Persistence of Gaussian stationary processes: A spectral perspective

Persistence and Ball Exponents for Gaussian Stationary Processes

“…A study of this result, has led the first two authors to introduce spectral conditions which ensure exponential bounds on persistence, requiring the spectral measure to have a polynomial decay and a bounded spectral density in a small vicinity of the origin [25]. This was extended together with Nitzan [27] to conditions under which persistence decays sub-or super-exponentially, providing also new examples for extremely fast decaying persistence probabilities. All of these conditions depend on the interplay between the spectral behavior near the origin and the decay of the spectral measure near infinity.…”

“…Here, we go beyond establishing exponential-type behavior of persistence: we show the existence of a persistence exponent and provide several new continuity results. We do so by combining and expanding the spectral method of [27] and the covariance method of [19].…”

“…This has been studied in particular for the critical case ℓ = 0 (c.f. [26,27,33,50]), with applications to statistical mechanics (c.f. [24,41,47,61]).…”

“…In recent years it became evident that it is possible to obtain a much more precise understanding of persistence in terms of the spectral measure of f (c.f. [25,26,27]). Our main result is that existence of a positive spectral density in the origin is sufficient for the existence of θ ℓ ρ .…”

Persistence probabilities of mixed FBM and other mixed processes

“…Thus, our result contributes to the amount of rather rare persistence results for stochastic processes violating both the properties of self-similarity and stationary increments. Self-similarity is a valuable property in the context of persistence as in this case, one is able to apply the so-called Lamperti transformation to get a stationary process and concerning persistence, many powerful tools are available for the class of stationary centred Gaussian processes, see [14], [18], [15], [10], [20], [8] and [19]. In particular, combined with non-negative (and non-degenerate) covariances, self-similarity always guarantees the existence of the persistence exponent.…”