  We consider a directed percolation process at its critical point. The
probability that the deviation of the global order parameter with respect to
its average has not changed its sign between 0 and t decays with t as a power
law. In space dimensions d<4 the global persistence exponent theta_p that
characterizes this decay is theta_p=2 while for d<4 its value is increased to
first order in epsilon = 4-d. Combining a method developed by Majumdar and Sire
with renormalization group techniques we compute the correction to theta_p to
first order in epsilon. The global persistence exponent is found to be a new
and independent exponent. We finally compare our results with existing
simulations.
