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 | References | Similar Articles | Additional Information

Abstract: We give a proof of Fernique's theorem that if *X* is a stationary Gaussian process and then *X* has continuous sample paths provided that, for some , where is any increasing function satisfying

() |

We prove the partial converse that if and is any increasing function not satisfying then the paths are not continuous. In particular, if is monotonic we may take and is then necessary and sufficient for sample path continuity. Our proof is based on an important lemma of Slepian.

Finally we show that if is monotonie and convex in then as iff the paths are *incrementally continuous*, meaning that for each monotonic bounded sequence , 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