  We study spontaneous symmetry breaking in a one-dimensional driven
two-species stochastic cellular automaton with parallel sublattice update and
open boundaries. The dynamics are symmetric with respect to interchange of
particles. Starting from an empty initial lattice, the system enters a symmetry
broken state after some time T_1 through an amplification loop of initial
fluctuations. It remains in the symmetry broken state for a time T_2 through a
traffic jam effect. Applying a simple martingale argument, we obtain rigorous
asymptotic estimates for the expected times <T_1> ~ L ln(L) and ln(<T_2>) ~ L,
where L is the system size. The actual value of T_1 depends strongly on the
initial fluctuation in the amplification loop. Numerical simulations suggest
that T_2 is exponentially distributed with a mean that grows exponentially in
system size. For the phase transition line we argue and confirm by simulations
that the flipping time between sign changes of the difference of particle
numbers approaches an algebraic distribution as the system size tends to
infinity.
