Sufficient conditions for the continuity of stationary gaussian processes and applications to random series of functions
Annales de l'Institut Fourier, Tome 24 (1974) no. 2, pp. 117-141.

Soit $\left\{X\left(t\right),\phantom{\rule{0.277778em}{0ex}}t\in \left[0,1{\right]}^{n}\right\}$ un processus gaussien séparable et stochastiquement continu, satisfaisant à la condition $E\left[X\left(t+h\right)-X\left(t\right){\right]}^{2}={\sigma }^{2}\left(|h|\right)$. On obtient une condition suffisante de continuité presque sûre de $X\left(t\right)$, mise en termes de ré-arrangement monotone de $\sigma$. On fait l’application de ce résultat aux séries des fonctions aléatoires, en particulier, aux séries aléatoires de Fourier.

Let $\left\{X\left(t\right),\phantom{\rule{0.277778em}{0ex}}t\in \left[0,1{\right]}^{n}\right\}$ be a stochastically continuous, separable, Gaussian process with $E\left[X\left(t+h\right)-X\left(t\right){\right]}^{2}={\sigma }^{2}\left(|h|\right)$. A sufficient condition, in terms of the monotone rearrangement of $\sigma$, is obtained for $X\left(t\right)$ to have continuous sample paths almost surely. This result is applied to a wide class of random series of functions, in particular, to random Fourier series.

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author = {Jain, Naresh C. and Marcus, Michael B.},
title = {Sufficient conditions for the continuity of stationary gaussian processes and applications to random series of functions},
journal = {Annales de l'Institut Fourier},
pages = {117--141},
publisher = {Institut Fourier},
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Jain, Naresh C.; Marcus, Michael B. Sufficient conditions for the continuity of stationary gaussian processes and applications to random series of functions. Annales de l'Institut Fourier, Tome 24 (1974) no. 2, pp. 117-141. doi : 10.5802/aif.508. http://www.numdam.org/articles/10.5802/aif.508/

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