Sufficient conditions for the continuity of stationary gaussian processes and applications to random series of functions

Jain, Naresh C.; Marcus, Michael B.

Annales de l'Institut Fourier, Tome 24 (1974) no. 2, pp. 117-141

Let $\left\{X\right(t),t\in [0,1{]}^n\}$ be a stochastically continuous, separable, Gaussian process with $E\left[X\right(t+h)-X(t){]}^2={\sigma }^2(\left|h\right|)$. 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.

Soit $\left\{X\right(t),t\in [0,1{]}^n\}$ un processus gaussien séparable et stochastiquement continu, satisfaisant à la condition $E\left[X\right(t+h)-X(t){]}^2={\sigma }^2(\left|h\right|)$. 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.

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