Survival probabilities of weighted random walks

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.