Persistence probabilities of weighted sums of stationary Gaussian sequences

With {xi i }i >= 0 being a centered stationary Gaussian sequence with non-negative correlation function rho(i) := E [xi 0 xi i] and {sigma(i)}i >= 1 a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum n-ary sumation li=1 sigma(i)xi i,l >= 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) similar to 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. (c) 2023 Elsevier B.V. All rights reserved.