  The extension of the statistical parton distributions to include their
transverse momentum dependence (TMD) is revisited by considering that the
proton target has a finite longitudinal momentum. The TMD will be generated by
means of a transverse energy sum rule. The new results are mainly relevant for
electron-proton inelastic collisions in the low $Q^2$ region. We take into
account the effects of the Melosh-Wigner rotation for the helicity
distributions.
