  A rigorous study is carried out for the randomly forced Burgers equation in
the inviscid limit. No closure approximations are made. Instead the probability
density functions of velocity and velocity gradient are related to the
statistics of quantities defined along the shocks. This method allows one to
compute the anomalies, as well as asymptotics for the structure functions and
the probability density functions. It is shown that the left tail for the
probability density function of the velocity gradient has to decay faster than
$|\xi|^{-3}$. A further argument confirms the prediction of E et al., Phys.
Rev. Lett. {\bf 78}, 1904 (1997), that it should decay as $|\xi|^{-7/2}$.
