On the Probability That a Stationary Gaussian Process With Spectral Gap Remains Non-negative on a Long Interval

Abstract

Let f be a zero mean continuous stationary Gaussian process on R whose spectral measure vanishes in a δ-neighborhood of the origin. Then the probability that f stays non-negative on an interval of length L is at most e −cδ 2 L 2 with some absolute c > 0 and the result is sharp without additional assumptions.1991 Mathematics Subject Classification. 60G10, 60G15.

Persistence Probabilities of Two-Sided (Integrated) Sums of Correlated Stationary Gaussian Sequences

“…Finally, the third term on the right hand side of (37) can be estimated as 1 − corr X k (s), X k (s + τ ) ρ,σ τ, which verifies (19), and hence completes the proof of the theorem.…”

“…This observation along with part (b) of Lemma 3.2 shows that θ(C∞,H) = 0. From this, the conclusion follows from part (a) of Lemma 3.10, once we verify (19). But this follows on noting that Cp,H (τ /p) ≥ e − p+H p τ .…”

“…The corresponding lower bound follows from the trivial bound Cp,H(τ ) ≥ e −(p+H)τ , giving lim H↓1/2 Cp,H(τ ) = e −(p+1/2)τ , which is the correlation function of the scaled Ornstein-Uhlenbeck process which has persistence exponent p + 1/2, by Remark 3.9, and satisfies (20). The desired conclusion will then follow from Lemma 3.6, once we verify the conditions (18) and (19) of the lemma. To this effect, (18) follows from (51) and Remark 3.7, and (19) follows on noting that Cp,H (τ ) ≥ e −(p+H)τ , and so the proof is complete.…”

“…There are sufficient conditions in the literature for truly exponential decay (cf. [11,16,18,19]), but in our understanding none of them are both necessary and sufficient.…”

“…Invoking (53) we have Cp,H (τ /(p + H)) H e −τ so that we get (18) via Remark 3.7. Further, (19) is obtained from the observation that Cp,H ( τ p+H ) ≥ e −τ , and so (14) follows.…”