  We study the probability of a real-valued stationary process to be positive
on a large interval $[0,N]$. We show that if in some neighborhood of the origin
the spectral measure of the process has density which is bounded away from zero
and infinity, then the decay of this probability is bounded between two
exponential functions in $N$. This generalizes similar bounds obtained for
particular cases, such as a recent result by Artezana, Buckley, Marzo, Olsen.
