Persistence probabilities of weighted sums of stationary Gaussian sequences

2020/03/02 by Aurzada, Frank, Mukherjee, Sumit · 2 citations
#60G15 #FOS: Mathematics #Primary: 60G22 #Probability (math.PR) #Secondary: 60G10

paper · doi:10.48550/arxiv.2003.01192

Abstract

With \ξ_i\_{i≥ 0} being a centered stationary Gaussian sequence with non-negative correlation function ρ(i):=𝔼[ ξ₀ξ_i] and \σ(i)\_{i≥ 1} a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum ∑_i=1^ℓ σ(i) ξ_i, 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)∼ 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 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.

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