  The statistical tools needed to obtain a mass function from realistic
collapse time estimates are presented. Collapse dynamics has been dealt with in
paper I of this series by means of the powerful Lagrangian perturbation theory
and the simple ellipsoidal collapse model. The basic quantity considered here
is the inverse collapse time F; it is a non-linear functional of the initial
potential, with a non-Gaussian distribution. In the case of sharp k-space
smoothing, it is demonstrated that the fraction of collapsed mass can be
determined by extending to the F process the diffusion formalism introduced by
Bond et al. (1991). The problem is then reduced to a random walk with a moving
absorbing barrier, and numerically solved; an accurate analytical fit, valid
for small and moderate resolutions, is found. For Gaussian smoothing, the F
trajectories are strongly correlated in resolution. In this case, an
approximation proposed by Peacock & Heavens (1990) can be used to determine the
mass functions. Gaussian smoothing is preferred, as it optimizes the
performances of dynamical predictions and stabilizes the F trajectories. The
relation between resolution and mass is treated at a heuristic level, and the
consequences of this approximation are discussed. The resulting mass functions,
compared to the classical Press & Schechter (1974) one, are shifted toward
large masses (confirming the findings of Monaco 1995), and tend to give more
intermediate-mass objects at the expense of small-mass objects. However, the
small-mass part of the mass function, which depends on uncertain dynamics and
is likely to be affected by uncertainties in the resolution--mass relation, is
not considered a robust prediction of this theory.
