Persistence and Ball Exponents for Gaussian Stationary Processes

Abstract

Consider a real Gaussian stationary process f ρ , indexed on either R or Z and admitting a spectral measure ρ. We study, the persistence exponent of f ρ . We show that, if ρ has a positive density at the origin, then the persistence exponent exists; moreover, if ρ has an absolutely continuous component, then θ ℓ ρ > 0 if and only if this spectral density at the origin is finite. We further establish continuity of θ ℓ ρ in ℓ, in ρ (under a suitable metric) and, if ρ is compactly supported, also in dense sampling. Analogous continuity properties are shown for ψ ℓ ρ = − lim, the ball exponent of f ρ , and it is shown to be positive if and only if ρ has an absolutely continuous component.

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 [8,11,18,19,[23][24][25]. In particular, combined with non-negative (and non-degenerate) covariances, self-similarity always guarantees the existence of the persistence exponent.…”

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.…”