  We introduce the concept of `discrete-time persistence', which deals with
zero-crossings of a continuous stochastic process, X(T), measured at discrete
times, T = n \Delta T. For a Gaussian Markov process with relaxation rate \mu,
we show that the persistence (no crossing) probability decays as \rho(a)^n for
large n, where a = \exp(-\mu \Delta T), and we compute \rho(a) to high
precision. We also define the concept of `alternating persistence', which
corresponds to a<0. For a>1, corresponding to motion in an unstable potential
(\mu<0), there is a nonzero probability of having no zero-crossings in infinite
time, and we show how to calculate it.
