  We discuss the level of natural shot noise in a proton bunch-driven plasma
accelerator. The required seeding for the plasma wake field must be larger than
the cumulative shot noise. This is the necessary condition for the axial
symmetry of the generated wake and the acceleration quality. We develop an
analytical theory of the noise field and compare it with multi-dimensional
simulations. It appears that the natural noise wake field generated in plasma
by the available at CERN super-protons-synchrotron (SPS) bunches is very low,
at the level of a few 10 kV/m. This fortunate fact eases the requirements on
the seed. Our three dimensional simulations show that even a few tens MeV
electron bunch precursor of a very moderate intensity is sufficient to seed the
proton bunch self-modulation in plasma.
