  We consider the problem of `discrete-time persistence', which deals with the
zero-crossings of a continuous stochastic process, X(T), measured at discrete
times, T = n(\Delta T). For a Gaussian Stationary Process the persistence (no
crossing) probability decays as exp(-\theta_D T) = [\rho(a)]^n for large n,
where a = \exp[-(\Delta T)/2], and the discrete persistence exponent, \theta_D,
is given by \theta_D = \ln(\rho)/2\ln(a). Using the `Independent Interval
Approximation', we show how \theta_D varies with (\Delta T) for small (\Delta
T) and conclude that experimental measurements of persistence for smooth
processes, such as diffusion, are less sensitive to the effects of discrete
sampling than measurements of a randomly accelerated particle or random walker.
We extend the matrix method developed by us previously [Phys. Rev. E 64,
015151(R) (2001)] to determine \rho(a) for a two-dimensional random walk and
the one-dimensional random acceleration problem. We also consider `alternating
persistence', which corresponds to a < 0, and calculate \rho(a) for this case.
