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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Continuity of Gaussian processes


Authors: M. B. Marcus and L. A. Shepp
Journal: Trans. Amer. Math. Soc. 151 (1970), 377-391
MSC: Primary 60.40
DOI: https://doi.org/10.1090/S0002-9947-1970-0264749-1
MathSciNet review: 0264749
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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

$\displaystyle \int_0^\varepsilon {\frac{{\psi (h)}}{{h{{(\log(1/h))}^{1/2}}}}} dh < \infty .$ ($ \ast$)

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.


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DOI: https://doi.org/10.1090/S0002-9947-1970-0264749-1
Article copyright: © Copyright 1970 American Mathematical Society