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On the Continuity of Stationary Gaussian Processes

Published online by Cambridge University Press:  22 January 2016

Makiko Nisio
Makiko Nisio*
Affiliation:
Kobe University
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Let us consider a stochastically continuous, separable and measurable stationary Gaussian process X = {X(t), − ∞ < t < ∞} with mean zero and with the covariance function p(t) = EX(t + s)X(s). The conditions for continuity of paths have been studied by many authors from various viewpoints. For example, Dudley [3] studied from the viewpoint of ε-entropy and Kahane [5] showed the necessary and sufficient condition in some special case, using the rather neat method of Fourier series.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 34 , March 1969 , pp. 89 - 104
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1969

References

[1] Belayev, Yu. K., Continuity and Hölder’s conditions for sample functions of stationary Gaussian processes, Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, 23–33.Google Scholar
[2] Delporte, J., Extension des conditions suffisantes pour la construction de functions aléatoires normales, presque sûrement continues, possédant une covariance donnée, C. R.Acad. Sci. Paris 256 (1963), 3816–3819.Google Scholar
[3] Dudley, R.M., The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, Jour, of Functional Analysis, 1 (1967), 290–330.Google Scholar
[4] Fernique, X., Continuité des processus Gaussiens, C. R. Acad. Sci, Paris 258 (1964), 6058–6060.Google Scholar
[5] Kahane, J.P., Series de Fourier aléatoires, Sém. Math. Super. Univ. de Montréal. (1963).Google Scholar
[6] Sirao, T. and Watanabe, H., On the Holder continuity of stationary Gaussian processes, (to appear).Google Scholar