On The Probability That A Stationary Gaussian Process With Spectral Gap Remains Non-Negative On A Long Interval

Let f be a zero mean continuous stationary Gaussian process on R whose spectral measure vanishes in a d-neighborhood of the origin. Then, the probability that f stays non-negative on an interval of length L is at most e(-c delta 2L2) with some absolute c > 0 and the result is sharp without additional assumptions.