can be generalized for the connected N-point distributions as well. We introduce generating functions for the cumulative distributions which are particularly important for the cluster correlation functions. We give a collection of formulae of widely used special probability densities with their discrete counterparts and expansions of some generating functions. With the availability of efficient computer algebra programs, like MATHEMATICA, these formulae provide a true simplification.