  We study the asymptotic behaviour of the probability that a weighted sum of
centered i.i.d. random variables X_k does not exceed a constant barrier. For
regular random walks, the results follow easily from classical fluctuation
theory, while this theory does not carry over to weighted random walks, where
essentially nothing seems to be known. First we discuss the case of a
polynomial weight function and determine the rate of decay of the above
probability for Gaussian X_k. This rate is shown to be universal over a larger
class of distributions that obey suitable moment conditions. Finally we discuss
the case of an exponential weight function. The mentioned universality does not
hold in this setup anymore so that the rate of decay has to be determined
separately for different distributions of the X_k. We present some results in
the Gaussian framework.
