Persistence probabilities of weighted sums of stationary Gaussian sequences

Abstract

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.