Continuity of Gaussian processes

Abstract:

We give a proof of Fernique’s theorem that if X is a stationary Gaussian process and ${\sigma ^2}(h) = E{(X(h) - X(0))^2}$ then X has continuous sample paths provided that, for some $\varepsilon > 0,\sigma (h) \leqq \psi (h),0 \leqq h \leqq \varepsilon$, where $\psi$ is any increasing function satisfying \begin{equation}\tag {$\ast $} \int _0^\varepsilon {\frac {{\psi (h)}}{{h{{(\log (1/h))}^{1/2}}}}} dh < \infty .\end{equation} We prove the partial converse that if $\sigma (h) \geqq \psi (h),0 \leqq h \leqq \varepsilon$ and $\psi$ is any increasing function not satisfying $( ^\ast )$ then the paths are not continuous. In particular, if $\sigma$ is monotonic we may take $\psi = \sigma$ and $(^\ast )$ is then necessary and sufficient for sample path continuity. Our proof is based on an important lemma of Slepian. Finally we show that if $\sigma$ is monotonie and convex in $[0,\varepsilon ]$ then $\sigma (h){(\log 1/h)^{1/2}} \to 0$ as $h \to 0$ iff the paths are incrementally continuous, meaning that for each monotonic bounded sequence $t = {t_1},{t_2}, \ldots ,X({t_{n + 1}}) - X({t_n}) \to 0$, w.p.l.