  With $\{\xi_i\}_{i\ge 0}$ being a centered stationary Gaussian sequence with
non-negative correlation function $\rho(i):=\mathbb{E}[ \xi_0\xi_i]$ and
$\{\sigma(i)\}_{i\ge 1}$ a sequence of positive reals, we study the asymptotics
of the persistence probability of the weighted sum $\sum_{i=1}^\ell \sigma(i)
\xi_i$, $\ell\ge 1$. For summable correlations $\rho$, we show that the
persistence exponent is universal. On the contrary, for non-summable $\rho$,
even for polynomial weight functions $\sigma(i)\sim i^p$ the persistence
exponent depends on the rate of decay of the correlations (encoded by a
parameter $H$) and on the polynomial rate $p$ of $\sigma$. In this case, we
show existence of the persistence exponent $\theta(H,p)$ and study its
properties as a function of $(p,H)$. During the course of our proofs, we
develop several tools for dealing with exit problems for Gaussian processes
with non-negative correlations -- e.g.\ a continuity result for persistence
exponents and a necessary and sufficient criterion for the persistence exponent
to be zero -- that might be of independent interest.
