  The highly nonlinear regime of gravitational clustering is characterized by
the presence of scale-invariant form of many-body correlation functions. Useful
insights can be obtained by investigating the consequences of a generic scaling
{\em ansatz}. Extending earlier studies by Bernardeau & Schaeffer (1992) we
calculate the detailed consequences of such scaling. We generalise the
two-point cumulant correlators to a hierarchy of multi-point cumulant
correlators (MCC). We introduce the concept of reduced cumulant correlators
(RCC) and their related generating functions. Using analytical and diagrammatic
method we show that every new vertex of the tree representation of higher-order
correlations has its own reduced cumulant correlator. In the limit of large
separations, MCCs of arbitrary order can be expressed in terms of RCCs of the
same and lower order. We relate the generating functions of RCCs with the
statistics of {\em collapsed} objects. We develop the hierarchy for the
correlation functions of overdense regions. In this vein, we compute the
lower-order cumulants and cumulant correlators for overdense regions. Our study
shows how they vary as a function of the initial power spectrum of primordial
density fluctuations (abridged).
