  Ultra-relativistic hollow electron beams can drive plasma wakefields ($\sim$
GV/m) suitable for positron acceleration. Stable propagation of hollow electron
beams for long distances in plasmas is required to accelerate positrons to high
energies by these plasma wakefields. In this work, we show by quasi-static
kinetic simulations using the code WAKE that an ultra-relativistic
azimuthally-symmetric hollow electron beam propagates in a plasma by developing
fish-bone like structure and shifting its bulk, differentially along its length
(rear part fastest), towards its axis due to the decrease in the betatron time
period of beam electrons from the beam-front to beam-rear. Hollow electron
beams with small radius collapse into their axis due to the pull by the
secondary wakefields generated by some of the beam electrons reaching the axis.
Hollow beams with sufficiently large radius, however, can propagate stably in
plasmas for several meters and be used for positron acceleration.
