  We present an analytical and numerical study of the velocity distribution
function of self gravitating collisionless particles, which include dark matter
and star clusters. We show that the velocity distribution derived through the
Eddington's formula is identical to the analytical one derived directly from
the generalized entropy of non-extensive statistical mechanics. This implies
that self gravitating collisionless structures are to be described by
non-extensive thermo-statistics. We identify a connection between the density
slope of dark matter structures, \gamma, from \rho ~ r^{-\gamma}, and the
entropic index, q, from the generalized entropy, S_q. Our numerical result
confirms the analytical findings of earlier studies and clarifies which is the
correct connection between the density slope and the entropic index. We use
this result to conclude that from a fundamental statistical mechanics point of
view the central density slope of self gravitating collisionless dark matter
structures is not constrained, and even cored dark matter structures are
allowed with \gamma = 0. We find that the outer density slope is bounded by
\gamma= 10/3.
