  We study the persistence probability for some discrete-time, time-reversible
processes. In particular, we deduce the persistence exponent in a number of
examples: first, we deal with random walks in random sceneries (RWRS) in any
dimension with Gaussian scenery. Second, we deal with sums of stationary
Gaussian sequences with correlations exhibiting long-range dependence. Apart
from the persistence probability we deal with the position of the maximum and
the time spent on the positive half-axis by the process.
