  (Abridged) We apply a very general statistical theorem introduced by Cramer
(1936) to study the origin of the deviations of the halo spin PDF from the
reference lognormal shape. We find that these deviations originate from
correlations between two quantities entering the definition of spin, namely the
ratio $J/M^{5/2}$ (which depends only on mass) and the total gravitational
binding energy $E$. To reach this conclusion, we have made usage of the results
deduced from two high spatial- and mass resolution simulations. Our simulations
cover a relatively small volume and produce a sample of more than 16.000
gravitationally bound halos, each traced by at least 300 particles. We verify
that our results are stable to different systematics, by comparing our results
with those derived by the GIF2 and by a more recent simulation performed by
Maccio' et al. We find that the spin probability distribution function shows
systematic deviations from a lognormal, at all redshifts z <= 1. These
deviations depend on mass and redshift: at small masses they change little with
redshift, and also the best lognormal fits are more stable. The J-M
relationship is well described by a power law of exponent $\alpha$ very near to
the linear theory prediction (alpha=5/3), but systematically lower than this at
z<= 0.3. We argue that the fact that deviations from a lognormal PDF are
present only for high-spin halos could point to a role of large-scale tidal
fields in the evolution of the spin PDF.
