  In this paper, we present the detailed calculation of the persistence
exponent $\theta$ for a nearly-Markovian Gaussian process $X(t)$, a problem
initially introduced in [Phys. Rev. Lett. 77, 1420 (1996)], describing the
probability that the walker never crosses the origin. New resummed perturbative
and non-perturbative expressions for $\theta$ are obtained, which suggest a
connection with the result of the alternative independent interval
approximation (IIA). The perturbation theory is extended to the calculation of
$\theta$ for non-Gaussian processes, by making a strong connection between the
problem of persistence and the calculation of the energy eigenfunctions of a
quantum mechanical problem. Finally, we give perturbative and non-perturbative
expressions for the persistence exponent $\theta(X_0)$, describing the
probability that the process remains bigger than $X_0\sqrt{<X^2(t)>}$.
