Persistence and Ball Exponents for Gaussian Stationary Processes

Abstract

Consider a real Gaussian stationary process $f_{\ddot{I}}$, indexed on either $\mathbb{R}$ or $\mathbb{Z}$ and admitting a spectral measure $\ddot{I}$. We study $\hat{I}{¸}_{\ddot{I}}{}^{\ell }=-\underset{T\to \infty }{\lim }\frac{1}{T}\log \mathbb{P}\left({\inf }_{t\in [0,T]}{f}_{\ddot{I}}(t)>\ell \right)$, the persistence exponent of $f_{\ddot{I}}$. We show that, if $\ddot{I}$ has a positive density at the origin, then the persistence exponent exists; moreover, if $\ddot{I}$ has an absolutely continuous component, then $\hat{I}{¸}_{\ddot{I}}{}^{\ell }>0$ if and only if this spectral density at the origin is finite. We further establish continuity of $\hat{I}{¸}_{\ddot{I}}{}^{\ell }$ in $\ell$, in $\ddot{I}$ (under a suitable metric) and, if $\ddot{I}$ is compactly supported, also in dense sampling. Analogous continuity properties are shown for $\ddot{I}{ˆ}_{\ddot{I}}{}^{\ell }=-\underset{T\to \infty }{\lim }\frac{1}{T}\log \mathbb{P}\left({\inf }_{t\in [0,T]}|f_{\ddot{I}}(t)|\le \ell \right)$, the ball exponent of $f_{\ddot{I}}$, and it is shown to be positive if and only if $\ddot{I}$ has an absolutely continuous component.

Fixed inaccuracies in Lemma 1.1 and 1.3 of the previous draft