Persistence probabilities of weighted sums of stationary Gaussian sequences

Abstract

With {ξi}i≥0 being a centered stationary Gaussian sequence with non-negative correlation function ρ(i)≔E[ξ0ξi] and {σ(i)}i≥1 a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum ∑i=1ℓσ(i)ξi, ℓ≥1. For summable correlations ρ, we show that the persistence exponent is universal. On the contrary, for non-summable ρ, even for polynomial weight functions σ(i)∼ip the persistence exponent depends on the rate of decay of the correlations (encoded by a parameter H) and on the polynomial rate p of σ. In this case, we show existence of the persistence exponent θ(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.